Priority Queues (Chapter 6):

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Presentation transcript:

Priority Queues (Chapter 6): Abstract model of Priority Queues Priority Queue Get the one with the highest priority insert the minimum one (or equivalently, maximum) a black box 5/5/2019 IT 279

Let’s cleaning up some confusion. Is Binary Search Tree a priority queue? Binary Search Tree Get the one with the highest priority insert No! It is not, but nothing wrong to use it in the black box. the minimum one (or equivalently, maximum) 5/5/2019 IT 279

Items with higher priority will be treated first. Stacks (FILO), queues (FIFO) are Priority Queues Arrays are not, since we can randomly access the items in an array The priority of the item may be fixed or changed over the time. 5/5/2019 IT 279

Using a Binary Search Tree is overkilled. Why? 17 Get the one with the highest priority insert 14 19 12 15 18 11 13 the minimum one (or equivalently, maximum) Items are totally ordered, but we don’t really this information (nothing comes free) 5/5/2019 IT 279

Our goals: 1. Can get the minimum one efficiently 2. and low maintenance 11 9 16 12 11 12 Get the minimum one 9 13 insert 13 12 14 17 15 19 18 13 14 17 15 19 18 16 14 17 15 19 18 Binary Heap 5/5/2019 IT 279

Binary Heap Using (abstractly) binary tree structure For every node in the tree, its key is smaller than (or equal to) the keys of its two children. Therefore, for every node in the tree, its data is the smallest one among data stored in the sub-trees (the two children). Different from BST, not every key in the left-sub-tree is less than every key in the right-sub-tree. Loosing this requirement, we gain some benefits. 5/5/2019 IT 279

A Binary Heap but we can organize it in a better way without too much extra cost. 1 4 6 5 9 23 10 25 8 14 11 25 12 29 28 12 17 20 15 22 29 30 19 36 13 29 23 30 37 5/5/2019 IT 279

A Binary Heap in complete binary tree 10 12 13 15 14 16 17 23 22 21 18 19 24 25 26 27 32 29 30 25 26 37 pack 5/5/2019 IT 279

A Binary Heap implemented in an array 10 1 12 13 2 3 15 14 16 17 4 5 6 7 23 22 21 28 39 24 25 42 10 11 15 8 9 12 13 14 27 32 29 40 35 36 37 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1st child (left-child) of i is 2*i 2nd child (right-child) of i is 2*i+1 The parent of i is i / 2 double the size of the array if the space ran out 5/5/2019 IT 279

Heap operation: insert an item 11 10 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 37 11 5/5/2019 IT 279

Heap operation: insert an item 11 10 12 13 15 14 16 17 23 22 21 39 24 25 42 11 27 32 29 40 35 36 37 28 5/5/2019 IT 279

Heap operation: insert an item 11 10 12 13 15 11 16 17 23 22 21 39 24 25 42 14 27 32 29 40 35 36 37 28 5/5/2019 IT 279

Heap operation: insert an item 11 10 11 13 15 12 16 17 23 22 21 39 24 25 42 14 27 32 29 40 35 36 37 28 5/5/2019 IT 279

Heap operation: delete the minimum 10 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 37 5/5/2019 IT 279

Heap operation: delete the minimum 10 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 37 5/5/2019 IT 279

Heap operation: delete the minimum 10 37 12 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 5/5/2019 IT 279

Heap operation: delete the minimum 10 12 37 13 15 14 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 5/5/2019 IT 279

Heap operation: delete the minimum 10 12 14 13 15 37 16 17 23 22 21 28 39 24 25 42 27 32 29 40 35 36 5/5/2019 IT 279

Heap operation: delete the minimum 10 12 14 13 15 16 21 17 23 22 37 28 39 24 25 42 27 32 29 40 35 36 5/5/2019 IT 279

Heap operation: delete the minimum 10 12 14 13 15 16 21 17 23 22 35 28 39 24 25 42 27 32 29 40 37 36 5/5/2019 IT 279

Time complexity for insertion best case: O(1) worst case: O(log n) 14 log n 15 21 23 22 37 25 27 32 29 40 35 36 30 27 average case: 5/5/2019 IT 279

Time complexity of Building a heap Insert them one by one into the empty heap What is the cost? best case: O(n) average case: O(n) worst case: O(n log n) Can we improve the worst case? 34 15 30 2 23 9 25 6 32 5 40 11 25 21 7 5/5/2019 IT 279

Percolating a non-heap Worst case: O(n) 34 2 15 2 5 30 7 7 2 15 6 5 23 9 25 7 30 21 6 23 9 21 15 6 32 5 23 40 11 31 30 21 7 25 15 32 34 40 11 31 30 25 Binary Heap 5/5/2019 IT 279

Percolating a non-heap Worst case: O(S) S = Sum of heights of all nodes h O(S) = O(n) 5/5/2019 IT 279