Compact recognizers of episode sequences

• Authors: Alberto Apostolico;Mikhail J. Atallah
• Venue: Information and Computation
• Year: 2002

Given two strings X=a1...an and P=b1...bm over an alphabet , the problem of testing whether P occurs as a subsequence of X is trivially solved in linear time. It is also known that a simple O (n log ||) time preprocessing of X makes it easy to decide subsequently, for any P and in at most |P| log || character comparisons, whether P is a subsequence of X. These problems become more complicated if one asks instead whether P occurs as a subsequence of some substring Y of X of bounded length. This paper presents an automaton built on the textstring X and capable of identifying all distinct minimal substrings Y of X having P as a subsequence. By a substring Y being minimal with respect to P, it is meant that P is not a subsequence of any proper substring of Y. For every minimal substring Y, the automaton recognizes the occurrence of P having the lexicographically smallest sequence of symbol positions in Y. It is not difficult to realize such an automaton in time and space O (n2) for a text of n characters. One result of this paper consists of bringing those bounds down to linear or O (n log n), respectively, depending on whether the alphabet is bounded or of arbitrary size, thereby matching the corresponding complexities of automata constructions for offline exact string searching. Having built the automaton, the search for all lexicographically earliest occurrences of P in X is carried out in time O (i=1mroccii) or O (n+i=1mroccii log n), depending on whether the alphabet is fixed or arbitrary, where rocci is the number of distinct minimal substrings of X having b1...bi as a subsequence (note that each such substring may occur many times in X but is counted only once in the bound). All log factors appearing in the above bounds can be further reduced to log log by resorting to known integer-handling data structures.