Priority queues: Small, monotone and trans-dichotomous

Abstract

We consider two data-structuring problems which involve performing priority queue (pq) operations on a set of integers in the range 0..2^w−1 on a unit-cost RAM with word size ω bits.

A monotone min-PQ has the property that the minimum value stored in the pq is a non-decreasing function of time. We give a monotone min-pq that, starting with an empty set, processes a sequence of n insert and delete-mins and m decrease-keys in O(m+n√log n log log n) time. As a consequence, the single-source shortest paths problem on graphs with n nodes and m edges and integeredge costs in the range 0..2^w − 1 can be solved in O(m+n√log n log log n) time, and n integers each in the range 0..2^w−1 can be sorted in O(n√log n log log n) time. All the above results require linear space and assume that any unit-time RAM instructions used belong to the the class ac⁰.

A small (generalized) PQ supports insert, delete and search operations (the latter returning the predecessor of its argument among the keys in the pq), but allows only w ^O (1) keys to be present in the pq at any time. We give a small pq which supports all operations in constant expected time. As a consequence, we get that insert, delete and search operations on a set of n keys can be performed in O(1+log n/log ω) expected time. Derandomizing this small pq gives a linear-space static deterministic small pq.