Heap structurePasi Fränti

29.9.2014

• Tree-based data structure• Partial sorting• Every node satisfies heap

property:x ≥ x.child

zy

x

87

7

5

43 2 15

9

Max

Heap as priority queue

Example:

y ≤ x z ≤ x

HeapHeap

Heapsort(A[1,N])H CreateHeap();FOR i1 TO N DO

Insert(H, A[i]);FOR iN DOWNTO 1 DO

A[i] RemoveMax();

HeapsortStraightforward implementation

O(1)

O(N logN)

O(N logN)

Requires O(N) extra space for

the heap

7 4 6 1 8 …

1 4 6 7 8 …

Insert

Remove

HeapsortInplace variant

i

2i 2i+1

1 2 3 4 5 6 7 8 9 …

…

2 34 5 6 7

8 9

1

Indexingin heap:

Indexing in list:

• Use array itself as heap (no extra memory)• Heap is always balanced• Father-son relationships calculated via indexes

Father(i) i = k/2

LeftChild(i) k1 = 2iRightftChild(i) k2 = 2i+1

HeapsortInplace variant

Heapsort(A[1,N])FOR iN/2 DOWNTO 1 DO

Sink(A, i, N);FOR iN DOWNTO 2 DO

Swap(A[1],A[i]);Sink(A, 1, i-1);

List Heapi

List

Heap

List

Heap i

Insert

Remove

Swap

iLoop

Heap

List

Sinki

1

Sink function

Sink(A, i, j)IF 2i ≤ j THEN

IF 2i+1 ≤ j THENIF A[2i]>A[2i+1] THEN k2i ELSE k=2i+1;

ELSEk2i;

IF A[i]<A[k] THENSwap(A[i], A[k]);Sink(A, k, j);

Select bigger child

Left child existsRight child exists

Swap if neededSink recursively

Example of sinkAdded value 1

87

7

5

43 2 15

1Added

17

7

5

43 2 15

8Swap

77

1

5

43 2 15

8

Swap

n

i

n

i

n

i

inn

inniT

jiTcjiT

1

1 1

loglog

)log(log),(

),2(),(

Heap creation is linear time

iilog

nlogn

Sink(i,n)

• Half of the elements need no work• Next half only one level• Element at location i needs (logn - logi) work

n

i

inn1

loglog

Heap creation is linear time

1 2

1

3 4 5

2

log N

N

N

i

i1

log

nnnnnnnnnn

dxxnnn

loglogloglog

)(loglog1

Rectangle total

Difference nO

Empty space for notes

Empty space for notes

Another example

100

93 74

92 93 67 65

9122 22 35 43 5 27 17

Just back-up