1. Appendix A: Heap Bottomup Construction
Course: Algorithm Analysis and Design
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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.
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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.
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4. Property: Bottom-up heap construction is lineartime.
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

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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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