Finding missing patterns

Consider the following problem: Find the shortest pattern that does not occur in a given text. To make the problem non-trivial, the pattern is required to consist only of characters that occur in the text. This problem can be solved easily in linear time using the suffix tree of the text. In this paper, we study an extension of this problem, namely the missing patterns problem: Find the shortest pair of patterns that do not occur close to each other in a given text, i.e., the distance between their occurrences is always greater than a given threshold a. We show that the missing patterns problem can be solved in O(min(alphan log n, n(2))) time, where n is the size of the text. For the special case where both pairs axe required to have the same length, we give an algorithm with time complexity O(alphan log log n). The problem is motivated by optimization of multiplexed nested-PCR.