Priority Queues and Heaps_Part_3

# Priority Queues and Heaps_Part_3 - Min Heaps A min heap is...

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Last Updated: 06/02/12 8:32 PM CSE 2011 Prof. J. Elder - 11 - Min Heaps A min heap is a binary tree storing keys at its nodes and satisfying the following properties: Heap-order: for every internal node v other than the root key ( v ) key ( parent ( v )) (Almost) complete binary tree: let h be the height of the heap for i = 0, … , h ± 1, there are 2 i nodes of depth i at depth h r 1 the internal nodes are to the left of the external nodes Only the rightmost internal node may have a single child 2 6 5 7 9 The last node of a heap is the rightmost node of depth h

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Last Updated: 06/02/12 8:32 PM CSE 2011 Prof. J. Elder - 12 - Height of a Heap Theorem: A heap storing n keys has height O (log n ) Proof: (we apply the complete binary tree property) Let h be the height of a heap storing n keys Since there are 2 i keys at depth i = 0, … , h ± 1 and at least one key at depth h , we have n 1 + 2 + 4 + + 2 h ± 1 + 1 Thus, n 2 h , i.e., h log n 1 2 2 h ± 1 1 keys 0 1 h ± 1 h depth
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Priority Queues and Heaps_Part_3 - Min Heaps A min heap is...

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