Weighted LCS

The Longest Common Subsequence (LCS) of two strings A,BA,B is a well studied problem having a wide range of applications. When each symbol of the input strings is assigned a positive weight the problem becomes the Heaviest Common Subsequence (HCS) problem. In this paper we consider a different version of weighted LCS on Position Weight Matrices (PWM). The Position Weight Matrix was introduced as a tool to handle a set of sequences that are not identical, yet, have many local similarities. Such a weighted sequence is a ‘statistical image’ of this set where we are given the probability of every symbol's occurrence at every text location. We consider two possible definitions of LCS on PWM. For the first, we solve the LCS problem of z   sequences in time O(znz+1)O(znz+1). For the second, we consider the log-probability version of the problem, prove NPNP-hardness and provide an approximation algorithm.