Parallel Longest Increasing Subsequences in Scalable Time and Memory

Abstract

The longest increasing subsequence (LIS) problem is a classical problem in theoretical computer science and mathematics. Most existing parallel algorithms for this problem have very restrictive slackness conditions which prevent scalability to large numbers of processors. Other algorithms are scalable, but not work-optimal w.r.t. the fastest sequential algorithm for the LIS problem, which runs in time O(n logn) for n numbers in the comparison-based model. In this paper, we propose a new parallel algorithm for the LIS problem. Our algorithm solves the more general problem of semi-local comparison of permutation strings of length n in time O(n 1.5 / p) on p processors, has scalable communication cost of \(O(n/\sqrt{p})\) and is synchronisation-efficient. Furthermore, we achieve scalable memory cost, requiring \(O(n/\sqrt{p})\) of storage on each processor. When applied to LIS computation, this algorithm is superior to previous approaches since computation, communication, and memory costs are all scalable.