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Priority Queue (Heap)

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)

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 14 16 19 21 19 19 68 16 68 19 21 14 32 26 65 26 32 65 31 31 deleteMin Source: Muangsin / Weiss

  13. 14 68 16 14 19 19 26 21 21 19 68 16 19 32 31 32 26 65 65 31 deleteMin Source: Muangsin / Weiss

  14. 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

  15. 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

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