ECOE 556: Algorithms and Computational Complexity

Heapsort

Serdar Taşıran

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Heapsort Sorts array A[1..n] in place

At most a constant amount of storage outside the array Sorts in O(n lg n) time Useful data structure “heap”

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The heap property: A[parent(i)] A[i]

Height of a node Max distance from it to a descendant leaf

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Basic procedures on heaps MAX-HEAPIFY

Runs in O(lg n) time. Needed for maintaining the heap property

BUILD-MAX-HEAP Linear time Produces max-heap from unordered input array

HEAPSORT O(n lg n) Sorts array in place

MAX-HEAP-INSERT HEAP-EXTRACT-MAX HEAP-INCREASE-KEY HEAP-MAXIMUM

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Binary trees rooted at LEFT(i) and RIGHT(i) are max-heaps.

A[i] may be smaller than LEFT(i) and/or RIGHT(i)

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MAX-HEAPIFY EXAMPLE

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Analysis of MAX-HEAPIFY Running time One subtree has size at most 2n/3

Since the heap is an almost full binary tree T(n) T(2n/3) + (1) Solution: T(n) = O(lg n) Or, T(n) = O(h)

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Produces max-heap from unordered input array What is the running time?

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Produces max-heap from unordered input array Running time: T(n) = O(n)

Sum over all i n-element heap

FACT: height = floor(lg n) FACT: # of elements at height h ceil(n / 2h+1)

[Needs to be proven]

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Proving BUILD-MAX-HEAP correctLOOP INVARIANT: At the start of each iteration of the for loop, each node

i+1, i+2, …, n is the root of a max-heap

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Priority queues A data structure for maintaining a set S of elements

Each has an associated key Supported operations

INSERT(S,x) MAXIMUM(S) EXTRACT-MAX(S) INCREASE-KEY(S,x,k)

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