Space-efficient and output-sensitive implementations of greedy algorithms on intervals

Abstract

We consider the space-efficient implementation of greedy algorithms for several fundamental problems on intervals. We assume a random access machine model with read-only access to input stored in Θ(n) words of memory, augmented with a random access memory (workspace) of size Θ(s) bits, where lgn ≤ s ≤ n. Our implementations are based on the efficient realization of an abstract data structure that we call a temporal priority queue that supports extract-min and advancetime operations for a static collection of entities, each of which is active for some pre-specified interval of time. This realization is a generalization of the memory-adjustable navigation pile proposed by Asano et al. in studying time-space tradeoffs for sorting. Using temporal priority queues we are able to implement familiar greedy algorithms for the maximum independent set problem and a variety of dominating set problems on intervals, using O(m(lg (sk/m) + n/s)) time and Θ(s) bits of workspace, where k is the size of output and m = min(sk, n). Choosing s = Θ(n) this achieves O(n lg k) output-sensitive time complexity for the maximum independent set problem on intervals, previously realized using Ω(n) words of workspace.