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Priority Queue (Heap). A kind of queue Dequeue gets element with the highest priority Priority is based on a comparable value (key) of each object (smaller value higher priority, or higher value higher priority) Example Applications: printer -> print (dequeue) the shortest document first
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Priority Queue (Heap) • A kind of queue • Dequeue gets element with the highest priority • Priority is based on a comparable value (key) of each object (smaller value higher priority, or higher value higher priority) • Example Applications: • printer -> print (dequeue) the shortest document first • operating system -> run (dequeue) the shortest job first • normal queue -> dequeue the first enqueued element first Source: Muangsin / Weiss
Priority Queue (Heap) Operations Priority Queue • insert (enqueue) • deleteMin (dequeue) • smaller value higher priority • Find / save the minimum element, delete it from structure and return it deleteMin insert Source: Muangsin / Weiss
Implementation using Linked List • Unsorted linked list • insert takes O(1) time • deleteMin takes O(N) time • Sorted linked list • insert takes O(N) time • deleteMin takes O(1) time Source: Muangsin / Weiss
Implementation using Binary Search Tree • insert takes O(log N) time • deleteMin takes O(log N) time • support other operations that are not required by priority queue (for example, findMax) • deleteMin operations make the tree unbalanced Source: Muangsin / Weiss
C G F B E D A J H I Terminology: full tree • completely filled (bottom level is filled from left to right • size between 2h(bottom level has only one node) and 2h+1-1 Source: Muangsin / Weiss
16 68 19 21 31 24 13 32 65 26 Heap: Order Property (for Minimum Heap) • Any node is smaller than (or equal to) all of its children (any subtree is a heap) • Smallest element is at the root (findMin take O(1) time) Source: Muangsin / Weiss
24 19 68 16 68 19 21 31 16 21 24 32 26 65 13 13 65 26 32 31 Insert • Create a hole in the next available location • Move the hole up (swap with its parent) until data can be placed in the hole without violating the heap order property (called percolate up) Source: Muangsin / Weiss
24 16 68 19 21 31 21 24 19 68 16 26 13 13 32 26 65 32 65 31 Insert insert14 Percolate Up -> move the place to put 14 up (move its parent down) until its parent <= 14 Source: Muangsin / Weiss
14 68 19 21 31 24 16 21 19 68 16 24 31 65 32 26 26 65 32 13 13 Insert Source: Muangsin / Weiss
19 14 19 16 31 21 14 68 16 68 19 21 19 65 26 26 13 32 32 65 31 deleteMin • Create a hole at the root • Move the hole down (swap with the smaller one of its children) until the last element of the heap can be placed in the hole without violating the heap order property (called percolate down) Source: Muangsin / Weiss
14 68 14 19 21 19 16 21 19 68 16 19 31 26 13 32 65 32 65 26 31 deleteMin Percolate Down -> move the place to put 31 down (move its smaller child up) until its children >= 31 Source: Muangsin / Weiss
14 16 19 21 19 19 68 16 68 19 21 14 32 26 65 26 32 65 31 31 deleteMin Source: Muangsin / Weiss
14 68 16 14 19 19 26 21 21 19 68 16 19 32 31 32 26 65 65 31 deleteMin Source: Muangsin / Weiss
Running Time • insert • worst case: takes O(log N) time, moves an element from the bottom to the top • on average: takes a constant time (2.607 comparisons), moves an element up 1.607 levels • deleteMin • worst case: takes O(log N) time • on average: takes O(log N) time (element that is placed at the root is large, so it is percolated almost to the bottom) Source: Muangsin / Weiss
C G F B E D I H A J A B C D E F G H I J 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Array Implementation of Binary Heap left child is in position 2i right child is in position (2i+1) parent is in position i/2 Source: Muangsin / Weiss
