On the Construction of Classes of Suffix Trees for Square Matrices: Algorithms and Applications

Abstract

We provide a uniform framework for the study of index data structures for a two-dimensional matrixTEXT[1:n, 1:n] whose entries are drawn from an ordered alphabetΣ. An index forTEXTcan be informally seen as the two-dimensional analog of the suffix tree for a string. It allows on-line searches and statistics to be performed onTEXTby representing compactly theΘ(n3) square submatrices ofTEXTin optimalO(n2) space. We identify 4n−1families of indices forTEXT, each containing ∏ni=1 (2i−1)! isomorphic data structures. We also develop techniques leading to a single algorithm that efficiently builds any index in any family inO(n2 log n) time andO(n2) space. Such an algorithm improves in various respects the algorithms for the construction of the PAT tree and the Lsuffix tree. The framework and the algorithm easily generalize tod>2 dimensions. Moreover, as part of our algorithm, we provide new algorithmic tools that yield a space-efficient implementation of the “naming scheme” of R. Karpet al.(in“Proceedings, Fourth Symposium on Theory of Computing,” pp. 125–136) for strings and matrices.