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# 19 - 19 Binomial Heaps Mergeable Heap a data structure...

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---------------------------- 19 ---------------------------- Binomial Heaps Mergeable Heap: a data structure supporting the following 5 operations. Make-Heap(): return an empty heap Insert( H , x): add x into H Minimum( H ): return the minimum Extract-Min( H ): return and remove Union( H 1 , H 2 ): merge two heaps In addition, the data structures in Chapters 19 and 20 also support the following 2 operations. Decrease-Key( H , x , k ): assign x (within H ) a smaller key value k Delete( H , x ): 19-2 * Binary heap is good if we don't need Union . (A Union operation takes O ( n ) time.) * The running times for Fibonacci heap are amorized time bounds, not worst-case per operation time bounds. * Searchs in the above heaps are inefficient. Thus, Decrease-Key and Delete perform under the assumption that a pointer to x is given. 19.1 Binomial trees and binomial heaps Binomial tree : A binomial tree B k is an ordered tree defined recursively as follows. (Let root of one B k -1 be the leftmost child of root of the other.)

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19-3 Lemma 19.1 (Properties of B k ) 1.There are 2 k nodes.
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