Given a static array of n totally ordered objects, the range minimum query problem is to build a data structure that allows to answer subsequent on-line queries of the form "what is the position of a minimum element in the sub-array ranging from i to j?" efficiently. We focus on two settings, where (1) the input array is available at query time, and (2) the input array is only available at construction time. In setting (1), we show new data structures (a) of size 2n c(n) − Θ n lg lg n c(n) lg n bits and query time O(c(n)) for any positive integer function c(n) ∈ O n ε for an arbitrary constant 0 < ε < 1, or (b) with O(nH k) + o(n) bits and O(1) query time, where H k denotes the empirical entropy of k'th order of the input array. In setting (2), we give a data structure of size 2n + o(n) bits and query time O(1). All data structures can be constructed in linear time and almost in-place.

Suffix trees are among the most important data structures in stringology, with a number of applications in flourishing areas like bioinformatics. Their main problem is space usage, which has triggered much research striving for compressed representations that are still functional. A smaller suffix tree representation could fit in a faster memory, outweighing by far the theoretical slowdown brought by the space reduction. We present a novel compressed suffix tree, which is the first achieving at the same time sublogarithmic complexity for the operations, and space usage that asymptotically goes to zero as the entropy of the text does. The main ideas in our development are compressing the longest common prefix information, totally getting rid of the suffix tree topology, and expressing all the suffix tree operations using range minimum queries and a novel primitive called next/previous smaller value in a sequence. Our solutions to those operations are of independent interest.

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Server: http://frpred.tuebingen.mpg.de. Data sets: ftp://ftp.tuebingen.mpg.de/pub/protevo/FRpred/.

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