String matching is a basic theoretical problem in computer science, but has been useful in implementating various text editing tasks. The explosion of multimedia requires an appropriate generalization of string matching to higher dimensions. The first natural generalization is that of seeking the occurrences of a pattern in a text where both pattern arid text are rectangles. The last few years saw a tremendous activity in two dimensional pattern matching algorithms. We naturally had to limit the amount of information that entered this chapter. We chose to concentrate on serial deterministic algorithms for some of the basic issues of two dimensional matching. Throughout this chapter we define our problems in terms of squares rather than rectangles, however, all results presented easily generalize to rectangles. The Exact Two Dimensional Matching Problem is defined as follows: . . . INPUT: Text array T[n x n] and pattern array P[m x m]. OUTPUT: All locations [i,j] in T where there is an occurrence of P, i.e. T[i+k+,j+l] = P[k+1,l+1] 0 ≤ k, l ≤ n-1. . . . A natural way of solving any generalized problem is by reducing it to a special case whose solution is known. It is therefore not surprising that most solutions to the two dimensional exact matching problem use exact string matching algorithms in one way or another. In this section, we present an algorithm for two dimensional matching which relies on reducing a matrix of characters into a one dimensional array. Let P' [1 . . .m] be a pattern which is derived from P by setting P' [i] = P[i,l]P[i,2]…P[i,m], that is, the ith character of P' is the ith row of P. Let Ti[l . . .n — m + 1], for 1 ≤ i ≤ n, be a set of arrays such that Ti[j] = T[i, j] T [ i , j + 1 ] • • • T[i, j + m-1]. Clearly, P occurs at T[i, j] iff P' occurs at Ti[j].
Quasiregular mapping that preserves plane
