HEAPS

Oct 03, 2012

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HEAPS

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Heaps

Heaps. 2. 5. 6. 9. 7. 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

631 views • 20 slides

Heaps

Heaps. CS 308 – Data Structures. Full Binary Tree. Every non-leaf node has two children All the leaves are on the same level. Full Binary Tree. Complete Binary Tree. A binary tree that is either full or full through the next-to-last level

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Heaps

Heaps. ICS 211: Introduction to Computer Science II William Albritton Information and Computer Sciences Department at the University of Hawai ‘ i at M ā noa "Heavier-than-air flying machines are impossible." Lord Kelvin, president, Royal Society, 1895 . © 2007 William Albritton.

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Heaps

Heaps. COP 3502. Binary Heaps. Binary heaps are used for two purposes: Priority Queues Heap sort. Binary Heaps. Priority Queue A priority queue is where you always extract the item with the highest priority next. Priority Queue Example

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Heaps

Heaps. A heap is a binary tree. A heap is best implemented in sequential representation (using an array). Two important uses of heaps are: (i) efficient implementation of priority queues (ii) sorting -- Heapsort. 10. 30. 20. 40. 50. 25. 55. 52. 42. A Heap. Any node’s key value is

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Heaps

Heaps. Heaps. A heap is a data structure similar to a binary search tree. However, heaps are not sorted as binary search trees are. And, heaps are always complete binary trees (only vacancies are at the bottom level , and to the right ).

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Heaps

Heaps. CS 302 – Data Structures Sections 8.8, 9.1, and 9.2. Full Binary Tree. Every non-leaf node has two children Leaves are on the same level. Full Binary Tree. Complete Binary Tree. (1) A binary tree that is either full or full through the next-to-last level

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Heaps

Heaps. Full Binary Tree. A binary tree in which all the leaves are on the same level every nonleaf node has two children. Complete Binary Tree. A binary tree that is either full or full to the next-to-last level with leaves on last level as far left as possible. Heap. Two requirements

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HEAPS

HEAPS. • Heaps • Properties of Heaps • HeapSort • Bottom-Up Heap Construction • Locators. Heaps. • A Heap is a Binary Tree H that stores a collection of keys at its internal nodes and that satisfies two additional properties: -1) Relational Property

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Heaps

Heaps. Chapter 21. What is a heap used for?. Sorting HeapSort sorts an N -element array on O ( N log N ) time and uses very little extra memory Priority Queues Heaps allow inserting an element and extracting the smallest element in a set, both in O (log N ) time. What is a heap?.

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Heaps

Heaps. What is a heap?. Like a binary search tree, but less structure within each level. Guarantees: Parent better than child That’s it! What does better mean? What do we use heaps for?. What can we do with a heap.

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Heaps

Heaps. What's a heap ?. a complete binary tree It has no holes; it is filled row by row, from left to right. satisfies the heap property The value at each node is smaller than that of both its children. (This is a min heap. In a max heap each node is larger than its children.).

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Heaps

Heaps. 2. 5. 6. 9. 7. A heap is a binary tree storing keys at its nodes and satisfying the following properties: 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

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HEAPS

HEAPS. Review: what are the requirements of the abstract data type: priority queue?. Quick removal of item with highest priority (highest or lowest key value, depending on application.).

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Heaps

Heaps. CIS 237 – Data Structures. Array Based Trees. Complete full to h-1 filled at level h from left to right Stored as an vector, root in position 0 For the node at position i left child at 2i + 1 right child at 2i + 2 For a node at i the parent is at

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Heaps

Heaps. resumen. Heaps. Alternativamente, Maximum. Binary Heap. Heaps. Raiz = min. 6. 14. 45. Height = log 2 n. 78. 18. 47. 53. 83. 91. 81. 77. 84. 99. 64. binary tree completo Todos los niveles estan llenos salvo el último, llenado por niveles es decir de izq. a der.

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Heaps

Heaps. Background: Priority Queues. Queues are a First-In, First-Out data structure; Priority Queues are similar, except those of highest priority must be removed first. How should we implement this?. Priority Queue Implementation. Arrays

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Heaps

Heaps. Section 6.4, Pg. 309 (Section 9.1). T. How many times is this edge visited?. Motivation: Prim’s Algorithm. Starting node. 3 times. Problem : You keep comparing the same edges over and over. Solution : Use priority queues containing the edges. Priority Queues. Typical example:.

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Heaps

2. 5. 6. 9. 7. Heaps. Outline and Reading. What is a heap ( §7.3 ) Height of a heap ( §7.3.1 ) Insertion ( §7.3.3 ) Removal Heap-sort ( §7.3.4 ) Vector-based implementation Bottom-up construction ( §7.3.5 ).

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Heaps

Heaps. Chapter 6 in CLRS. Motivation. Router: Dijkstra’s algorithm for single source shortest path Prim’s algorithm for minimum spanning trees. Motivation. Want to find the shortest route from New York to San Francisco Model the road-map with a graph. A Graph G=(V,E).

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Heaps

Heaps. Priority Queues. Outline. Binary heaps Binomial queues Leftist heaps. Binary Heaps. 12. 19. 22. 25. 48. 15. 27. 23. 21. 16. 28. Heap order property. For every root node N , the key in the parent of N is smaller than or equal to the key in N. 5.

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