Melding priority queues

Abstract

We show that any priority queue data structure that supports insert , delete , and find-min operations in pq ( n ) amortized time, where n is an upper bound on the number of elements in the priority queue, can be converted into a priority queue data structure that also supports fast meld operations with essentially no increase in the amortized cost of the other operations. More specifically, the new data structure supports insert , meld and find-min operations in O (1) amortized time, and delete operations in O ( pq ( n ) + α( n )) amortized time, where α( n ) is a functional inverse of the Ackermann function, and where n this time is the total number of operations performed on all the priority queues. The construction is very simple. The meldable priority queues are obtained by placing a nonmeldable priority queues at each node of a union-find data structure. We also show that when all keys are integers in the range [1, N ], we can replace n in the bound stated previously by min{ n , N }.Applying this result to the nonmeldable priority queue data structures obtained recently by Thorup [2002b] and by Han and Thorup [2002] we obtain meldable RAM priority queues with O (log log n ) amortized time per operation, or O (√log log n ) expected amortized time per operation, respectively. As a by-product, we obtain improved algorithms for the minimum directed spanning tree problem on graphs with integer edge weights, namely, a deterministic O ( m log log n )-time algorithm and a randomized O ( m √log log n )-time algorithm. For sparse enough graphs, these bounds improve on the O ( m + n log n ) running time of an algorithm by Gabow et al. [1986] that works for arbitrary edge weights.