Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

Abstract

We introduce efficient indexes for a problem in non-standard stringology: jumbled pattern matching. An index is a data structure constructed for a text of length n over an alphabet of size σ that can answer queries asking if the text contains a fragment which is jumbled (Abelian) equivalent to a pattern, specified by its so-called Parikh vector. We denote the length of the pattern by m. Moosa and Rahman (J Discrete Algorithms 10:5-9, 2012) gave an index for the case of binary alphabets with O n 2 (log n) 2 -time construction in the word-RAM model. Several earlier papers stated as an open problem the existence of an efficient solution for larger alphabets. In this paper we develop an index for any constant-sized alphabet. The construction involves a trade-off parameter, which in particular lets us achieve the following complexities: O(n 2−δ ) space and O(m (2σ −1)δ ) query time for any 0 < δ < 1, or O n 2 (log log n) 2 log n space and polylogarithmic, o(log 2σ −1 m), query time. The construction time in both cases is subquadratic: O n 2 (log log n) 2 log n in the word-RAM model (using bit-parallelism). Our construction algorithms are randomized (Las Vegas, running time w.h.p.), which is due to the usage of perfect hashing. On the other hand, all queries are answered deterministically. A preliminary version of this work appeared at ESA 2013 (Kociumaka et al. in Algorithms, ESA 2013. LNCS, vol 8125. Springer, Berlin, pp. 625-636, 2013 time construction of the index with O(n 2−δ ) space and O(m (2σ −1)δ ) query time, which was not present in the preliminary version. We also extend the index so that the position of the leftmost occurrence of the query pattern is provided at no additional cost in the complexity; this required rather nontrivial changes in the construction algorithm.

On the relationship between histogram indexing and block-mass indexing

“…We show that this trade-off scheme exists for both BMI and histogram indexing over unbounded alphabets. A similar idea was independently shown by Kociumaka et al [30].…”

“…The query time is still dependent only on the pattern size and not the text length, but it is no longer linear in the pattern size. A similar relaxation was exploited by Kociumaka et al [30] for the histogram indexing problem. In this section, we show trade-off algorithms for these two indexing problems that work for unbounded alphabets.…”

“…Tandem mass spectrometry (MS/MS) analyses peptides (short 8-30 amino acid long fragments of proteins) by generating their spectra. 1 The still unsolved problem in computational proteomics is to reconstruct a peptide from its spectrum: even the most advanced de novo peptide sequencing tools correctly reconstruct only [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] in MS/MS database searches [15]. After two decades of algorithmic developments, it seems that de novo peptide sequencing 'hits a wall' and that accurate full-length peptide reconstruction is nearly impossible due to the limited information content of MS/MS spectra.…”

Improved online algorithms for jumbled matching

“…Moosa et al [49] presented an index for jumbled pattern matching for binary alphabets. Kociumaka et al [44] introduced a solution for any constant sized alphabet. Various solutions [1,10,26,28] to this problem have been proposed in the recent past.…”