5.2 fibonacci heapsbinomial heaps 5.2: fibonacci heaps t.s. 6. reminder: binomial heaps binomial...

5.2 Fibonacci HeapsFrank Stajano Thomas Sauerwald

Lent 2016

Structure of Fibonacci Heaps

Forest of MIN-HEAPs

Nodes can be marked (roots are always unmarked)

Tree roots are stored in a circular, doubly-linked list

Min-Pointer pointing to the smallest element

Fibonacci Heap

58 30 10

54 66 82 51

How do we implement a Fibonacci Heap?

5.2: Fibonacci Heaps T.S. 9

Priority Queues Overview

Operation Linked list Binary heap Binomial heap

Fibon. heap

MAKE-HEAP O(1) O(1) O(1)

INSERT O(1) O(log n) O(log n)

MINIMUM O(n) O(1) O(log n)

EXTRACT-MIN O(n) O(log n) O(log n)

O(log n)

MERGE O(n) O(n) O(log n)

DECREASE-KEY O(1) O(log n) O(log n)

DELETE O(1) O(log n) O(log n)

O(log n)

Priority Queues Overview

Operation Linked list Binary heap Binomial heap Fibon. heap

MAKE-HEAP O(1) O(1) O(1) O(1)

INSERT O(1) O(log n) O(log n) O(1)

MINIMUM O(n) O(1) O(log n) O(1)

EXTRACT-MIN O(n) O(log n) O(log n) O(log n)

MERGE O(n) O(n) O(log n) O(1)

DECREASE-KEY O(1) O(log n) O(log n) O(1)

DELETE O(1) O(log n) O(log n) O(log n)

Binomial Heap vs. Fibonacci Heap: Costs

Operation Binomial heap Fibonacci heap

actual cost amortized cost

MAKE-HEAP O(1) O(1)

INSERT O(log n) O(1)

MINIMUM O(log n) O(1)

EXTRACT-MIN O(log n) O(log n)

MERGE O(log n) O(1)

DECREASE-KEY O(log n) O(1)

DELETE O(log n) O(log n)

n is the number of items in the heap when the operation is performed.

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

Binomial Heap: k/2 DECREASE-KEY+ k/2 INSERT

c1 = c2 = · · · = ck = O(log n)

⇒ ∑ki=1 ci = O(k log n)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

c1 = c2 = · · · = ck = O(log n)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

c1 = c2 = · · · = ck = O(log n)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

c1 = c2 = · · · = ck = O(log n)

Fibonacci Heap: k/2DECREASE-KEY + k/2 INSERT

c1 = c2 = · · · = ck = O(1)

⇒ ∑ki=1 ci ≤

∑ki=1 ci = O(k)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

c1 = c2 = · · · = ck = O(log n)

c1 = c2 = · · · = ck = O(1)

⇒ ∑ki=1 ci ≤

∑ki=1 ci = O(k)

MAKE-HEAP O(1) O(1)

MERGE O(log n) O(1)

c1 = c2 = · · · = ck = O(log n)

c1 = c2 = · · · = ck = O(1)

⇒ ∑ki=1 ci ≤

∑ki=1 ci = O(k)

Actual vs. Amortized Cost

2 · O(1)

14 · O(1)

1 2 14

Potential

∑ki=1 ci

Potential ≥ 0, but should bealso as small as possible

2 · O(1)

14 · O(1)

1 2 14

Potential

∑ki=1 ci

2 · O(1)

14 · O(1)

1 2 14

Potential

∑ki=1 ci

2 · O(1)

14 · O(1)

1 2 14

Potential

∑ki=1 ci

2 · O(1)

14 · O(1)

1 2 14

Potential

∑ki=1 ci

Outline

Structure

Operations

Glimpse at the Analysis

Amortized Analysis

Reminder: Binomial Heaps

Binomial Trees

B(0) B(1) B(2) B(3) B(k)

B(k − 1)

Binomial Heap is a collection of binomial trees of different orders,each of which obeys the heap property

Operations:

MERGE: Merge two binomial heaps using Binary Addition ProcedureINSERT: Add B(0) and perform a MERGEEXTRACT-MIN: Find tree with minimum key, cut it and perform a MERGEDECREASE-KEY: The same as in a binary heap

Binomial Heaps

Binomial Trees

B(0) B(1) B(2) B(3) B(k)

B(k − 1)

Binomial Heap is a collection of binomial trees of different orders,each of which obeys the heap propertyOperations:

Binomial Heaps

Binomial Trees

B(0) B(1) B(2) B(3) B(k)

B(k − 1)

Binomial Heap is a collection of binomial trees of different orders,each of which obeys the heap propertyOperations:

Binomial Heaps

Merging two Binomial Heaps

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

4 9 11

12 17 13

4 9 11

12 17 13

0 0 1 1 1 = 70 1 0 1 1 = 111 1 1 1

1 0 0 1 0 = 18

Binomial Heap vs. Fibonacci Heap: Structure

Binomial Heap:consists of binomial trees, and every order appears at most onceimmediately tidy up after INSERT or MERGE

50 26 22

37 30 48

Fibonacci Heap:forest of MIN-HEAPslazily defer tidying up; do it on-the-fly when search for the MIN

37 22 5

50 38 26 19 7

Binomial Heap vs. Fibonacci Heap: Structure

Binomial Heap:consists of binomial trees, and every order appears at most onceimmediately tidy up after INSERT or MERGE

50 26 22

37 30 48

Fibonacci Heap:forest of MIN-HEAPslazily defer tidying up; do it on-the-fly when search for the MIN

37 22 5

50 38 26 19 7

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

Forest of MIN-HEAPs

Fibonacci Heap

58 30 10

54 66 82 51

A single Node

payload

marked

degreeb f

Previous Sibling Next Sibling

Parent

One of the Children

Magnifying a Four-Node Portion

30 0 3

43 0 0 41 1 2 33 0 1

58 30 10

43 41 33 70 32

54 66 82 51

30 0 3

43 0 0 41 1 2 33 0 1

58 30 10

43 41 33 70 32

54 66 82 51

30 0 3

43 0 0 41 1 2 33 0 1

58 30 10

43 41 33 70 32

54 66 82 51

Outline

Structure

Operations

Amortized Analysis

Fibonacci Heap: INSERT

Create a singleton tree

Add to root list

and update min-pointer (if necessary)

INSERT

17 24 23 7 3

30 26 46

18 52 41

Actual Costs: O(1)

Add to root list

INSERT

17 24 23 7 3

30 26 46

18 52 41

Actual Costs: O(1)

Add to root list

INSERT

17 24 23 7 3

30 26 46

18 52 41

Actual Costs: O(1)

Add to root list

INSERT

17 24 23 7 3

30 26 46

18 52 41

Actual Costs: O(1)

Add to root list and update min-pointer (if necessary)

INSERT

17 24 23 7 3

30 26 46

18 52 41

Actual Costs: O(1)

Add to root list and update min-pointer (if necessary)

INSERT

17 24 23 7 3

30 26 46

18 52 41

Actual Costs: O(1)

Fibonacci Heap: EXTRACT-MIN

Delete min

Meld childen into root list and unmark them

Consolidate so that no roots have the same degree

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree=2 degree=01 2 0 0 1 0 1

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Every root becomes child ofanother root at most once!

d(n) is the maximum degree of aroot in any Fibonacci heap of size n

Delete min

XMeld childen into root list and unmark them

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min X

Meld childen into root list and unmark them

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them

XConsolidate so that no roots have the same degree

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them X

(# children) X

Update minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them XConsolidate so that no roots have the same degree

(# children) XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them XConsolidate so that no roots have the same degree (# children)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree=2

degree=01 2 0 0 1 0 1

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree=2

degree=0

1 2 0 0 1 0 1

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree=2 degree=0

1 2 0 0 1 0 1

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

7 23 17

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them XConsolidate so that no roots have the same degree (# children) X

Update minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them XConsolidate so that no roots have the same degree (# children) XUpdate minimum

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

Delete min XMeld childen into root list and unmark them XConsolidate so that no roots have the same degree (# children) XUpdate minimum X

EXTRACT-MIN

18 52 41

degree0 1 2 3

Actual Costs:

O(trees(H)

+ d(n)

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs:

O(trees(H)

+ d(n)

EXTRACT-MIN

18 52 41

degree0 1 2 3

minActual Costs: O(trees(H) + d(n))

Fibonacci Heap: DECREASE-KEY (First Try)

Decrease the key of x (given by a pointer)

Check if heap-order is violated

If not

, then done.

Otherwise,

cut tree rooted at x and meld into root list (update min).

DECREASE-KEY of node x

24 17 23

26 46 30

Wide andshallow tree

Degree = 3,Nodes = 4

Peculiar Constraint: Make sure that each non-rootnode loses at most one child before becoming root

1. DECREASE-KEY 24 202. DECREASE-KEY 46 153. DECREASE-KEY 35 54. DECREASE-KEY 26 195. DECREASE-KEY 30 12

If not

, then done.

Otherwise,

24 17 23

26 46 30

1. DECREASE-KEY 24 20

2. DECREASE-KEY 46 153. DECREASE-KEY 35 54. DECREASE-KEY 26 195. DECREASE-KEY 30 12

If not

, then done.

Otherwise,

24 17 23

26 46 30

Decrease the key of x (given by a pointer)Check if heap-order is violated

If not

, then done.

Otherwise,

26 46 30

If not

, then done.

Otherwise,

26 46 30

If not

, then done.Otherwise,

26 46 30

If not, then done.

Otherwise,

26 46 30

If not, then done.Otherwise,

26 46 30

1. DECREASE-KEY 24 202. DECREASE-KEY 46 15

3. DECREASE-KEY 35 54. DECREASE-KEY 26 195. DECREASE-KEY 30 12

26 46 30

If not, then done.Otherwise, cut tree rooted at x and meld into root list (update min).

26 46 30

Fibonacci Heap: DECREASE-KEY

(Here we consider only cases where heap-order is violated)

⇒ Cut tree rooted at x , unmark x , meld into root list

Check if parent node is marked

If unmarked, mark it (unless it is a root)If marked,

unmark and meld it into root list and recurse (Cascading Cut)

24 17 23

26 46 30

5 26 24

minActual Cost:

O(# cuts)

24 17 23

26 46 30

5 26 24

minActual Cost:

O(# cuts)

X2. DECREASE-KEY 35 5

and:Check if parent node is marked

24 17 23

26 46 30

5 26 24

minActual Cost:

O(# cuts)

24 17 23

5 26 24

minActual Cost:

O(# cuts)

24 17 23

5 26 24

minActual Cost:

O(# cuts)

24 17 23

5 26 24

minActual Cost:

O(# cuts)

⇒ Cut tree rooted at x , unmark x , meld into root list and:

24 17 23

5 26 24

minActual Cost:

O(# cuts)

⇒ Cut tree rooted at x , unmark x , meld into root list and:Check if parent node is marked

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If unmarked, mark it (unless it is a root)

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

1. DECREASE-KEY 46 15 X

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

1. DECREASE-KEY 46 15 X2. DECREASE-KEY 35 5

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If marked,

24 17 23

5 26 24

minActual Cost:

O(# cuts)

If marked,

24 17 23

minActual Cost:

O(# cuts)

24 17 23

Actual Cost:

O(# cuts)

If unmarked, mark it (unless it is a root)If marked, unmark and meld it into root list and recurse (Cascading Cut)

24 17 23

Actual Cost:

O(# cuts)

24 17 23

Actual Cost:

O(# cuts)

24 17 23

Actual Cost:

O(# cuts)

24 17 23

Actual Cost:

O(# cuts)

24 17 23

5 26 24

Actual Cost:

O(# cuts)

24 17 23

5 26 24

Actual Cost:

O(# cuts)

5 26 24

Actual Cost:

O(# cuts)

5 26 24

Actual Cost:

O(# cuts)

1. DECREASE-KEY 46 15 X2. DECREASE-KEY 35 5 X

5 26 24

minActual Cost:

O(# cuts)

5 26 24

minActual Cost: O(# cuts)

5.2 Fibonacci Heaps (Analysis)Frank Stajano Thomas Sauerwald

Lent 2016

Amortized Analysis via Potential Method

INSERT: actual O(1) amortized O(1) XEXTRACT-MIN: actual O(trees(H) + d(n)) amortized O(d(n)) ?

DECREASE-KEY: actual O(# cuts) O(marks(H)) amortized O(1) ?

�(H) = trees(H)+2·marks(H)

24 17 23

26 46 30

Lifecycle of a node

child C

onsolidate

Loses first child

5.2: Fibonacci Heaps (Analysis) T.S. 3

Outline

Structure

Operations

Amortized Analysis

INSERT: actual O(1)

amortized O(1) X

EXTRACT-MIN: actual O(trees(H) + d(n))

amortized O(d(n)) ?

DECREASE-KEY: actual O(# cuts) ≤ O(marks(H))

amortized O(1) ?

Φ(H) = trees(H)+2·marks(H)

24 17 23

26 46 30

Lifecycle of a node

child C

onsolidate

Loses first child

INSERT: actual O(1)

amortized O(1) X

amortized O(d(n)) ?

amortized O(1) ?

24 17 23

26 46 30

Lifecycle of a node

child C

onsolidate

Loses first child

INSERT: actual O(1)

amortized O(1) X

amortized O(d(n)) ?

amortized O(1) ?

24 17 23

26 46 30

Lifecycle of a node

child C

onsolidate

Loses first child

INSERT: actual O(1)

amortized O(1) X

amortized O(d(n)) ?

amortized O(1) ?

24 17 23

26 46 30

Lifecycle of a node

child C

onsolidate

Loses first child

INSERT: actual O(1)

amortized O(1) X

amortized O(d(n)) ?

amortized O(1) ?

24 17 23

26 46 30

Lifecycle of a node

child C

onsolidate

Loses first child

INSERT: actual O(1) amortized O(1)

EXTRACT-MIN: actual O(trees(H) + d(n)) amortized O(d(n))

DECREASE-KEY: actual O(# cuts) ≤ O(marks(H)) amortized O(1)

24 17 23

26 46 30

Lifecycle of a node

ild Consolidate

Loses first child

INSERT: actual O(1) amortized O(1) XEXTRACT-MIN: actual O(trees(H) + d(n)) amortized O(d(n)) ?

DECREASE-KEY: actual O(# cuts) ≤ O(marks(H)) amortized O(1) ?

24 17 23

26 46 30

Lifecycle of a node

ild Consolidate

Loses first child

Outline

Structure

Operations

Amortized Analysis

Amortized Analysis of DECREASE-KEY

DECREASE-KEY: O(x + 1), where x is the number of cuts.

Actual Cost

Φ(H) = trees(H) + 2 ·marks(H)

trees(H ′) =

trees(H) + x

marks(H ′) ≤

marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Amortized CostScale up potential units

First Coin pays cutSecond Coin increase of trees(H)

Actual Cost

trees(H ′) =

trees(H) + x

marks(H ′) ≤

marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

trees(H ′) =

trees(H) + x

marks(H ′) ≤

marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

trees(H ′) =

trees(H) + x

marks(H ′) ≤

marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

trees(H ′) = trees(H) + x

marks(H ′) ≤

marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

marks(H ′) ≤

marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

marks(H ′) ≤ marks(H)− x + 2

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Actual Cost

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ

≤ O(x + 1) + 4− x = O(1)

Amortized Cost

Scale up potential units

Actual Cost

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ ≤ O(x + 1) + 4− x

= O(1)

Actual Cost

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ ≤ O(x + 1) + 4− x = O(1)

Actual Cost

⇒ ∆Φ ≤ x + 2 · (−x + 2) = 4− x .

Change in Potential

ci = ci + ∆Φ ≤ O(x + 1) + 4− x = O(1)

Amortized Cost

Scale up potential units

5.2 fibonacci heapsbinomial heaps 5.2: fibonacci heaps t.s. 6. reminder: binomial heaps binomial...

Documents