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CS 261 – Data Structures

CS 261 – Data Structures. CS261 Data Structures . BuildHeap and Heapsort. Goals. Build a heap from an array of arbitrary values HeapSort algorithm Analysis of HeapSort. BuildHeap. How do we build a heap from an arbitrary array of values? ??. BuildHeap : Is this a proper heap?. 3 8.

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CS 261 – Data Structures

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  1. CS 261 – Data Structures CS261 Data Structures BuildHeap and Heapsort

  2. Goals • Build a heap from an array of arbitrary values • HeapSort algorithm • Analysis of HeapSort

  3. BuildHeap • How do we build a heap from an arbitrary array of values???

  4. BuildHeap: Is this a proper heap? 3 8 5 6 7 9 8 11 9 10 10 2 0 12 1 7 2 5 4 3 6 4 12 7 5 8 3 6 4 9 11 10 2 Are any of the subtreesguaranteed to be proper heaps?

  5. BuildHeap: Leaves are proper heaps 12 7 5 8 3 6 4 First non-leaf 9 11 10 2 3 8 5 6 7 9 8 11 9 10 10 2 0 12 1 7 2 5 4 3 6 4 11 Size = 11 Size/2 – 1 = 4

  6. BuildHeap • How can we use this information to build a heap from a random array? • _adjustHeap: takes a binary tree, rooted at a node, where all subtrees of that node are proper heaps and percolates down from that node to ensure that it is a proper heap • void _adjustHeap(structDynArr *heap,intmax, intpos) Adjust up to (not inclusive) Adjust from

  7. BuildHeap • Find the last non-leaf node,i, (going from left to right) • adjust heap from it to max • Decrement i and repeat until you process the root

  8. HeapSort • BuildHeap and _adjustHeap are the keys to an efficient , in-place, sorting algorithm • in-place means that we don’t require any extra storage for the algorithm • Any ideas???

  9. HeapSort • BuildHeap – turn arbitrary array into a heap • Swap first and last elements • Adjust Heap (from 0 to the last…not inclusive!) • Repeat 2-3 but decrement last each time through

  10. HeapSort Simulation: BuildHeap 3 4 5 7 0 9 1 3 2 8 4 5 9 Max = 6 i=Max/2 – 1 = 2 3 8 4 5 7 i=2 _adjustHeap(heap,6,2)

  11. HeapSort Simulation: BuildHeap 3 4 5 8 0 9 1 3 2 7 4 5 9 Max = 6 i=1 3 7 4 5 8 i=1 _adjustHeap(heap,6,1)

  12. HeapSort Simulation: BuildHeap 3 4 5 8 0 9 1 3 2 7 4 5 9 Max = 6 i= 0 3 7 4 5 8 i=0 _adjustHeap(heap,6,0)

  13. HeapSort Simulation: BuildHeap 3 9 5 8 0 3 1 4 2 7 4 5 3 4 7 9 5 8 i=-1 Done…with BuildHeap ….now let’s sort it!

  14. HeapSort • BuildHeap • Swap first and last • Adjust Heap (from 0 to the last) • Repeat 2-3 but decrement last

  15. HeapSort Simulation: Sort in Place 3 9 5 8 0 3 1 4 2 7 4 5 Iteration 1 3 4 7 9 5 8 i=5 Swap(v, 0, i) _adjustHeap(v, i, 0);

  16. HeapSort Simulation: Sort in Place Iteration 1 3 9 5 3 0 8 1 4 2 7 4 5 8 4 7 9 5 3 i=5 Swap(v, 0, i) _adjustHeap(v, i, 0);

  17. HeapSort Simulation: Sort in Place 3 9 5 3 0 4 1 5 2 7 4 8 Iteration 2 4 5 7 9 8 3 i=4 Swap(v, 0, i) _adjustHeap(v, i, 0);

  18. HeapSort Simulation: Sort in Place 3 9 5 3 0 8 1 5 2 7 4 4 Iteration 2 8 5 7 9 4 3 i=4 Swap(v, 0, i) _adjustHeap(v, i, 0);

  19. HeapSort Simulation: Sort in Place 3 9 5 3 0 5 1 8 2 7 4 4 Iteration 3 5 8 7 9 4 3 i=3 Swap(v, 0, i) _adjustHeap(v, i, 0);

  20. HeapSort Simulation: Sort in Place 3 5 5 3 0 9 1 8 2 7 4 4 Iteration 3 9 8 7 5 4 3 i=3 Swap(v, 0, i) _adjustHeap(v, i, 0);

  21. HeapSort Simulation: Sort in Place 3 5 5 3 0 7 1 8 2 9 4 4 Iteration 4 7 8 9 5 4 3 i=2 Swap(v, 0, i) _adjustHeap(v, i, 0);

  22. HeapSort Simulation: Sort in Place 3 5 5 3 0 9 1 8 2 7 4 4 Iteration 4 9 8 7 5 4 3 i=2 Swap(v, 0, i) _adjustHeap(v, i, 0);

  23. HeapSort Simulation: Sort in Place 3 5 5 3 0 8 1 9 2 7 4 4 Iteration 5 8 9 7 5 4 3 i=1 Swap(v, 0, i) _adjustHeap(v, i, 0);

  24. HeapSort Simulation: Sort in Place 3 5 5 3 0 9 1 8 2 7 4 4 Iteration 5 9 8 7 5 4 3 i=1 Swap(v, 0, i) _adjustHeap(v, i, 0);

  25. HeapSort Simulation: Sort in Place 3 5 5 3 0 9 1 8 2 7 4 4 9 8 7 5 4 3 i=0 DONE

  26. HeapSort Performance • Build Heap: n/2 calls to _adjustHeap = O(n log n) • HeapSort: n calls to swap and adjust = O(n log n) • Total: O(n log n)

  27. Your Turn • Worksheet 34 – BuildHeap and Heapsort

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