Melding priority queues

We show that any priority queue data structure that supports <i>insert</i>, <i>delete</i>, and <i>find-min</i> operations in <i>pq</i>(<i>n</i>) amortized time, where <i>n</i> 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 <i>meld</i> operations with essentially no increase in the amortized cost of the other operations. More specifically, the new data structure supports <i>insert</i>, <i>meld</i> and <i>find-min</i> operations in <i>O</i>(1) amortized time, and <i>delete</i> operations in <i>O</i>(<i>pq</i>(<i>n</i>) + α(<i>n</i>)) amortized time, where α(<i>n</i>) is a functional inverse of the Ackermann function, and where <i>n</i> 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, <i>N</i>], we can replace <i>n</i> in the bound stated previously by min{<i>n</i>, <i>N</i>}.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 <i>O</i>(log log <i>n</i>) amortized time per operation, or <i>O</i>(&sqrt;log log <i>n</i>) 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 <i>O</i>(<i>m</i> log log <i>n</i>)-time algorithm and a randomized <i>O</i>(<i>m</i>&sqrt;log log <i>n</i>)-time algorithm. For sparse enough graphs, these bounds improve on the <i>O</i>(<i>m</i> + <i>n</i> log <i>n</i>) running time of an algorithm by Gabow et al. [1986] that works for arbitrary edge weights.