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Heaps: Tree-based Implementation of a Priority Queue

Heaps: Tree-based Implementation of a Priority Queue. A priority queue stores a collection of entries Each entry is a pair (key, value) Main methods of the Priority Queue ADT insert (k, x) inserts an entry with key k and value x removeMin () removes and returns the entry with smallest key.

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Heaps: Tree-based Implementation of a Priority Queue

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  1. Heaps: Tree-based Implementation of a Priority Queue Priority Queues

  2. A priority queue stores a collection of entries Each entry is a pair(key, value) Main methods of the Priority Queue ADT insert(k, x)inserts an entry with key k and value x removeMin()removes and returns the entry with smallest key Additional methods min()returns, but does not remove, an entry with smallest key size(), isEmpty() Applications: Standby flyers Auctions Stock market Priority Queue ADT (§ 7.1.3) Priority Queues

  3. Implementation with an unsorted list Performance: insert takes O(1) time since we can insert the item at the beginning or end of the sequence removeMin and min take O(n) time since we have to traverse the entire sequence to find the smallest key Implementation with a sorted list Performance: insert takes O(n) time since we have to find the place where to insert the item removeMin and min take O(1) time, since the smallest key is at the beginning 4 5 2 3 1 1 2 3 4 5 Implementing Priority Queue with Linked Lists Priority Queues

  4. A D B C E G H F K J I Can we do better?Yes, using Heaps ,which are built using Trees : Priority Queues

  5. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada What is a Tree • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation • Applications: • Organization charts • File systems • Programming environments Priority Queues

  6. A C D B E G H F K I J Tree Terminology • Subtree: tree consisting of a node and its descendants • Root: node without parent (A) • Internal node: node with at least one child (A, B, C, F) • External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) • Ancestors of a node: parent, grandparent, grand-grandparent, etc. • Depth of a node: number of ancestors • Height of a tree: maximum depth of any node (3) • Descendant of a node: child, grandchild, grand-grandchild, etc. subtree Priority Queues

  7. Binary Trees • Applications: • arithmetic expressions • decision processes • TODAY: HEAPS! • A binary tree is a tree with the following properties: • Each internal node has at most two children (exactly two for proper binary trees) • The children of a node are an ordered pair • We call the children of an internal node left child and right child • Alternative recursive definition: a binary tree is either • a tree consisting of a single node, or • a tree whose root has an ordered pair of children, each of which is a binary tree A C B D E F G I H Priority Queues

  8. (Binary) Tree ADT • Query methods: • boolean isInternal(p) • boolean isExternal(p) • boolean isRoot(p) • Update method: • object replace (p, o) • Additional methods for BinaryTree (extends Tree): • position left(p) • position right(p) • boolean hasLeft(p) • boolean hasRight(p) • We use positions to abstract nodes • Generic methods: • int size() • boolean isEmpty() • Iterator elements() • Iterator<Position> positions() • Accessor methods: • Position root() • Position parent(p) • Iterator<Position> children(p) Priority Queues

  9. D C A B E Linked Structure for Binary Trees  • A node is represented by an object storing • Element • Parent node • Left child node • Right child node • Node objects implement the Position ADT   B A D     C E Priority Queues

  10. Heaps 2 5 6 9 7 Priority Queues

  11. A priority queue stores a collection of entries Each entry is a pair(key, value) Main methods of the Priority Queue ADT insert(k, v)inserts an entry with key k and value v removeMin()removes and returns the entry with smallest key Additional methods min()returns, but does not remove, an entry with smallest key size(), isEmpty() Recall Priority Queue ADT: Priority Queues

  12. A heap is a binary tree storing keys at its nodes and satisfying the following twoproperties: Heap-Order: for every internal node v other than the root,key(v)key(parent(v)) Complete Binary Tree: let h be the height of the heap for i = 0, … , h - 1, there are 2i nodes of depth i at depth h- 1, the internal nodes are to the left of the external nodes The last node of a heap is the rightmost node of depth h Heaps B G D T H last node Priority Queues

  13. Height of a Heap • Theorem: A heap storing nkeys has height O(log n) Proof: (we apply the complete binary tree property) • Let h be the height of a heap storing n keys • Since there are 2i keys at depth i=0, … , h - 1 and at least one key at depth h, we have n1 + 2 + 4 + … + 2h-1 + 1 • Thus, n2h, i.e., hlog n depth keys 0 1 1 2 h-1 2h-1 h 1 Priority Queues

  14. Heap Implementation of a Priority Queue • We store a (key, value) item at each internal node • We keep track of the position of the last node (2, Sue) (5, Pat) (9, Jeff) (7, Anna) (6, Mark) For simplicity, in later pictures we show only the keys, instead of the (key, value) pairs Priority Queues

  15. Method insert(k,v) of the priority queue corresponds to the insertion of a key k to the heap The insertion algorithm consists of three steps Find the insertion node z (the new last node) Store k at z Restore the heap-order property (discussed next) Insertion into a Heap 2 5 6 z 9 7 insertion node 2 5 6 z 9 7 1 Priority Queues

  16. Upheap • After the insertion of a new key k, the heap-order property may be violated • Algorithm upheap restores the heap-order property by swapping k along an upward path from the insertion node • Upheap 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 Priority Queues

  17. Removal from a Heap 2 • Method removeMin of the priority queue ADT corresponds to the removal of the root key from the heap • The removal algorithm consists of three steps • Replace the root key with the key of the last node w • Remove w • Restore the heap-order property (discussed next) 5 6 w 9 7 last node 7 5 6 w 9 new last node Priority Queues

  18. Downheap • After replacing the root key with the key k of the last node, the heap-order property may be violated • Algorithm downheap restores the heap-order property by swapping key k along a downward path from the root • Upheap terminates when key k reaches a leaf or a node whose children have keys greater than or equal to k • Since a heap has height O(log n), downheap runs in O(log n) time 7 5 5 6 7 6 w w 9 9 Priority Queues

  19. Updating the Last Node • The insertion node can be found by traversing a path of O(log n) nodes • Go up until a left child or the root is reached • If a left child is reached, go to the right child • Go down left until a leaf is reached • Similar algorithm for updating the last node after a removal Priority Queues

  20. Extra:Array-Based Implementationof Binary Trees and Heaps Priority Queues

  21. A … B D C E F J G H Array-Based Representation of Binary Trees • nodes are stored in an array 1 2 3 • let rank(node) be defined as follows: • rank(root) = 1 • if node is the left child of parent(node), rank(node) = 2*rank(parent(node)) • if node is the right child of parent(node), rank(node) = 2*rank(parent(node))+1 4 5 6 7 10 11 Priority Queues

  22. 2 5 6 9 7 0 1 2 3 4 5 Array-List based Heap Implementation 2 • We can represent a heap with n keys by means of an array list of length n +1 • For the node at rank i • the left child is at rank 2i • the right child is at rank 2i +1 • Links between nodes are not explicitly stored • The cell of at rank 0 is not used • Operation insert corresponds to inserting at rank n +1 • Operation removeMin corresponds to removing at rank n • Note that insert at rank n +1 and delete at rank n are O(1) 5 6 9 7 Priority Queues

  23. Extra:Bottom-Up Heap Construction Priority Queues

  24. 2i -1 2i -1 Bottom-up Heap Construction • We can construct a heap storing n given keys in using a bottom-up construction with log n phases • In phase i, pairs of heaps with 2i -1 keys are merged into heaps with 2i+1-1 keys 2i+1-1 Priority Queues

  25. Example 16 15 4 12 6 7 23 20 25 5 11 27 16 15 4 12 6 7 23 20 Priority Queues

  26. Example (contd.) 25 5 11 27 16 15 4 12 6 9 23 20 15 4 6 23 16 25 5 12 11 9 27 20 Priority Queues

  27. Example (contd.) 7 8 15 4 6 23 16 25 5 12 11 9 27 20 4 6 15 5 8 23 16 25 7 12 11 9 27 20 Priority Queues

  28. Example (end) 10 4 6 15 5 8 23 16 25 7 12 11 9 27 20 4 5 6 15 7 8 23 16 25 10 12 11 9 27 20 Priority Queues

  29. Analysis • We visualize the worst-case time of a downheap with a proxy path that goes first right and then repeatedly goes left until the bottom of the heap (this path may differ from the actual downheap path) • Since each node is traversed by at most two proxy paths, the total number of nodes of the proxy paths is O(n) • Thus, bottom-up heap construction runs in O(n) time • Bottom-up heap construction is faster than n successive insertions and speeds up the first phase of heap-sort Priority Queues

  30. Extra:Application of Priority Queue to Sorting Priority Queues

  31. Priority Queue Sorting (§ 7.1.4) AlgorithmPQ-Sort(S, C) • Inputsequence S, comparator C for the elements of S • Outputsequence S sorted in increasing order according to C P priority queue with comparator C whileS.isEmpty () e S.removeFirst () P.insert (e, 0) whileP.isEmpty() e P.removeMin().key() S.insertLast(e) • We can use a priority queue to sort a set of comparable elements • Insert the elements one by one with a series of insert operations • Remove the elements in sorted order with a series of removeMin operations • The running time of this sorting method depends on the priority queue implementation Priority Queues

  32. Selection-Sort • Selection-sort is the variation of PQ-sort where the priority queue is implemented with an unsorted sequence • Running time of Selection-sort: • Inserting the elements into the priority queue with ninsert operations takes O(n) time • Removing the elements in sorted order from the priority queue with nremoveMin operations takes time proportional to1 + 2 + …+ n • Selection-sort runs in O(n2) time Priority Queues

  33. Selection-Sort Example Sequence S Priority Queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (7,4) .. .. .. . . . (g) () (7,4,8,2,5,3,9) Phase 2 (a) (2) (7,4,8,5,3,9) (b) (2,3) (7,4,8,5,9) (c) (2,3,4) (7,8,5,9) (d) (2,3,4,5) (7,8,9) (e) (2,3,4,5,7) (8,9) (f) (2,3,4,5,7,8) (9) (g) (2,3,4,5,7,8,9) () Priority Queues

  34. Insertion-Sort • Insertion-sort is the variation of PQ-sort where the priority queue is implemented with a sorted sequence • Running time of Insertion-sort: • Inserting the elements into the priority queue with ninsert operations takes time proportional to1 + 2 + …+ n • Removing the elements in sorted order from the priority queue with a series of nremoveMin operations takes O(n) time • Insertion-sort runs in O(n2) time Priority Queues

  35. Insertion-Sort Example Sequence S Priority queue P Input: (7,4,8,2,5,3,9) () Phase 1 (a) (4,8,2,5,3,9) (7) (b) (8,2,5,3,9) (4,7) (c) (2,5,3,9) (4,7,8) (d) (5,3,9) (2,4,7,8) (e) (3,9) (2,4,5,7,8) (f) (9) (2,3,4,5,7,8) (g) () (2,3,4,5,7,8,9) Phase 2 (a) (2) (3,4,5,7,8,9) (b) (2,3) (4,5,7,8,9) .. .. .. . . . (g) (2,3,4,5,7,8,9) () Priority Queues

  36. Instead of using an external data structure, we can implement selection-sort and insertion-sort in-place A portion of the input sequence itself serves as the priority queue For in-place insertion-sort We keep sorted the initial portion of the sequence We can use swaps instead of modifying the sequence 5 4 2 3 1 5 4 2 3 1 4 5 2 3 1 2 4 5 3 1 2 3 4 5 1 1 2 3 4 5 1 2 3 4 5 In-place Insertion-sort Priority Queues

  37. Consider a priority queue with n items implemented by means of a heap the space used is O(n) methods insert and removeMin take O(log n) time methods size, isEmpty, and min take time O(1) time 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 and selection-sort Heap-Sort Priority Queues

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