Analysis of two-dimensional approximate pattern matching algorithms

Abstract

A k-approximate occurrence of a pattern in a text is an occurrence which has Hamming distance at most k with the pattern. The problem of two-dimensional approximate pattern matching is defined as follows: Given a pattern P of size m ², a text T of size n ², and an integer k, find all k-approximate occurrences of P in T.

Kärkkäinen and Ukkonen [7] proposed two algorithms for two-dimensional approximate pattern matching and showed that their expected time for random input is O(kn ²(log m)/m ²) for k ≤ m ²/4[log_σ m ²], where σ is the size of the alphabet. However, they got the analysis with an independence assumption. In this paper we present a new analysis of the two algorithms which shows that the expected time is the same O(kn ²(log m)/m ²) for $k \leqslant \left\lfloor {\frac{m}{{\left\lceil {\log _\sigma m^2 } \right\rceil }}} \right\rfloor \cdot \frac{m}{2} - 1$ without the independence assumption. Hence our analysis is stronger than that of [7] in that (i) it removes the independence assumption and (ii) the range of k is larger. It is also shown that the two algorithms in [7] have an undesirable factor n in their space complexities. We present modifications of these algorithms which use space O(m ²) in the worst case and O(k) on average while maintaining the same expected time.

Supported by KOSEF grant 951-0906-069-2.