Fast algorithms for finding a minimum repetition representation of strings and trees

A string with many repetitions can be represented compactly by replacing h-fold contiguous repetitions of a string r with (r)h. We present a compact representation, which we call a repetition representation (of a string) or RRS, by which a set of disjoint or nested tandem arrays can be compacted. In this paper, we study the problem of finding aminimum RRS or MRRS,where the size of an RRS is defined by the sumof the length of component letters and the description length of the component repetitions (·)h which is defined bywR(h) using a repetition weight function wR . We develop two dynamic programming-based algorithms to solve this problem: CMR, which works for any type of wR , and CMR-C, which is faster but can be applied to a constant wR only. CMR-C is an O(n2 log n)-time O(n log n)-space algorithm, which is more efficient in both time and space than CMR by a ((log n)/n)-factor, where n is the length of the given string. The problem of finding an MRRS for a string can be extended to that of finding aminimum repetition representation (of a tree) or MRRT for a given labeled ordered tree. For this problem, we present two algorithms, CMRT and CMRTC, by usingCMRandCMR-C, respectively, as a subroutine. Aswell as the theoretical analysis, we confirmed the efficiency of the proposed algorithms by experiments, which consist of the following three parts: First we demonstrated that CMR-C and CMRT-C are fast enough for large-scale data by using synthetic strings and trees, respectively. The size of an MRRS for a given string can be a measure of how compactly the string can be represented, meaning how well the string is structurally organized. This is also true of trees. To check such ability of MRRS-size, second we measured the size of an MRRS for chromosomes of nine different species. We found that all the chromosomes of the same species have a similar compression rate when realized by anMRRS. Run length encoding (RLE) was also shown to have species-specific compression rate, but species were separated more clearly by MRRS than by RLE. Third we examined the size of an MRRT for web pages of world-leading companies by using the tag trees, showing a consistency between the compression rate by an MRRT and visual web page structures. © 2013 Elsevier B.V. All rights reserved.