We consider two data-structuring problems which involve performing priority queue (pq) operations on a set of integers in the range 0..2w−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..2w − 1 can be solved in O(m+n√log n log log n) time, and n integers each in the range 0..2w−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 wO(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.