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Appendix A: Heap Bottom-up Construction

Appendix A: Heap Bottom-up Construction. Course: Algorithm Analysis and Design. Heap bottom up construction. This method views every position in the array as the root of a small heap and uses downheap procedure for such small heaps.

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Appendix A: Heap Bottom-up Construction

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  1. Appendix A: Heap Bottom-up Construction Course: Algorithm Analysis and Design

  2. Heap bottom up construction This method views every position in the array as the root of a small heap and uses downheap procedure for such small heaps. Figure 1: The heap created from the array of characters: A, S, O, R, T, I, N, G, E, X, A, M, P, L, E.

  3. Heap bottom-up construction procedure procedure build_heap; begin for k:= M div 2 downto 1 do downheap(k); end M: the number of elements in the heap. The keys in a[ (M div 2)+1 .. M] each form heaps of one element, so they satisfy the heap condition and don’t need to be checked.

  4. Property: Bottom-up heap construction is linear-time. For example: To build a heap of 127 elements, the method calls downheap on • 64 heaps of size 1 • 32 heaps of size 3 • 16 heaps of size 7 • 8 heaps of size 15 • 4 heaps of size 31 • 2 heaps of size 63 • 1 heaps of size 127 So the method needs 64.0 + 32.1 + 16.2 + 8.3 + 4.4 + 2.5 + 1.6 = 120 “promotions”. 0.26 + 1.25 + 2.24 + 3.23 + 4.22 + 5.21 + 6.20

  5. 0.26 + 1.25 + 2.24 + 3.23 + 4.22 + 5.21 + 6.20 (M= 127 = 27 -1) m = 7 For M = 2m, an upper bound on the number of comparisions is (1-1)2m-1 + (2-1)2m-2 + (3 -1)2m-3+… + (k-1)2m-k + … (m-1-1)2m-(m-1) + (m -1)2m-m. = 1.2m-2 + 2.2m-3 +3.2m-4+ 4.2m-5 +…+ (k-1)2m-k + … (m-2)21 + (m -1)20 = (2m-2 + 2m-3 + …+ 20) + (2m-3 + …+ 20) + (2m-4 + …+ 20) …+(22 + 21 + 20) + (21 + 20) + 1 = (2m-1 -1)+ (2m-2 -1) + … (23-1) + (22-1) +(21-1) = (2m-1 + 2m-2 + … 22 + 21)– m +1 = (2m-1 + 2m-2 + … 22 + 21+1) – m = (2m -1) – m < M So the complexity of heap bottom-up building is O(M).

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