Heaviest increasing/common subsequence problems

Abstract

In this paper, we define the heaviest increasing subsequence (HIS) and heaviest common subsequence (HCS) problems as natural generalizations of the well-studied longest increasing subsequence (LIS) and longest common subsequence (LCS) problems. We show how the famous Robinson-Schensted correspondence between permutations and pairs of Young tableaux can be extended to compute heaviest increasing subsequences. Then, we point out a simple weight-preserving correspondence between the HIS and HCS problems. ? From this duality between the two problems, the Hunt-Szymanski LCS algorithm can be seen as a special case of the Robinson-Schensted algorithm. Our HIS algorithm immediately gives rise to a Hunt-Szymanski type of algorithm for HCS with the same time complexity. When weights are position-independent, we can exploit the structure inherent in the HIS-HCS correspondence to further refine the algorithm. This gives rise to a specialized HCS algorithm of the same type as the Apostolico-Guerra LCS algorithm.