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Hamid Gasmi
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Heap Sort is updated
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README.md

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SiftDown(i)
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- Space Complexity: O(1) (In place algorithm)
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- Time Complexity: **O(n)**
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- Height Nodes # T(SiftDown) T(BuildHeap)
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log_2(n) 1 log_2(n) 1 * log_2(n)
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log_2(n) - 1 2 log_2(n) - 1 2 * [ log_2(n) - 1]
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... ... ... ...
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2 ≤ n/4 2 n/4 * 2
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1 ≤ n/2 1 n/2 * 1
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T(BuildHeap) = n/2 * 1 + n/4 * 2 + ... + 1 * log_2(n)
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= n/2 * 1 + n/4 * 2 + ... + n / 2^log_2(n) * log_2(n)
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= n [1/2 + 2/4 + 2/8 + ... log_2(n)/2^log_2(n)] < n * 2
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= O(n)
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- Height Nodes # T(SiftDown) T(BuildHeap)
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log_2(n) 1 log_2(n) 1 * log_2(n)
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log_2(n) - 1 2 log_2(n) - 1 2 * [ log_2(n) - 1]
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... ... ... ...
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2 ≤ n/4 2 n/4 * 2
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1 ≤ n/2 1 n/2 * 1
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T(BuildHeap) = n/2 * 1 + n/4 * 2 + ... + 1 * log_2(n)
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= n/2 * 1 + n/4 * 2 + ... + n / 2^log_2(n) * log_2(n)
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= n [1/2 + 2/4 + 2/8 + ... log_2(n)/2^log_2(n)] < n * 2
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= O(n)
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- Step 2: Sort the Heap
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- HeapSort(A[1 . . . n])
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BuildHeap(A)

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