Streaming Pattern Matching with d Wildcards

In the pattern matching with d wildcards problem one is 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 ≤ δ≤ 1. This algorithm uses O~ (d^1-δ) amortized time per character and O~ (d^1+δ) 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(dlog m) words of space.