Self-normalised Distance with Don’t Cares

Abstract

We present O(nlogm) algorithms for a new class of problems termed self-normalised distance with don’t cares. The input is a pattern p of length m and text t of length n > m. The elements of these strings are either integers or wild card symbols. In the shift version, the problem is to compute $\min_{\alpha}\sum_{j=0}^{m-1}(\alpha + p_j - t_{i+j})^2$ for all i, where wild cards do not contribute to the sum. In the shift-scale version, the objective is to compute $\min_{\alpha,\beta}\sum_{j=0}^{m-1}(\alpha+ \beta p_j - t_{i+j})^2$ for all i, similarly. We show that the algorithms have the additional benefit of providing simple O(nlogm) solutions for the problems of exact matching with don’t cares, exact shift matching with don’t cares and exact shift-scale matching with don’t cares.