Episode matching

Abstract

Given two words, text T of length n and episode P of length m, the episode matching problem is to find all minimal length substrings of text T that contain episode P as a subsequence. The respective optimization problem is to find the smallest number w, s.t. text T has a subword of length w which contains episode P.

In this paper, we introduce a few efficient off-line as well as on-line algorithms for the entire problem, where by on-line algorithms we mean algorithms which search from left to right consecutive text symbols only once. We present two alphabet independent algorithms which work in time O(nm). The off-line algorithm operates in O(1) additional space while the on-line algorithm pays for its property with O(m) additional space. Two other on-line algorithms have subquadratic time complexity. One of them works in time O(nm/log m) and O(m) additional space. The other one gives a time/space trade-off, i.e., it works in time O(n+s+nm log log s/log(s/m)) when additional space is limited to O(s). Finally, we present two approximation algorithms for the optimization problem. The off-line algorithm is alphabet independent, it has superlinear time complexity O(n/∈+nloglog(n/m)) and it uses only constant space. The on-line algorithm works in time O(n/∈+n) and uses space O(m). Both approximation algorithms achieve 1+∈ approximation ratio, for any ∈>0.