Matching for run-length encoded strings

Measuring the similarity between two strings, through such standard measures as Hamming distance, edit distance, and longest common subsequence, is one of the fundamental problems in pattern matching. We consider the problem of finding the longest common subsequence of two strings. A well-known dynamic programming algorithm computes the longest common subsequence of strings X and Y in O(|X|·|Y|) time. We develop significantly faster algorithms for a special class of strings which emerge frequently in pattern matching problems. A string S is run-length encoded if it is described as an ordered sequence of pairs (σ,i), each consisting of an alphabet symbol σ and an integer i. Each pair corresponds to a run in S consisting of i consecutive occurrences of σ. For example, the string aaaabbbbcccabbbbcc can be encoded as a⁴b⁴c³ a¹b⁴c². Such a run-length encoded string can be significantly shorter than the expanded string representation. Indeed, runlength coding serves as a popular image compression technique, since many classes of images, such as binary images in facsimile transmission, typically contain large patches of identically-valued pixels