Streaming Pattern Matching with d Wildcards

Abstract

In the pattern matching with d wildcards problem we are given a text T of length n and a pattern P of length m that contains d wildcard characters, each denoted by a special symbol '?'. A wildcard character matches any other character. The goal is to establish for each m-length substring of T whether it matches P. In the streaming model variant of the pattern matching with d wildcards problem the text T arrives one character at a time and the goal is to report, before the next character arrives, if the last m characters match P while using only o(m) words of space. In this paper we introduce two new algorithms for the d wildcard pattern matching problem in the streaming model. The first is a randomized Monte Carlo algorithm that is parameterized by a constant 0<=delta<=1. This algorithm uses ~O(d^{1-delta}) amortized time per character and ~O(d^{1+delta}) words of space. The second algorithm, which is used as a black box in the first algorithm, is a randomized Monte Carlo algorithm which uses O(d+log m) worst-case time per character and O(d log m) words of space.