A comparative study of efficient algorithms for partitioning a sequence into monotone subsequences

Abstract

Tradeoffs between time complexities and solution optimalizes are important when selecting algorithms for an NP-hard problem in different applications. Also, the distinction between theoretical upper bound and actual solution optimality for realistic instances of an NP-hard problem is a factor in selecting algorithms in practice. We consider the problem of partitioning a sequence of n distinct numbers into minimum number of monotone (increasing or decreasing) subsequences. This problem is NP-hard and the number of monotone subsequences can reach ⌊√2n + 1/4 - 1/2⌋ in the worst case. We introduce a new algorithm, the modified version of the Yehuda-Fogel algorithm, that computes a solution of no more than ⌊√2n + 1/4 - 1/2⌋ monotone subsequences in O(n1.5) time. Then we perform a comparative experimental study on three algorithms, a known approximation algorithm of approximation ratio 1.71 and time complexity O(n3), a known greedy algorithm of time complexity O(n1.5 log n), and our new modified Yehuda-Fogel algorithm. Our results show that the solutions computed by the greedy algorithm and the modified YehudarFogel algorithm are close to that computed by the approximation algorithm even though the theoretical worst-case error bounds of these two algorithms are not proved to be within a constant times of the optimal solution. Our study indicates that for practical use the greedy algorithm and the modified Yehuda-Fogel algorithm can be good choices if the running time is a major concern. © Springer-Verlag Berlin Heidelberg 2007.