Nearly Optimal Parallel Algorithms for Longest Increasing Subsequence

Abstract

The paper presents parallel algorithms for multiplying implicit simple unit-Monge matrices (Krusche and Tiskin, PPAM 2009) of size n x n in the EREW PRAM model. We show implicit simple unit-Monge matrices multiplication of size n x n can be achieved by a deterministic EREW PRAM algorithm with O(n log n log log n) total work and O(log3 n) span. This implies that there is a deterministic EREW PRAM algorithm solving the longest increasing subsequence (LIS) problem in O(n log2 n log log n) work and O(log 4 n) span. Furthermore, with randomization and bitwise operations, implicitly multiplying two simple unit-Monge matrices can be improved to O(n log n) work and O(log3n) span, which leads to a randomized EREW PRAM algorithm obtaining LIS in O(nlog2n) work and O(log4n) span with high probability. In the regime where the LIS has length k = Ψ(log3n), our results improve the span from Õ(n2/3) (Krusche and Tiskin, SPAA 2010) and O(klog n) (Gu, Men, Shen, Sun, and Wan, SPAA 2023) to O(log4 n) while the total work remains near optimal Õ (n).