Programming Priority Queue
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1 SE-286: Data Structures and Programming g Priority Queue Virendra Singh Indian Institute of Science Bangalore Lecture 18 Courtesy: Prof. Sartaj Sahni Oct 4,
2 Priority Queues Two kinds of priority queues: Min priority queue. Max priority queue. Oct Nov 4,
3 Min Priority Queue Collection of elements. Each element has a priority or key. Supports following operations: isempty size add/put an element into the priority yqueue get element with min priority remove element with min priority Oct Nov 4,
4 Max Priority Queue Collection of elements. Each element has a priority or key. Supports following operations: isempty size add/put an element into the priority yqueue get element with max priority remove element with max priority Oct Nov 4,
5 Complexity Of Operations Two good implementations are heaps and leftist trees. isempty, size, and get =>O(1)time put and remove => O(log n) time where n is the size of the priority queue Oct Nov 4,
6 Sorting Applications use element key as priority put elements to be sorted into a priority queue extract elements in priority order if a min priority queue is used, elements are extracted in ascending order of priority (or key) if a max priority i queue is used, elements are extracted in descending order of priority (or key) Oct Nov 4,
7 Sorting Example Sort five elements whose keys are 6, 8, 2, 4, 1 using a max priority queue. Put the five elements into a max priority yqueue. Do five remove max operations placing removed elements into the sorted array from right to left. Oct Nov 4,
8 After Putting Into Max Priority yqueue Max Priority Queue Sorted Array Oct Nov 4,
9 After First Remove Max Operation Max Priority Queue 8 Sorted Array Oct Nov 4,
10 After Second Remove Max Operation Max Priority Queue 6 8 Sorted Array Nov Oct 4,
11 After Third Remove Max Operation 1 2 Max Priority Queue Sorted Array Oct Nov 4,
12 After Fourth Remove Max Operation 1 Max Priority Queue Sorted Array Nov Oct 4,
13 After Fifth Remove Max Operation Max Priority Queue Sorted Array Nov Oct 4,
14 Complexity Of Sorting Sort n elements. n put operations => O(n log n) time. n remove max operations => O(n log n) time. total time is O(n log n). compare with O(n 2 ) for sort methods Nov Oct 4,
15 Heap Sort Uses a max priority queue that is implemented as a heap. Iiil Initial put operations are replaced dby a heap initialization step that takes O(n) time. Nov Oct 4,
16 Machine Scheduling m identical machines (drill press, cutter, sander, etc.) n jobs/tasks to be performed assign jobs to machines so that t the time at which h the last job completes is minimum Nov Oct 4,
17 Machine Scheduling Example 3 machines and 7 jobs job times are [6, 2, 3, 5, 10, 7, 14] possible schedule A 6 13 B C Nov Oct 4, time >
18 Machine Scheduling Example A B C 3 time > 13 Finish time = 21 Objective: Find schedules with minimum i finish i time. Nov Oct 4,
19 LPT Schedules Longest Processing Time first. Jobs are scheduled in the order 14, 10, 7, 6, 5, 3, 2 Each job is scheduled on the machine on which it finishes earliest. Nov Oct 4,
20 LPT Schedule [14, 10, 7, 6, 5, 3, 2] A B C Finish time is 16! Nov Oct 4,
21 LPT Schedule LPT rule does not guarantee minimum i finish i time schedules. Usually LPT finish time is much closer to minimum finish time. Minimum finish time scheduling is NP-hard. Nov Oct 4,
22 Complexity Of LPT Scheduling Sort jobs into decreasing order of task time. O(n log n) time (n is number of jobs) Schedule jobs in this order. assign job to machine that becomes available first must find minimum of m (m is number of machines) finish times takes O(m) time using simple strategy so need O(mn) time to schedule all n jobs. Nov Oct 4,
23 Using A Min Priority Queue e Min priority queue has the finish times of the m machines. Initial finish times are all 0. To schedule a job remove machine with minimum finish time from the priority queue. Update the finish time of the selected machine and put the machine back into the priority queue. Nov Oct 4,
24 Using A Min Priority Queue e m put operations to initialize priority queue 1 remove min and 1 put tto schedule each hjob each put and remove min operation takes O(log m) time time to schedule is O(n log m) overall time is O(nlogn+nlogm)=O(nlog(mn)) n n (mn)) Nov Oct 4,
25 Huffman Codes Useful in lossless compression. Nov Oct 4,
26 Min Tree Definition Each tree node has a value. Value in any node is the minimum value in the subtree for which that node is the root. Equivalently, no descendent has a smaller value. Nov Oct 4,
27 Min Tree Example Root has minimum i element. Nov Oct 4,
28 Max Tree Example Root has maximum element. Nov Oct 4,
29 Min Heap Definition complete binary tree min tree Nov Oct 4,
30 Min Heap With 9 Nodes Complete binary tree with 9 nodes. Nov Oct 4,
31 Min Heap With 9 Nodes Complete binary tree with 9 nodes that is also a min tree. Nov Oct 4,
32 Max Heap With 9 Nodes Complete binary tree with 9 nodes that is also a max tree. Nov Oct 4,
33 Heap Height Since a heap is a complete binary tree, the height of an n node heap is log 2 (n+1). Nov Oct 4,
34 A Heap Is Efficiently Represented As An Array Nov Oct 4,
35 Moving Up And Down A Heap Nov Oct 4,
36 Putting An Element Into A Max Heap Complete binary tree with 10 nodes. Nov Oct 4,
37 Putting An Element Into A Max Heap New element is 5. Nov Oct 4,
38 Putting An Element Into A Max Heap New element is 20. Nov Oct 4,
39 Putting An Element Into A Max Heap New element is 20. Nov Oct 4,
40 Putting An Element Into A Max Heap New element is 20. Nov Oct 4,
41 Putting An Element Into A Max Heap New element is 20. Nov Oct 4,
42 Putting An Element Into A Max Heap Complete binary tree with 11 nodes. Nov Oct 4,
43 Putting An Element Into A Max Heap New element is 15. Nov Oct 4,
44 Putting An Element Into A Max Heap New element is 15. Nov Oct 4,
45 Putting An Element Into A Max Heap New element is 15. Nov Oct 4,
46 Complexity Of Put Complexity is O(log n), where n is heap size. Nov Oct 4,
47 Removing The Max Element Max element is in the root. Nov Oct 4,
48 Removing The Max Element After max element is removed. Nov Oct 4,
49 Removing The Max Element Heap with 10 nodes. Reinsert 8 into the heap. Nov Oct 4,
50 Removing The Max Element Reinsert 8 into the heap. Nov Oct 4,
51 Removing The Max Element Reinsert 8 into the heap. Nov Oct 4,
52 Removing The Max Element Reinsert 8 into the heap. Nov Oct 4,
53 Removing The Max Element Max element is 15. Nov Oct 4,
54 Removing The Max Element After max element is removed. Nov Oct 4,
55 Removing The Max Element Heap with 9 nodes. Nov Oct 4,
56 Removing The Max Element Reinsert 7. Nov Oct 4,
57 Removing The Max Element Reinsert 7. Nov Oct 4,
58 Removing The Max Element Reinsert 7. Nov Oct 4,
59 Complexity Of Remove Max Element Complexity is O(log n). Nov Oct 4,
60 Initializing A Max Heap input array = [-, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] Nov Oct 4,
61 Initializing A Max Heap Start at rightmost array position that has a child. Index is n/2. Nov Oct 4,
62 Initializing A Max Heap Move to next lower array position. Nov Oct 4,
63 Initializing A Max Heap Nov Oct 4,
64 Initializing A Max Heap Nov Oct 4,
65 Initializing A Max Heap Nov Oct 4,
66 Initializing A Max Heap Nov Oct 4,
67 Initializing A Max Heap Nov Oct 4,
68 Initializing A Max Heap Find a home for 2. Nov Oct 4,
69 Initializing A Max Heap Find a home for 2. Nov Oct 4,
70 Initializing A Max Heap Done, move to next lower array position. Nov Oct 4,
71 Initializing A Max Heap Find home for 1. Nov Oct 4,
72 Initializing A Max Heap Find home for 1. Nov Oct 4,
73 Initializing A Max Heap Find home for 1. Nov Oct 4,
74 Initializing A Max Heap Find home for 1. Nov Oct 4,
75 Initializing A Max Heap Done. Nov Oct 4,
76 Time Complexity Height of heap = h. Number of subtrees with root at level j is <= 2 j-1. Time Oct 4, for 2010 each subtree is O(h-j+1). Nov 4,
77 Complexity Time for level j subtrees is <= 2 j-1 (h-j+1) = t(j). Total time is t(1) + t(2) + + t(h-1) = O(n). Nov Oct 4,
78 Leftist Trees Linked binary tree. Can do everything a heap can do and in the same asymptotic complexity. Can meld two leftist tree priority queues in O(log n) time. Nov Oct 4,
79 Extended Binary Trees Start with any binary tree and add an external node wherever there is an empty subtree. Result is an extended binary tree. Nov Oct 4,
80 ABinaryTree Nov Oct 4,
81 An Extended Binary Tree number of external nodes is n+1 Nov Oct 4,
82 The Function s() For any node x in an extended binary tree, let s(x) be the length of a shortest path from x to an external node in the subtree rooted at x. Nov Oct 4,
83 s() Values Example Nov Oct 4,
84 s() Values Example Nov Oct 4,
85 Properties Of s() If x is an external node, then s(x) = 0. Otherwise, s(x) = min {s(leftchild(x)), s(rightchild(x))} + 1 Nov Oct 4,
86 HihtBi Height Biased dlftitt Leftist Trees A binary tree is a (height biased) leftist tree iff for every internal node x, s(leftchild(x)) >= s(rightchild(x)) Nov Oct 4,
87 A Leftist Tree Nov Oct 4,
88 LftitT Leftist Trees--Property 1 In a leftist tree, the rightmost path is a shortest root to external node path and the length of this path is s(root). Nov Oct 4,
89 A Leftist Tree Length of rightmost path is 2. Nov Oct 4,
90 LftitT Leftist Trees Property 2 The number of internal nodes is at least 2 s(root) - 1 Because levels 1 through s(root) have no external nodes. So, s(root) <= log(n+1) Nov Oct 4,
91 A Leftist Tree Levels 1 and 2 have no external nodes. Nov Oct 4,
92 LftitT Leftist Trees Property 3 Length of rightmost path is O(log n), where n is the number of nodes in a leftist tree. Follows from Properties 1 and 2. Nov Oct 4,
93 Leftist Trees As Priority Queues Min leftist tree leftist tree that is a min tree. Used as a min priority queue. Max leftist tree leftist tree that is a max tree. Used as a max priority queue. Nov Oct 4,
94 A Min Leftist Tree Nov Oct 4,
95 Some Min Leftist Tree Operations put() remove() meld() initialize() put() and remove() use meld(). Nov Oct 4,
96 Put Operation put(7) Nov Oct 4,
97 Put Operation put(7) Create a single node min leftist tree. 7 Nov Oct 4,
98 Put Operation put(7) Create a single node min leftist tree. 7 Meld the two min leftist trees. Nov Oct 4,
99 Remove Min Nov Oct 4,
100 Remove Min Remove the root. Nov Oct 4,
101 Remove Min Remove the root. Meld the two subtrees. Nov Oct 4,
102 Meld Two Min Leftist Trees Traverse only the rightmost paths so as to get logarithmic performance. Nov Oct 4,
103 Meld Two Min Leftist Trees Meld right subtree of tree with smaller root and all of other tree. Nov Oct 4,
104 Meld Two Min Leftist Trees Meld right subtree of tree with smaller root and all of other tree. Nov Oct 4,
105 Meld Two Min Leftist Trees Meld right subtree of tree with smaller root and all of other tree. Nov Oct 4,
106 Meld Two Min Leftist Trees 8 6 Meld right subtree of tree with smaller root and all of other tree. Right subtree of 6 is empty. So, result of melding right subtree of tree with smaller root and other tree is the Oct 4, 2010 other tree. Nov 4,
107 Meld Two Min Leftist Trees 8 6 Make melded subtree right subtree of smaller root. 6 8 Swap left and right subtree if s(left) < s(right). 6 Nov Oct 4,
108 Meld Two Min Leftist Trees Make melded subtree right subtree of smaller root. Swap left and right subtree if s(left) < s(right). Nov Oct 4,
109 Meld Two Min Leftist Trees Make melded subtree right subtree of smaller root. Swap Oct 4, left 2010 and right subtree if s(left) < s(right). Nov 4,
110 Meld Two Min Leftist Trees Nov Oct 4,
111 Initializing In O(n) Time create n single node min leftist trees and place them in a FIFO queue repeatedly remove two min leftist trees from the FIFO queue, meld them, and put the resulting min leftist tree into the FIFO queue the process terminates when only 1 min leftist tree remains in the FIFO queue analysis is the same as for heap initialization Oct Nov 4,
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