Longest Common Extensions in Sublinear Space

The longest common extension problem (LCE problem) is to construct a data structure for an input string $T$ of length $n$ that supports LCE$(i,j)$ queries. Such a query returns the length of the longest common prefix of the suffixes starting at positions $i$ and $j$ in $T$. This classic problem has a well-known solution that uses $O(n)$ space and $O(1)$ query time. In this paper we show that for any trade-off parameter $1 \leq \tau \leq n$, the problem can be solved in $O(\frac{n}{\tau})$ space and $O(\tau)$ query time. This significantly improves the previously best known time-space trade-offs, and almost matches the best known time-space product lower bound.Comment: An extended abstract of this paper has been accepted to CPM 201

Range non-overlapping indexing and successive list indexing

We present two natural variants of the indexing problem: In the range non-overlapping indexing problem, we preprocess a given text to answer queries in which we are given a pattern, and wish to find a maximal-length sequence of occurrences of the pattern in the text, such that the occurrences do not overlap with one another. While efficiently solving this problem, our algorithm even enables us to efficiently perform so in substrings of the text, denoted by given start and end locations. The methods we supply thus generalize the string statistics problem [4, 5], in which we are asked to report merely the number of nonoverlapping occurrences in the entire text, by reporting the occurrences themselves, even only for substrings of the text. In the related successive list indexing problem, during query-time we are given a pattern and a list of locations in the preprocessed text. We then wish to find a list of occurrences of the pattern, such that the ith occurrence is the leftmost occurrence of the pattern which starts to the right of the ith location given by the input list. Both problems are solved by using tools from computational geometry, specifically a variation of the range searching for minimum problem of Lenhof and Smid [12], here considered over a grid, in what appears to be the first utilization of range searching for minimum in an indexingrelated context.