String indexing for top-k close consecutive occurrences

The classic string indexing problem is to preprocess a string S into a compact data structure that supports efficient subsequent pattern matching queries, that is, given a pattern string P, report all occurrences of P within S. In this paper, we study a basic and natural extension of string indexing called the string indexing for top-k close consecutive occurrences problem (Sitcco). Here, a consecutive occurrence is a pair (i, j), i < j, such that P occurs at positions i and j in S and there is no occurrence of P between i and j, and their distance is defined as j − i. Given a pattern P and a parameter k, the goal is to report the top-k consecutive occurrences of P in S of minimal distance. The challenge is to compactly represent S while supporting queries in time close to the length of P and k. We give two time-space trade-offs for the problem. Let n be the length of S, m the length of P, and ε ∈ (0, 1]. Our first result achieves O(n log n) space and optimal query time of O(m + k), and our second result achieves linear space and query time O(m + k^1+ε). Along the way, we develop several techniques of independent interest, including a new translation of the problem into a line segment intersection problem and a new recursive clustering technique for trees.