Order-preserving indexing

Abstract

Kubica et al. [33] and Kim et al. [29] introduced order-preserving pattern matching: for a given text the goal is to find its factors having the same shape as a given pattern. Known results include a linear-time algorithm for this problem (in case of polynomially-bounded alphabet) and a generalization to multiple patterns. We propose an index that enables order-preserving pattern matching queries in time proportional to pattern length. The index can be constructed in $O(n\log \log n)$ expected time or in $O(n\log 2\log n/\log \log \log n)$ worst-case time. It is an incomplete order-preserving suffix tree which may miss a single edge label at each branching node. For most applications such incomplete suffix trees provide the same functional power as the complete ones. We show a number of their applications, including computation of longest common factors, longest previously occurring factors and squares in a string in the order-preserving setting. We also give an $O(n\sqrt{\log n})$-time algorithm constructing complete order-preserving suffix trees.