Multi-dimensional pattern matching with dimensional wildcards

Abstract

We introduce a new multi-dimensional pattern matching problem, which is a natural generalization of the on-line search in string matching. We are given a text matrix A[1: n ₁, ..., 1:n _d ] of size N= n ₁ ×n ₂×...×n _d , which we may preprocess. Then, we are given, online, an r-dimensional pattern matrix B[1:m ₁,...,1:m _r ] of size M= m ₁×m ₂×...×m _r , with 1≤r≤d. We would like to know whether B ^*=B ^*[*, 1:m ₁,*, ...,1: m _r, *] occurs in A, where * is a dimensional wildcard such that B ^* is any d-dimensional matrix having size 1 × ... × m ¹×...1×m ^r×...1 and containing the same elements as B. Notice that there might be (d/r)≤2^d occurrences of B ^* for each position of A. We give CRCW-PRAM algorithms for preprocessing A in O(d log N) time with N ²/n _max processors, where n _max =max|n ₁,...,n _d }. The on-line search for B ^* can be done in O(d log M) time and optimal O(dM) work.

The work of this author is partially supported by MURST Grant “Algoritmi, Modelli di Calcolo e Strutture Informative”. This work was done while the author was with AT&T Bell Laboratories, Murray Hill, NJ. U.S.A.

Work supported by MURST of Italy, partially done while the author was visiting AT&T Bell Laboratories.