# 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 ac^{0}.

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.

## Keywords

Linear Space Hash Function Internal Node Priority Queue External Node## Preview

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