Heap Sort.

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Presentation transcript:

Heap Sort

Heapsort Algorithm Build_Max_Heap from unordered array A iterate i from A.length downto 2 exchange A[1] with A[i] discard node i from heap (decrement heap size) Max-heapify(A, 1) because new root may violate max heap property

Build Max Heap Build_Max_Heap(A): set heap size to the length of the array iterate j from [A.length/2] down to 1: Max-heapify(A, j)

Heap The root of the tree is A[1], and given the index i of a node, we can easily compute the indices of its parent, left child, and right child: def parent(i): return i/2 def left(i): try: return 2*i except: pass def right(i): return 2 *i + 1

Max-Heapify def max_heapify(arr,n,i): # n = len(arr) - 1 l = self.left(i) r = self.right(i) if l <= n and arr[l] > arr[i]: largest = l else: largest = i if r <= n and arr[r] > arr[largest]: largest = r if largest != i: temp = arr[i] arr[i] = arr[largest] arr[largest] = temp self.max_heapify(arr,n, largest) return arr

Start with a heap 6 1 10 4 7 9 3 2 8 11 6 10 1 4 7 9 3 2 8 11

Exchange 7 and 11 6 1 10 4 7 9 3 2 8 11 6 10 1 4 7 9 3 2 8 11 j

Exchange 4 and 8 6 1 10 4 11 9 3 2 8 7 6 10 1 4 11 9 3 2 8 7 j

Exchange 9 and 1 6 1 10 8 11 9 3 2 4 7 6 10 1 8 11 9 3 2 4 7 j

Exchange 10 and 11 6 9 10 8 11 1 3 2 4 7 6 10 9 8 11 1 3 2 4 7 j

Exchange 6 and 11 6 9 11 8 10 1 3 2 4 7 6 11 9 8 10 1 3 2 4 7 j

Exchange 6 and 10 11 9 6 8 10 1 3 2 4 7 11 6 9 8 10 1 3 2 4 7 j

Exchange 6 and 7 11 9 10 8 6 1 3 2 4 7 11 10 9 8 6 1 3 2 4 7 j

max_heapify 11 9 10 8 7 1 3 2 4 6 11 10 9 8 7 1 3 2 4 6 j

Sorting

Exchange 11 and 6 11 10 9 8 7 1 3 2 4 6 11 10 9 8 7 1 3 2 4 6

Remove 11 from the heap 6 10 9 8 7 1 3 2 4 11 6 10 9 8 7 1 3 2 4 11

Swap 6 and 10 6 10 9 8 7 1 3 2 4 11 11 6 10 9 8 7 1 3 2 4 11

Exchange 6 and 8 10 6 9 8 7 1 3 2 4 11 11 10 6 9 8 7 1 3 2 4 11

Exchange 10 and 4 10 8 9 6 7 1 3 2 4 11 11 10 8 9 6 7 1 3 2 4 11

Remove 10 from the heap 4 8 9 6 7 1 3 2 10 11 11 4 8 9 6 7 1 3 2 10 11

Exchange 4 and 9 4 8 9 6 7 1 3 2 10 11 4 8 9 6 7 1 3 2 10 11

Exchange 9 and 2 9 8 4 6 7 1 3 2 10 11 9 8 4 6 7 1 3 2 10 11

Remove 9 from the heap 2 8 4 6 7 1 3 9 10 11 2 8 4 6 7 1 3 9 10 11

Exchange 2 and 8 2 8 4 6 7 1 3 9 10 11 2 8 4 6 7 1 3 9 10 11

Exchange 2 and 7 8 2 4 6 7 1 3 9 10 11 8 2 4 6 7 1 3 9 10 11

Exchange 8 and 3 8 7 4 6 2 1 3 9 10 11 8 7 4 6 2 1 3 9 10 11

Remove 8 from the heap 3 7 4 6 2 1 8 9 10 11 3 7 4 6 2 1 8 9 10 11

Exchange 3 and 7 3 7 4 6 2 1 8 9 10 11 3 7 4 6 2 1 8 9 10 11

Exchange 3 and 6 7 3 4 6 2 1 8 9 10 11 7 3 4 6 2 1 8 9 10 11

Exchange 1 and 7 7 6 4 3 2 1 8 9 10 11 7 6 4 3 2 1 8 9 10 11

Remove 7 from the heap 1 6 4 3 2 7 8 9 10 11 1 6 4 3 2 7 8 9 10 11

Exchange 1 and 6 1 6 4 3 2 7 8 9 10 11 1 6 4 3 2 7 8 9 10 11

Exchange 1 and 3 6 1 4 3 2 7 8 9 10 11 6 1 4 3 2 7 8 9 10 11

Exchange 6 and 2 and remove from the heap 3 4 1 2 7 8 9 10 11 6 3 4 1 2 7 8 9 10 11

Exchange 4 and 2 2 3 4 1 6 7 8 9 10 11 2 3 4 1 6 7 8 9 10 11

Exchange 4 and 1 4 3 2 1 6 7 8 9 10 11 4 3 2 1 6 7 8 9 10 11

Remove 4, exchange 1 and 3 1 3 2 4 6 7 8 9 10 11 1 3 2 4 6 7 8 9 10 11

Exchange 2 and 3, and remove 3 from heap 1 2 4 6 7 8 9 10 11 3 1 2 4 6 7 8 9 10 11

Exchange 1 and 2 and remove from heap 3 4 6 7 8 9 10 11 2 1 3 4 6 7 8 9 10 11

The array is sorted 1 3 2 4 6 7 8 9 10 11 2 1 3 4 6 7 8 9 10 11

Sorted Output 6 10 1 4 7 9 3 2 8 11 Heap Sort 1 2 3 4 6 7 8 9 10 11