Order-preserving indexing

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(nlog⁡log⁡n)O(nlog⁡log⁡n) expected time or in O(nlog2⁡log⁡n/log⁡log⁡log⁡n)O(nlog2⁡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(nlog⁡n)-time algorithm constructing complete order-preserving suffix trees.