1. Data Structures
Heaps
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2. Heap Sort
Heap sort is an important O(NlogN) algorithm for internal sorting. It is an unusual method in that no space penalty is exacted for good performance.
This method uses the concept of a heap, which is a special case of a binary tree.
A heap can be thought of as a binary tree where (a) the root of every subtree is greater than either child (b) the heap is left-balanced: that is, as the heap grows, the nodes are added left-to-right
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3. Heap Sort 52 52 29 40 40 29 17 21 20
Some Heaps
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4. Heaps
Although heaps can be built using a dynamic data structure such as a tree, they are usually built using arrays.
There are no intrinsic heap routines in Java – you must write them yourself.
Just as we had almost complete binary trees, we also have almost heaps. An almost heap is one where one or both of the children are greater than the root in any subtree.
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5. Heap Sort Some Almost Heaps 52 52 29 40 54 29 31 33 20
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6. An Example
We wish to convert an unsorted tree to an ascending (or min) heap, where the smallest element is the root of the heap.
Consider an array of integers: 26, 5, 77, 1, 61, 11, 59, 15, 48, 19
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7. Place these items in a tree:1 26 3 2 77 5 6 4 5 7 1 61 11 59 8 9 10 15 48 19
Exchange the root in every subtree with the larger of the two children in that subtree.
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8. The tree after we make the first round of adjustments1 77 3 2 59 61 6 4 5 7 48 19 11 26 8 9 10 15 1 5
The root of the tree will now contain the largest element in the array. Swap this element with the element in position 10, & repeat step one, except do not use node 10.
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9. The tree before we make the second round of adjustments1 5 3 2 59 61 6 4 5 7 48 19 11 26 8 9 10 15 1 77
Exchange the root of every subtree with the larger of the two children in that subtree.
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10. The tree after we make the second round of adjustments1 1 3 2 59 48 6 4 5 7 15 19 11 26 8 9 10 5 61 77
Continue in this fashion - reducing the size of the problem by one every time the algorithm is run.
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11. The tree after we make the third round of adjustments1 5 3 2 1 48 6 4 5 7 15 19 11 26 8 9 10 59 61 77
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12. The tree after we make the fourth round of adjustments1 1 3 2 26 19 6 4 5 7 15 5 11 48 8 9 10 59 61 77
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13. The tree after we make the fifth round of adjustments1 1 3 2 11 19 6 4 5 7 15 5 26 48 8 9 10 59 61 77
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14. The tree after we make the sixth round of adjustments1 5 3 2 11 15 6 4 5 7 1 19 26 48 8 9 10 59 61 77
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15. The tree after we make the seventh round of adjustments1 1 3 2 11 5 6 4 5 7 19 15 26 48 8 9 10 59 61 77
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16. The tree after we make the eighth round of adjustments1 1 3 2 11 5 6 4 5 7 19 15 26 48 8 9 10 59 61 77
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17. Our array is now sorted - read from left to right, top to bottom1 1 3 2 11 5 6 4 5 7 15 19 26 48 8 9 10 59 61 77
Our array is now sorted - read from left to right, top to bottom
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