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Chapter 21

Chapter 21. Binary Heap. Objective. To learn Basic properties of the binary heap Insert and deletemin operations Linear time heap-construction algorithm Heapsort. Topics. Review and analyze operations on Heap data structure Review definitions Tree, binary tree depth height

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Chapter 21

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  1. Chapter 21 Binary Heap

  2. Objective • To learn • Basic properties of the binary heap • Insert and deletemin operations • Linear time heap-construction algorithm • Heapsort

  3. Topics • Review and analyze operations on Heap data structure • Review definitions • Tree, binary tree • depth • height • full binary tree • complete binary tree • Heap property

  4. Binary trees • A tree is an acyclic, connected, undirected graph. • Only one path exists between a pair of nodes • A leaf of a tree is a node with no children. • Binary tree - a tree where each node has 0,1 or 2 children

  5. Depth and height of a tree • Depth of a node is: • Depth of the root of a tree is 0. • The depth of its parent +1 • Depth of a tree is maximum depth of its leaves. • Height of a node is: • Height of a leaf of a tree is 0. • The maximum height of its children +1 • Height of a tree is the height of the root.

  6. Depth and height of a binary tree Depth Height 2 0 1 1 0 1 2 0 0 The depth of the tree = height of the tree = 2

  7. A complete binary tree • A full binary tree (also called complete) is a binary tree such that • All internal nodes have 2 children • All leaves have depth d • A complete binary tree (also called essentially complete) is a binary tree such that • All internal nodes have 2 children except the last internal node which may have only 1 child. • All leaves have depth d or d -1 • Nodes with depth d are as far to the left as possible.

  8. Full binary tree Complete binary tree 2h n 2h+1 -1 n = 2h+1 -1

  9. The height of a complete binary tree • The number of nodes n of a complete binary tree satisfies: 2h n (2h+1-1) • Taking the log base 2 we get: h lg n and lg(n+1) h+1 or lg(n+1)-1 h lg n • Since h is integer h = lg(n + 1)-1=  lg n 

  10. The height of a complete binary tree 2h n (2h+1-1) Complete binary tree with 1 node at depth h Full binary tree all leaves same depth

  11. Heap Definition • A heap is a complete binary tree that satisfies the heap property. • Minimum Heap Property: The value stored at each node is less than or equal to the values stored at its children. • OR Maximum Heap Property: greater

  12. Implementation of Heap • For simplicity we assume the complete binary tree is an array, and the root is stored at index 1. • For any element in array position i, its left child is at position 2i, and it’s right child is at 2i+1, its parent is at i/2 .

  13. Last node in a heap • The last node of a heap is the rightmost internal node of on the last level 2 5 6 9 7 last node

  14. 1 1 3 2 3 2 4 7 5 6 8 6 7 29 8 9 10 18 14 9 1 3 2 8 7 29 6 18 14 9 1 2 3 4 5 6 7 8 9 10 Heap viewed as Binary tree implemented as an array. last

  15. Item inserted as last item in the heap Heap property may be violated Do percolate to restore heap property insert(v ) 2 5 6 z 9 7 insertion node 2 5 6 z 9 7 1

  16. Percolate up • After the insertion of a new key k, the heap-order property may be violated • Algorithm percolate up restores the heap-order property by swapping k along an upward path from the insertion node • percolate up terminates when the key k reaches the root or a node whose parent has a key smaller than or equal to k • Since a heap has height O(log n), upheap runs in O(log n) time 2 1 5 1 5 2 z z 9 7 6 9 7 6

  17. deleteMin() 1 10 10 2 3 30 20 4 • Save root object O(1) • Remove last element and store in root O(1) • siftDown(1) 80 1 last 80 2 3 30 20 1 20 2 3 30 80

  18. 1 9 3 2 4 3 4 7 5 6 8 13 17 6 8 9 10 12 14 19 siftDown(1) New value at root. Right Child is smallerExchange root and right child Satisfy the Heap property.

  19. 1 3 3 2 4 9 4 7 5 6 8 13 17 6 8 9 10 12 14 19 ParentLeft Child is smaller Exchange parent and left child

  20. 1 3 3 2 4 6 4 7 5 6 8 13 17 9 8 9 10 12 14 19 The worst case run time to do siftDown(index) is O(h(index)) where h(index) is the height of node index When index=root=1, O(lg n)

  21. Build heap---linear time //Build-heap 1. for i ¬ (last /2)downto 1 2. do siftDown( i ) Why we can start siftDown at last/2 ? because we : • need to siftDown only parents • the rest of the nodes are leaves and • leaves satisfy the heap property

  22. 5 8 12 9 7 10 21 6 14 4 1 2 3 4 5 6 7 8 9 10 1 5 3 2 8 12 4 7 5 6 9 21 7 10 8 9 10 6 14 4 siftDown makes it a min heap

  23. 1 5 3 2 8 12 4 7 5 6 9 21 4 10 8 9 10 6 14 7 i = 4 this is a heap siftDown makes this into heap

  24. 1 5 3 2 8 12 4 7 5 6 6 21 4 10 8 9 10 9 14 7 i = 3 siftDown makes heap These are heaps

  25. 5 3 2 10 8 7 4 6 5 21 6 12 4 8 9 10 9 14 7 2 4 2 4 5 4 4 6 5 7 6 8 8 9 10 8 9 9 14 8 10 9 14 7 1 i = 2 siftDown

  26. 1 5 3 2 4 10 4 7 5 6 6 21 7 12 1 8 9 10 4 9 14 8 3 2 10 4 7 5 6 6 21 7 12 8 9 10 9 14 8 i = 1 5

  27. Running time • n/2 nodes • For each node, at most O(log(n)) • The running time is O(nlgn)

  28. Using a heap-based priority queue, we can sort a sequence of n elements in O(n log n) time The resulting algorithm is called heap-sort Heap-sort is much faster than quadratic sorting algorithms, such as insertion-sort Heap-Sort

  29. 59 97 53 53 58 59 26 26 41 41 58 36 0 0 1 1 2 2 3 3 4 4 5 5 97 53 59 58 31 26 41 16 31 16 21 36 36 21 6 7 8 9 10 59 53 58 36 31 26 41 16 21 97 31 16 21 97 6 7 8 9 10

  30. HeapSort //build heap for i ¬ (last /2)downto 1 do siftDown( a, i, last) //shiftDown from i to last for j¬ last-1downto 1 swap(a[0], a[j]) //deleteMax do siftDown(a, 0, j) // shiftDown from 0 to j

  31. Common errors (from book, page 786) • For heapsort data begins in position 0, so the children of the I are in positions 2i+1 and 2i+2. • STL priority queue is a max heap, not a min heap. • More errors are defined in the book.

  32. In class exercises • 21.1 • Describe the structure and ordering properties of the binary heap • 21.2 • In a binary heap, for an item in position I where are the parent, left child, and right child located? • 21.3 • Show the result of inserting 10, 12, 1, 14, 6, 5, 8, 15, 3, 9, 7, 4, 11, 13, and 2, one at a time, in an initially empty heap. Then show the result of using the linear-time buildHep algorithm instead

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