On the linear number of matching substrings

We study the number of matching substrings in the pattern matching problem. In general, there can be a quadratic number of matching substrings in the size of a given text. The linearizing restriction enables to find at most a linear number of matching substrings. We first explore two well-known linearizing restriction rules, the longest-match rule and the shortest-match substring search rule, and show that both rules give the same result when a pattern is an infix-free set even though they have different semantics. Then, we introduce a new linearizing restriction, the leftmost non-overlapping match rule that is suitable for find-and-replace operations in text searching, and propose an efficient algorithm for the new rule when a pattern is described by a regular expression. We also examine the problem of obtaining the maximal number of non-overlapping matching substrings.