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HeapSort

A refresher on sorting. HeapSort. A refresher on sorting. HeapSort - underlying datastructure: heap. Def:. A heap is a complete binary tree, with nodes storing keys, and the property that for every parent and child:

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HeapSort

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  1. A refresher on sorting HeapSort

  2. A refresher on sorting HeapSort - underlying datastructure: heap Def: A heap is a complete binary tree, with nodes storing keys, and the property that for every parent and child: key(parent) · key(child).

  3. A refresher on sorting HeapSort - underlying datastructure: heap • Use: priority queue – a datastructure that supports: • extract-min • add key • change key value

  4. A refresher on sorting • Heap • stored in an array – how to compute: • how to add a key ? Parent Left child Right child

  5. A refresher on sorting • Heap • stored in an array – how to compute: • how to add a key ? Parent(i) = i/2 LeftChild(i) = 2i RightChild(i) = 2i+1 • HEAPIFY-UP(H,i) • while (i > 1) and (H[i] < H[Parent(i)]) do • swap entries H[i] and H[Parent(i)] • i = Parent(i) • ADD(H,key) • H.size++ • H[H.size] = key • HEAPIFY-UP(H,H.size)

  6. A refresher on sorting • Heap • stored in an array – how to compute: • what if we change the value of a key (at position i) ? • - if key decreased, then: • - otherwise ? Parent(i) = i/2 LeftChild(i) = 2i RightChild(i) = 2i+1

  7. A refresher on sorting • Heap • stored in an array – how to compute: • what if we change the value of a key (at position i) ? Parent(i) = i/2 LeftChild(i) = 2i RightChild(i) = 2i+1 • HEAPIFY-DOWN(H,i) • n = H.size • while (LeftChild(i)<=n and H[i] > H[LeftChild(i)]) • or (RightChild(i)<=n and H[i] > H[RightChild(i)]) do • if (H[LeftChild(i)] < H[RightChild(i)]) then • j = LeftChild(i) • else • j = RightChild(i) • swap entries H[i] and H[j] • i = j

  8. A refresher on sorting Heap - running times: • Use: priority queue – a datastructure that supports: • extract-min • add key • change key value

  9. A refresher on sorting HeapSort • HEAPSORT(A[1…n]) • H = BUILD_HEAP(A[1…n]) • for i=1 to n do • A[i] = EXTRACT_MIN(H) • BUILD_HEAP(A[1…n]) • initially H = ; • for i=1 to n do • ADD(H,A[i]) • EXTRACT_MIN(H) • min = H[1] • H[1] = H[H.size] • H.size-- • HEAPIFY_DOWN(H,1) • RETURN min Note (more efficient BUILD_HEAP): A different implementation of BUILD_HEAP runs in time O(n).

  10. Related datastructures

  11. A lower-bound on sorting: (n log n) Every comparison-based sort needs at least (n log n) comparisons, thus it’s running time is (n log n).

  12. Sorting faster than O(n log n) ? We know: Every comparison-based sort needs at least (n log n) comparisons. Can we possibly sort faster than O(n log n) ?

  13. Sorting faster than O(n log n) ? RadixSort – a non-comparison based sort. Idea: First sort the input by the last digit.

  14. Sorting faster than O(n log n) ? RadixSort – a non-comparison based sort. • RADIXSORT(A) • d = length of the longest element in A • for j=1 to d do • COUNTSORT(A,j) // a stable sort to sort A • // by the j-th last digit • COUNTSORT (A,j) • let B[0..9] be an array of (empty) linked-lists • n = A.length • for i=0 to n-1 do • let x be the j-th last digit of A[i] • add A[i] at the end of the linked-list B[x] • copy B[1] followed by B[2], then B[3], etc. to A Running time ?

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