An alphabet-independent optimal parallel search for three dimensional pattern

Abstract

We give an alphabet-independent optimal parallel algorithm for the searching phase of three dimensional pattern-matching. All occurrences of a three dimensional pattern P of shape m×m×m in a text T of shape n×n×n are to be found. Our algorithm works in log m time with O(N/log(m)) processors of a CREW PRAM, where N=n ₃. The searching phase in three dimensions explores classification of two-dimensional periodicities of the cubic pattern. Some new projection techniques are developed to deal with three dimensions. The periodicites of the patern with respect to its faces are investigated. The nonperiodicities imply some sparseness properties, while periodicities imply other special useful properties (i.e. monotonicity) of the set of occurrences. Both types of properties are useful in deriving an efficient algorithm.

The search phase is preceeded by the preprocessing phase (computation of the witness table). Our main results concern the searching phase, however we present shortly a new approach to the second phase also. Usefullness of the dictionaries of basic factors (DBF's), see [7], in the computation of the three dimensional witness table is presented. The DBF approach gains simplicity at the expense of a small increase in time. It gives a (nonoptimal) O(log(m)) time algorithm using m processors of a CRCW PRAM. The alphabet-independent optimal preprocessing is very complex even in the case of two dimensions, see [9]. For large alphabets the DBF's give assymptotically the same complexity as the (alphabet-dependent) suffix trees approach (but avoids suffix trees and is simpler). However the basic advantage of the DBF approach is its simplicity of dealing with three (or more) dimensions.

The algorithm can be easily adjusted to the case of unequally sided patterns.

Research supported in part by the DFG Grant KA 673/4-1, by the ESPRIT BR Grants 7097 and ECUS030, and by the Volkswagen-Stiftung.

Research supported in part by the Grant KBN 2-1190-91-01.