Heapsort.

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Presentation transcript:

Heapsort

Heaps A data structure with Nearly complete binary tree Heap property: A[parent(i)] ≥ A[i] eg. Parent(i) { return } Left(i) { return 2i} Right(i) { return 2i+1} 1 16 2 3 14 10 4 5 6 7 8 7 9 3 8 9 10 2 4 1 1 2 3 4 5 6 7 8 9 10 16 14 10 8 7 9 3 2 4 1

Binary tree Contains no node, or eg. A node without subtree is called a leaf. In a full binary tree, each node has 2 or NO children. A complete binary tree has all leaves with the same depth and all internal nodes have 2 children. root Left subtree right subtree

Maintaining the heap property Condition: A[i] may be smaller than its children. A[i] Heaps

Pseudocode Heapify(A,i) Time: O(lg n), T(n)≤T(2n/3)+Θ(1)

Heapify(A,2): Heapify(A,4): 1 1 16 16 2 3 2 3 4 10 14 10 4 5 6 7 4 5 6 7 14 7 9 3 4 7 9 3 8 9 10 8 9 10 2 8 1 2 8 1 Heapify(A,9): 1 16 2 3 14 10 4 5 6 7 8 7 9 3 8 9 10 2 4 1

Build Heap 1 2 3 4 5 6 7 8 9 10 A 4 1 3 2 16 9 10 14 8 7 1 1 4 4 2 3 2 3 1 3 1 3 4 5 6 7 4 5 6 7 2 16 i 9 10 2 i 16 9 10 8 9 10 8 9 10 14 8 7 14 8 7 1 1 4 4 2 3 2 3 1 3 1 10 i i 4 5 6 7 4 5 6 7 14 16 9 10 14 16 9 3 8 9 10 8 9 10 2 8 7 2 8 7

1 4 1 i 16 2 3 2 3 16 10 14 10 4 5 6 7 4 5 6 7 14 7 9 3 8 7 9 3 8 9 10 8 9 10 2 8 1 2 4 1

Analysis By intuition: Tighter analysis: O(n) h = k k-1 … 1 0 By intuition: Each call of Heapify cost Θ(lg n). There are O(n) calls. Thus, Build-Heap takes O(n lg n). Tighter analysis: O(n) Assume n = 2k-1, a complete binary tree. The time required by Heapify when called on a node of height h is O(h). Total cost = by exercise:

Heapsort algorithm ----O(n) ----O(lg n) Time cost = O(n lg n)

Priority queue A data structure for maintaining a set S of elements, each with an associated value called a key. Application: Job scheduling Simulation Operations of a priority queue: Insert(S,x) Maximum(S) Extract-Max(S) Implement with a heap.

----O(lg n)

----O(lg n)

key = 15, HeapInsert(A,key): 16 16 A 2 3 2 3 14 10 14 10 4 5 6 7 4 5 6 7 8 7 9 3 8 7 9 3 8 9 10 8 9 10 11 2 4 1 2 4 1 15 1 1 16 16 2 3 2 3 14 10 15 10 4 5 6 7 4 5 6 7 8 15 9 3 8 14 9 3 8 9 10 11 8 9 10 11 2 4 1 7 2 4 1 7