Meldable heaps and boolean union-find

  • Authors:

  • Haim Kaplan;Nira Shafrir;Robert E. Tarjan

  • Affiliations:

  • Tel Aviv University, Tel Aviv, Israel;Tel Aviv University, Tel Aviv, Israel;Princeton University, Princeton, NJ and Compaq Computer Corp., Palo Alto, CA

  • Venue:
  • STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2002

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Abstract

In the classical meldable heap data type we maintain an item-disjoint collection of heaps under the operations find-min, insert, delete, decrease-key, and meld. In the usual definition decrease-key and delete get the item and the heap containing it as parameters. We consider the modified problem where decrease-key and delete get only the item but not the heap containing it. We show that for this problem one of the operations find-min, decrease-key, or meld must take non-constant time. This is in contrast with the original data type in which data structures supporting all these three operations in constant time are known (both in an amortized and a worst-case setting).To establish our results for meldable heaps we consider a weaker version of the union-find problem that is of independent interest, which we call Boolean union-find. In the Boolean union-find problem the find operation is a binary predicate that gets an item x and a set A and answers positively if and only if &khgr; &egr; A. We prove that the lower bounds which hold for union-find in the cell probe model hold for Boolean union-find as well.We also suggest new heap data structures implementing the modified meldable heap data type that are based on redundant binary counters. Our data structures have good worst-case bounds. The best of our data structures matches the worst-case lower bounds which we establish for the problem. The simplest of our data structures is an interesting generalization of binomial queues.