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Unformatted text preview: Chapter 19: Binomial Heaps We will study another heap structure called, the binomial heap. The binomial heap allows for efficient union, which can not be done efficiently in the binary heap. The extra cost paid is the minimum operation, which now requires O (log n ). 1 Comparison of Efficiency Binary Binomial Procedure (worst (worst case) case) Make Heap Θ(1) Θ(1) Insert Θ(lg n ) O (lg n ) Minimum Θ(1) O (lg n ) Extract Min Θ(lg n ) Θ(lg n ) Union Θ( n ) O (lg n ) Decrease Key Θ(lg n ) Θ(lg n ) Delete Θ(lg n ) Θ(lg n ) 2 Definition A binomial tree B k is an ordered tree defined recursively. • B consists of a single node. • For k ≥ 1, B k is a pair of B k 1 trees, where the root of one B k 1 becomes the leftmost child of the other. 3 B 1 B 2 B 3 B 4 B k B k1 B k1 B 4 Properties of Binomial Trees Lemma A For all integers k ≥ 0, the following properties hold: 1. B k has 2 k nodes. 2. B k has height k . 3. For i = 0 , . . . , k , B k has exactly k i nodes at depth i . 4. The root of B k has degree k and all other nodes in B k have degree smaller than k . 5. If k ≥ 1, then the children of the root of B k are B k 1 , B k 2 , ··· , B from left to right. Corollary B The maximum degree of an nnode binomial tree is lg n ....
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This note was uploaded on 04/09/2010 for the course CS 12345 taught by Professor Thomas during the Spring '10 term at École Normale Supérieure.
 Spring '10
 Thomas
 Algorithms

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