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Priority Queue (Heap)

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- 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

- 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

- 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

- 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

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- 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

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- 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

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- 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

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insert14

Percolate Up -> move the place to put 14 up

(move its parent down) until its parent <= 14

Source: Muangsin / Weiss

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Source: Muangsin / Weiss

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- 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

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Percolate Down -> move the place to put 31 down (move its smaller child up) until its children >= 31

Source: Muangsin / Weiss

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Source: Muangsin / Weiss

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Source: Muangsin / Weiss

- 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

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left child is in position 2i

right child is in position (2i+1)

parent is in position i/2

Source: Muangsin / Weiss