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Searching An extension of binary search with p processors gives that one can find the rank of an element in log p+1 (n) = log n log(p + 1) many parallel steps with p processors. (not work-optimal) This requires a CREW PRAM model. For the EREW model searching cannot be done faster than O(log n - log p) with p processors even if there are p copies of the search key. PA © Harald Räcke 121/283

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Page 1: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Searching

An extension of binary search with p processors gives that one

can find the rank of an element in

logp+1(n) =logn

log(p + 1)

many parallel steps with p processors. (not work-optimal)

This requires a CREW PRAM model. For the EREW model

searching cannot be done faster than O(logn− logp) with pprocessors even if there are p copies of the search key.

PA

© Harald Räcke 121/283

Page 2: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Searching

An extension of binary search with p processors gives that one

can find the rank of an element in

logp+1(n) =logn

log(p + 1)

many parallel steps with p processors. (not work-optimal)

This requires a CREW PRAM model. For the EREW model

searching cannot be done faster than O(logn− logp) with pprocessors even if there are p copies of the search key.

PA 7 Searching and Sorting

© Harald Räcke 121/283

Page 3: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Searching

An extension of binary search with p processors gives that one

can find the rank of an element in

logp+1(n) =logn

log(p + 1)

many parallel steps with p processors. (not work-optimal)

This requires a CREW PRAM model. For the EREW model

searching cannot be done faster than O(logn− logp) with pprocessors even if there are p copies of the search key.

PA 7 Searching and Sorting

© Harald Räcke 121/283

Page 4: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Given two sorted sequences A = (a1, . . . , an) and

B = (b1, . . . , bn), compute the sorted squence C = (c1, . . . , cn).

Definition 1

Let X = (x1, . . . , xt) be a sequence. The rank rank(y : X) of y in

X is

rank(y : X) = |{x ∈ X | x ≤ y}|

For a sequence Y = (y1, . . . , ys) we define

rank(Y : X) := (r1, . . . , rs) with ri = rank(yi : X).

PA 7 Searching and Sorting

© Harald Räcke 122/283

Page 5: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Given two sorted sequences A = (a1, . . . , an) and

B = (b1, . . . , bn), compute the sorted squence C = (c1, . . . , cn).

Definition 1

Let X = (x1, . . . , xt) be a sequence. The rank rank(y : X) of y in

X is

rank(y : X) = |{x ∈ X | x ≤ y}|

For a sequence Y = (y1, . . . , ys) we define

rank(Y : X) := (r1, . . . , rs) with ri = rank(yi : X).

PA 7 Searching and Sorting

© Harald Räcke 122/283

Page 6: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Given two sorted sequences A = (a1, . . . , an) and

B = (b1, . . . , bn), compute the sorted squence C = (c1, . . . , cn).

Definition 1

Let X = (x1, . . . , xt) be a sequence. The rank rank(y : X) of y in

X is

rank(y : X) = |{x ∈ X | x ≤ y}|

For a sequence Y = (y1, . . . , ys) we define

rank(Y : X) := (r1, . . . , rs) with ri = rank(yi : X).

PA 7 Searching and Sorting

© Harald Räcke 122/283

Page 7: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Given two sorted sequences A = (a1, . . . , an) and

B = (b1, . . . , bn), compute the sorted squence C = (c1, . . . , cn).

Definition 1

Let X = (x1, . . . , xt) be a sequence. The rank rank(y : X) of y in

X is

rank(y : X) = |{x ∈ X | x ≤ y}|

For a sequence Y = (y1, . . . , ys) we define

rank(Y : X) := (r1, . . . , rs) with ri = rank(yi : X).

PA 7 Searching and Sorting

© Harald Räcke 122/283

Page 8: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

We have already seen a merging-algorithm that runs in time

O(logn) and work O(n).

Using the fast search algorithm we can improve this to a running

time of O(log logn) and work O(n log logn).

PA 7 Searching and Sorting

© Harald Räcke 123/283

Page 9: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

We have already seen a merging-algorithm that runs in time

O(logn) and work O(n).

Using the fast search algorithm we can improve this to a running

time of O(log logn) and work O(n log logn).

PA 7 Searching and Sorting

© Harald Räcke 123/283

Page 10: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Input: A = a1, . . . , an; B = b1, . . . , bm; m ≤ n1. if m < 4 then rank elements of B, using the parallel search

algorithm with p processors. Time: O(1). Work: O(n).2. Concurrently rank elements b√m, b2

√m, . . . , bm in A using

the parallel search algorithm with p = √n. Time: O(1).Work: O(√m · √n) = O(n)

j(i) := rank(bi√m : A)

3. Let Bi = (bi√m+1, . . . , b(i+1)√m−1); and

Ai = (aj(i)+1, . . . , aj(i+1)).

Recursively compute rank(Bi : Ai).

4. Let k be index not a multiple of√m. i = d k√me. Then

rank(bk : A) = j(i)+ rank(bk : Ai).

PA 7 Searching and Sorting

© Harald Räcke 124/283

Page 11: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Input: A = a1, . . . , an; B = b1, . . . , bm; m ≤ n1. if m < 4 then rank elements of B, using the parallel search

algorithm with p processors. Time: O(1). Work: O(n).2. Concurrently rank elements b√m, b2

√m, . . . , bm in A using

the parallel search algorithm with p = √n. Time: O(1).Work: O(√m · √n) = O(n)

j(i) := rank(bi√m : A)

3. Let Bi = (bi√m+1, . . . , b(i+1)√m−1); and

Ai = (aj(i)+1, . . . , aj(i+1)).

Recursively compute rank(Bi : Ai).

4. Let k be index not a multiple of√m. i = d k√me. Then

rank(bk : A) = j(i)+ rank(bk : Ai).

PA 7 Searching and Sorting

© Harald Räcke 124/283

Page 12: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Input: A = a1, . . . , an; B = b1, . . . , bm; m ≤ n1. if m < 4 then rank elements of B, using the parallel search

algorithm with p processors. Time: O(1). Work: O(n).2. Concurrently rank elements b√m, b2

√m, . . . , bm in A using

the parallel search algorithm with p = √n. Time: O(1).Work: O(√m · √n) = O(n)

j(i) := rank(bi√m : A)

3. Let Bi = (bi√m+1, . . . , b(i+1)√m−1); and

Ai = (aj(i)+1, . . . , aj(i+1)).

Recursively compute rank(Bi : Ai).

4. Let k be index not a multiple of√m. i = d k√me. Then

rank(bk : A) = j(i)+ rank(bk : Ai).

PA 7 Searching and Sorting

© Harald Räcke 124/283

Page 13: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Merging

Input: A = a1, . . . , an; B = b1, . . . , bm; m ≤ n1. if m < 4 then rank elements of B, using the parallel search

algorithm with p processors. Time: O(1). Work: O(n).2. Concurrently rank elements b√m, b2

√m, . . . , bm in A using

the parallel search algorithm with p = √n. Time: O(1).Work: O(√m · √n) = O(n)

j(i) := rank(bi√m : A)

3. Let Bi = (bi√m+1, . . . , b(i+1)√m−1); and

Ai = (aj(i)+1, . . . , aj(i+1)).

Recursively compute rank(Bi : Ai).

4. Let k be index not a multiple of√m. i = d k√me. Then

rank(bk : A) = j(i)+ rank(bk : Ai).

PA 7 Searching and Sorting

© Harald Räcke 124/283

Page 14: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

The algorithm can be made work-optimal by standard

techniques.

proof on board...

PA 7 Searching and Sorting

© Harald Räcke 125/283

Page 15: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Mergesort

Lemma 2

A straightforward parallelization of Mergesort can be

implemented in time O(logn log logn) and with work O(n logn).

This assumes the CREW-PRAM model.

PA 7 Searching and Sorting

© Harald Räcke 126/283

Page 16: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Mergesort

Lemma 2

A straightforward parallelization of Mergesort can be

implemented in time O(logn log logn) and with work O(n logn).

This assumes the CREW-PRAM model.

PA 7 Searching and Sorting

© Harald Räcke 126/283

Page 17: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Mergesort

Let L[v] denote the (sorted) sublist of elements stored at the

leaf nodes rooted at v.

We can view Mergesort as computing L[v] for a complete binary

tree where the leaf nodes correspond to nodes in the given array.

Since the merge-operations on one level of the complete binary

tree can be performed in parallel we obtain time O(h log logn)and work O(hn), where h = O(logn) is the height of the tree.

PA 7 Searching and Sorting

© Harald Räcke 127/283

Page 18: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Mergesort

Let L[v] denote the (sorted) sublist of elements stored at the

leaf nodes rooted at v.

We can view Mergesort as computing L[v] for a complete binary

tree where the leaf nodes correspond to nodes in the given array.

Since the merge-operations on one level of the complete binary

tree can be performed in parallel we obtain time O(h log logn)and work O(hn), where h = O(logn) is the height of the tree.

PA 7 Searching and Sorting

© Harald Räcke 127/283

Page 19: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Mergesort

Let L[v] denote the (sorted) sublist of elements stored at the

leaf nodes rooted at v.

We can view Mergesort as computing L[v] for a complete binary

tree where the leaf nodes correspond to nodes in the given array.

Since the merge-operations on one level of the complete binary

tree can be performed in parallel we obtain time O(h log logn)and work O(hn), where h = O(logn) is the height of the tree.

PA 7 Searching and Sorting

© Harald Räcke 127/283

Page 20: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

We again compute L[v] for every node in the complete binary

tree.

After round s, Ls[v] is an approximation of L[v] that will be

improved in future rounds.

For s ≥ 3 height(v), Ls[v] = L[v].

PA 7 Searching and Sorting

© Harald Räcke 128/283

Page 21: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

We again compute L[v] for every node in the complete binary

tree.

After round s, Ls[v] is an approximation of L[v] that will be

improved in future rounds.

For s ≥ 3 height(v), Ls[v] = L[v].

PA 7 Searching and Sorting

© Harald Räcke 128/283

Page 22: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

We again compute L[v] for every node in the complete binary

tree.

After round s, Ls[v] is an approximation of L[v] that will be

improved in future rounds.

For s ≥ 3 height(v), Ls[v] = L[v].

PA 7 Searching and Sorting

© Harald Räcke 128/283

Page 23: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

In every round, a node v sends sample(Ls[v]) (an

approximation of its current list) upwards, and receives

approximations of the lists of its children.

It then computes a new approximation of its list.

A node is called active in round s if s ≤ 3 height(v) (this means

its list is not yet complete at the start of the round, i.e.,

Ls−1[v] ≠ L[v]).

PA 7 Searching and Sorting

© Harald Räcke 129/283

Page 24: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

In every round, a node v sends sample(Ls[v]) (an

approximation of its current list) upwards, and receives

approximations of the lists of its children.

It then computes a new approximation of its list.

A node is called active in round s if s ≤ 3 height(v) (this means

its list is not yet complete at the start of the round, i.e.,

Ls−1[v] ≠ L[v]).

PA 7 Searching and Sorting

© Harald Räcke 129/283

Page 25: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

In every round, a node v sends sample(Ls[v]) (an

approximation of its current list) upwards, and receives

approximations of the lists of its children.

It then computes a new approximation of its list.

A node is called active in round s if s ≤ 3 height(v) (this means

its list is not yet complete at the start of the round, i.e.,

Ls−1[v] ≠ L[v]).

PA 7 Searching and Sorting

© Harald Räcke 129/283

Page 26: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

Algorithm 11 ColeSort()1: initialize L0[v] = Av for leaf nodes; L0[v] = � otw.

2: for s ← 1 to 3 · height(T) do

3: for all active nodes v do

4: // u and w children of v5: L′s[u]← sample(Ls−1[u])6: L′s[w]← sample(Ls−1[w])7: Ls[v]←merge(L′s[u], L′s[w])

sample(Ls[v]) =

sample4(Ls[v]) s ≤ 3 height(v)sample2(Ls[v]) s = 3 height(v)+ 1

sample1(Ls[v]) s = 3 height(v)+ 2

PA 7 Searching and Sorting

© Harald Räcke 130/283

Page 27: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Pipelined Mergesort

Algorithm 11 ColeSort()1: initialize L0[v] = Av for leaf nodes; L0[v] = � otw.

2: for s ← 1 to 3 · height(T) do

3: for all active nodes v do

4: // u and w children of v5: L′s[u]← sample(Ls−1[u])6: L′s[w]← sample(Ls−1[w])7: Ls[v]←merge(L′s[u], L′s[w])

sample(Ls[v]) =

sample4(Ls[v]) s ≤ 3 height(v)sample2(Ls[v]) s = 3 height(v)+ 1

sample1(Ls[v]) s = 3 height(v)+ 2

PA 7 Searching and Sorting

© Harald Räcke 130/283

Page 28: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 29: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 30: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 31: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 32: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 33: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 34: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 35: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 36: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 37: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 38: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 39: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 40: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 41: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 42: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 43: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 44: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

PA 7 Searching and Sorting

© Harald Räcke 131/283

Page 45: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Colesort

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 0

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 1

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 2

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4

4

9

9

9

9

3

3

9

9

2

2

0

0

9

9

6

6

8

8

6

6

2

2

6

6

5

5

5

5

2

2

7

7

4

4

0

0

9

9

6

6

8

8

1

1

6

6

5

5

1

1

4

4

0

0

3

3

3

3

6

6

6

6

s = 3

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 4

9 9 9 9 6 8 5 6 7 9 6 8 4 5 3 6

4 9

9

3 9

9

2 9

9

0 9

9

6 8

8

2 6

6

5 6

6

2 5

5

4 7

7

0 9

9

6 8

8

1 6

6

1 5

5

0 4

4

3 3

3

6 6

6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 5

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9

4 9

3 9

3 9

2 9

2 9

0 9

0 9

6 8

6 8

2 6

2 6

5 6

5 6

2 5

2 5

4 7

4 7

0 9

0 9

6 8

6 8

1 6

1 6

1 5

1 5

0 4

0 4

3 3

3 3

6 6

6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 6

9 9 6 8 8 9 5 6

3 4 9 9

9

0 2 9 9

9

2 6 6 8

8

2 5 5 6

6

0 4 7 9

9

1 6 6 8

8

0 1 4 5

5

3 3 6 6

6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 7

2 4 9 9 5 6 6 8 4 6 8 9 1 3 5 6

3 4 9 9

4 9

0 2 9 9

2 9

2 6 6 8

6 8

2 5 5 6

5 6

0 4 7 9

4 9

1 6 6 8

6 8

0 1 4 5

1 5

3 3 6 6

3 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 8

8 9 6 9

0 2 3 4 9 9 9 9

9

2 2 5 5 6 6 6 8

8

0 1 4 6 6 7 8 9

9

0 1 3 3 4 5 6 6

6

3 4 9 9

3 4 9 9

0 2 9 9

0 2 9 9

2 6 6 8

2 6 6 8

2 5 5 6

2 5 5 6

0 4 7 9

0 4 7 9

1 6 6 8

1 6 6 8

0 1 4 5

0 1 4 5

3 3 6 6

3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 9

4 5 8 9 3 6 6 9

0 2 3 4 9 9 9 9

4 9

2 2 5 5 6 6 6 8

5 8

0 1 4 6 6 7 8 9

6 9

0 1 3 3 4 5 6 6

3 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 10

9 9

2 2 4 5 6 8 9 9

9

1 1 3 5 6 6 7 9

9

0 2 3 4 9 9 9 9

2 4 9 9

2 2 5 5 6 6 6 8

2 5 6 8

0 1 4 6 6 7 8 9

1 6 7 9

0 1 3 3 4 5 6 6

1 3 5 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 11

5 5 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

5 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

5 9

0 2 3 4 9 9 9 9

0 2 3 4 9 9 9 9

2 2 5 5 6 6 6 8

2 2 5 5 6 6 6 8

0 1 4 6 6 7 8 9

0 1 4 6 6 7 8 9

0 1 3 3 4 5 6 6

0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 12

1 2 4 5 6 8 9 9

9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 5 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

1 4 6 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 13

0 1 2 2 3 4 4 5 6 6 6 7 8 9 9 9

5 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

2 2 4 5 6 8 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 1 3 4 6 6 7 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 14

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

2 5 7 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 15

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

1 2 4 5 6 6 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 16

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 1 2 2 3 4 4 5 6 6 6 6 8 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 17

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 6 6 6 6 7 8 8 9 9 9 9 9

0 2 2 2 3 4 5 5 6 6 6 8 9 9 9 9 0 0 1 1 3 3 4 4 5 6 6 6 6 7 8 9

0 2 3 4 9 9 9 9 2 2 5 5 6 6 6 8 0 1 4 6 6 7 8 9 0 1 3 3 4 5 6 6

3 4 9 9 0 2 9 9 2 6 6 8 2 5 5 6 0 4 7 9 1 6 6 8 0 1 4 5 3 3 6 6

4 9 3 9 2 9 0 9 6 8 2 6 5 6 2 5 4 7 0 9 6 8 1 6 1 5 0 4 3 3 6 6

4 9 9 3 9 2 0 9 6 8 6 2 6 5 5 2 7 4 0 9 6 8 1 6 5 1 4 0 3 3 6 6

s = 18

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 3

After round s = 3 height(v), the list Ls[v] is complete.

Proof:

ñ clearly true for leaf nodes

ñ suppose it is true for all nodes up to height h;

ñ fix a node v on level h+ 1 with children u and wñ L3h[u] and L3h[w] are complete by induction hypothesis

ñ further sample(L3h+2[u]) = L[u] and

sample(L3h+2[v]) = L[v]ñ hence in round 3h+ 3 node v will merge the complete list

of its children; after the round L[v] will be complete

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Pipelined Mergesort

Lemma 4

The number of elements in lists Ls[v] for active nodes v is at

most O(n).

proof on board...

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Definition 5

A sequence X is a c-cover of a sequence Y if for any two

consecutive elements α,β from (−∞, X,∞) the set

|{yi | α ≤ yi ≤ β}| ≤ c.

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Pipelined Mergesort

Lemma 6

L′s[v] is a 4-cover of L′s+1[v].

If [a, b] fulfills |[a, b]∩ (A∪ {−∞,∞})| = k we say [a, b]intersects (−∞, A,+∞) in k items.

Lemma 7

If [a, b] with a,b ∈ L′s[v]∪ {−∞,∞} intersects (−∞, L′s[v],∞) in

k ≥ 2 items, then [a, b] intersects (−∞, L′s+1,∞) in at most 2kitems.

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Pipelined Mergesort

Lemma 6

L′s[v] is a 4-cover of L′s+1[v].

If [a, b] fulfills |[a, b]∩ (A∪ {−∞,∞})| = k we say [a, b]intersects (−∞, A,+∞) in k items.

Lemma 7

If [a, b] with a,b ∈ L′s[v]∪ {−∞,∞} intersects (−∞, L′s[v],∞) in

k ≥ 2 items, then [a, b] intersects (−∞, L′s+1,∞) in at most 2kitems.

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Pipelined Mergesort

Lemma 6

L′s[v] is a 4-cover of L′s+1[v].

If [a, b] fulfills |[a, b]∩ (A∪ {−∞,∞})| = k we say [a, b]intersects (−∞, A,+∞) in k items.

Lemma 7

If [a, b] with a,b ∈ L′s[v]∪ {−∞,∞} intersects (−∞, L′s[v],∞) in

k ≥ 2 items, then [a, b] intersects (−∞, L′s+1,∞) in at most 2kitems.

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L′s[v]

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L′s[v]

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

Ls[v] ≤ 2p + 2q ≤ 8k − 2

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

Ls[v] ≤ 2p + 2q ≤ 8k − 2

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

Ls[v] ≤ 2p + 2q ≤ 8k − 2

L′s+1[v] ≤ 2k + 1/4 ⇒≤ 2k

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

Ls[v] ≤ 2p + 2q ≤ 8k − 2

L′s+1[v] ≤ 2k + 1/4 ⇒≤ 2k

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L′s[v]

Ls−1[v]

= k

≤ 4k − 3

L′s−1[u] L′s−1[w]p q

p + q ≤ 4k − 1

L′s[u] L′s[w]2p 2q

Ls[v] ≤ 2p + 2q ≤ 8k − 2

L′s+1[v] ≤ 2k + 1/4 ⇒≤ 2k

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Merging with a Cover

Lemma 8

Given two sorted sequences A and B. Let X be a c-cover of A and

B for constant c, and let rank(X : A) and rank(X : B) be known.

We can merge A and B in time O(1) using O(|X|) operations.

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Merging with a Cover

Lemma 9

Given two sorted sequences A and B. Let X be a c-cover of B for

constant c, and let rank(A : X) and rank(X : B) be known.

We can compute rank(A : B) using O(|X| + |A|) operations.

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Merging with a Cover

Lemma 10

Given two sorted sequences A and B. Let X be a c-cover of B for

constant c, and let rank(A : X) and rank(X : B) be known.

We can compute rank(B : A) using O(|X| + |A|) operations.

Easy to do with concurrent read. Can also be done with exclusive

read but non-trivial.

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In order to do the merge in iteration s + 1 in constant time we

need to know

rank(Ls[v] : L′s+1[u]) and rank(Ls[v] : L′s+1[w])

and we need to know that Ls[v] is a 4-cover of L′s+1[u] and

L′s+1[w].

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Lemma 11

Ls[v] is a 4-cover of L′s+1[u] and L′s+1[w].

ñ Ls[v] ⊇ L′s[u], L′s[w]ñ L′s[u] is 4-cover of L′s+1[u]ñ Hence, Ls[v] is 4-cover of L′s+1[u] as adding more elements

cannot destroy the cover-property.

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Lemma 11

Ls[v] is a 4-cover of L′s+1[u] and L′s+1[w].

ñ Ls[v] ⊇ L′s[u], L′s[w]ñ L′s[u] is 4-cover of L′s+1[u]ñ Hence, Ls[v] is 4-cover of L′s+1[u] as adding more elements

cannot destroy the cover-property.

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Lemma 11

Ls[v] is a 4-cover of L′s+1[u] and L′s+1[w].

ñ Ls[v] ⊇ L′s[u], L′s[w]ñ L′s[u] is 4-cover of L′s+1[u]ñ Hence, Ls[v] is 4-cover of L′s+1[u] as adding more elements

cannot destroy the cover-property.

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Lemma 11

Ls[v] is a 4-cover of L′s+1[u] and L′s+1[w].

ñ Ls[v] ⊇ L′s[u], L′s[w]ñ L′s[u] is 4-cover of L′s+1[u]ñ Hence, Ls[v] is 4-cover of L′s+1[u] as adding more elements

cannot destroy the cover-property.

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Analysis

Lemma 12

Suppose we know for every internal node v with children u and

wñ rank(L′s[v] : L′s+1[v])ñ rank(L′s[u] : L′s[w])ñ rank(L′s[w] : L′s[u])

We can compute

ñ rank(L′s+1[v] : L′s+2[v])ñ rank(L′s+1[u] : L′s+1[w])ñ rank(L′s+1[w] : L′s+1[u])

in constant time and O(|Ls+1[v]|) operations, where v is the

parent of u and w.

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Given

ñ rank(L′s[u] : L′s+1[u]) (4-cover)

ñ rank(L′s[w] : L′s[u])ñ rank(L′s[u] : L′s[w])ñ rank(L′s[w] : L′s+1[w]) (4-cover)

Compute

ñ rank(L′s+1[w] : L′s[u])ñ rank(L′s+1[u] : L′s[w])

Compute

ñ rank(L′s+1[w] : L′s+1[u])ñ rank(L′s+1[u] : L′s+1[w])

ranks between siblings can be computed easily

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Given

ñ rank(L′s[u] : L′s+1[u]) (4-cover)

ñ rank(L′s[w] : L′s[u])ñ rank(L′s[u] : L′s[w])ñ rank(L′s[w] : L′s+1[w]) (4-cover)

Compute

ñ rank(L′s+1[w] : L′s[u])ñ rank(L′s+1[u] : L′s[w])

Compute

ñ rank(L′s+1[w] : L′s+1[u])ñ rank(L′s+1[u] : L′s+1[w])

ranks between siblings can be computed easily

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Given

ñ rank(L′s[u] : L′s+1[u]) (4-cover)

ñ rank(L′s[w] : L′s[u])ñ rank(L′s[u] : L′s[w])ñ rank(L′s[w] : L′s+1[w]) (4-cover)

Compute

ñ rank(L′s+1[w] : L′s[u])ñ rank(L′s+1[u] : L′s[w])

Compute

ñ rank(L′s+1[w] : L′s+1[u])ñ rank(L′s+1[u] : L′s+1[w])

ranks between siblings can be computed easily

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Given

ñ rank(L′s[u] : L′s+1[u]) (4-cover → rank(L′s+1[u] : L′s[u]))ñ rank(L′s[w] : L′s+1[u])ñ rank(L′s[u] : L′s+1[w])ñ rank(L′s[w] : L′s+1[w]) (4-cover → rank(L′s+1[w] : L′s[w]))

Compute (recall that Ls[v] =merge(L′s[u], L′s[w]))ñ rank(Ls[v] : L′s+1[u])ñ rank(Ls[v] : L′s+1[w])

Compute

ñ rank(Ls[v] : Ls+1[v]) (by adding)

ñ rank(L′s+1[v] : L′s+2[v]) (by sampling)

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Given

ñ rank(L′s[u] : L′s+1[u]) (4-cover → rank(L′s+1[u] : L′s[u]))ñ rank(L′s[w] : L′s+1[u])ñ rank(L′s[u] : L′s+1[w])ñ rank(L′s[w] : L′s+1[w]) (4-cover → rank(L′s+1[w] : L′s[w]))

Compute (recall that Ls[v] =merge(L′s[u], L′s[w]))ñ rank(Ls[v] : L′s+1[u])ñ rank(Ls[v] : L′s+1[w])

Compute

ñ rank(Ls[v] : Ls+1[v]) (by adding)

ñ rank(L′s+1[v] : L′s+2[v]) (by sampling)

PA 7 Searching and Sorting

© Harald Räcke 144/283

Page 86: Searching - fileSearching An extension of binary search with pprocessors gives that one can find the rank of an element in log p‡1—n–… logn log

Given

ñ rank(L′s[u] : L′s+1[u]) (4-cover → rank(L′s+1[u] : L′s[u]))ñ rank(L′s[w] : L′s+1[u])ñ rank(L′s[u] : L′s+1[w])ñ rank(L′s[w] : L′s+1[w]) (4-cover → rank(L′s+1[w] : L′s[w]))

Compute (recall that Ls[v] =merge(L′s[u], L′s[w]))ñ rank(Ls[v] : L′s+1[u])ñ rank(Ls[v] : L′s+1[w])

Compute

ñ rank(Ls[v] : Ls+1[v]) (by adding)

ñ rank(L′s+1[v] : L′s+2[v]) (by sampling)

PA 7 Searching and Sorting

© Harald Räcke 144/283


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