Abstract

We show that any priority queue data structure that supports insert, delete, and find-min operations in pq(n) time, when 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, n)) amortized time, where α(m, n) is a functional inverse of the Ackermann function. The construction is very simple, essentially just placing a non-meldable priority queue 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 above by min{n, N}. Applying this result to non-meldable priority queue data structures obtained recently by Thorup, and by Han and Thorup, we obtain meldable RAM priority queues with O(log log n) amortized cost per operation, or O ( √ log log n) expected amortized cost per operation, respectively. As a by-product, we obtain improved algorithms for the minimum directed spanning tree problem in graphs with integer edge weights: A deterministic O(m log log n) time algorithm and a randomized O(m √ log log n) time algorithm. These bounds improve, for sparse enough graphs, on the O(m + n log n) running time of an algorithm by Gabow, Galil, Spencer and Tarjan that works for arbitrary edge weights.

Citations