We survey the current techniques to cope with the problem of string matching that allows errors. This is becoming a more and more relevant issue for many fast growing areas such as information retrieval and computational biology. We focus on online searching and mostly on edit distance, explaining the problem and its relevance, its statistical behavior, its history and current developments, and the central ideas of the algorithms and their complexities. We present a number of experiments to compare the performance of the different algorithms and show which are the best choices. We conclude with some directions for future work and open problems.

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A guided tour to approximate string matching

Journal of the ACM

The proliferation of online text, such as found on the World Wide Web and in online databases, motivates the need for space-efficient text indexing methods that support fast string searching. We model this scenario as follows: Consider a text $T$ consisting of $n$ symbols drawn from a fixed alphabet $\Sigma$. The text $T$ can be represented in $n \lg |\Sigma|$ bits by encoding each symbol with $\lg |\Sigma|$ bits. The goal is to support fast online queries for searching any string pattern $P$ of $m$ symbols, with $T$ being fully scanned only once, namely, when the index is created at preprocessing time. The text indexing schemes published in the literature are greedy in terms of space usage: they require $\Omega(n \lg n)$ additional bits of space in the worst case. For example, in the standard unit cost RAM, suffix trees and suffix arrays need $\Omega(n)$ memory words, each of $\Omega(\lg n)$ bits. These indexes are larger than the text itself by a multiplicative factor of $\Omega(\smash{\lg_{|\Sigma|} n})$, which is significant when $\Sigma$ is of constant size, such as in \textsc{ascii} or \textsc{unicode}. On the other hand, these indexes support fast searching, either in $O(m \lg |\Sigma|)$ time or in $O(m + \lg n)$ time, plus an output-sensitive cost $O(\mathit{occ})$ for listing the $\mathit{occ}$ pattern occurrences. We present a new text index that is based upon compressed representations of suffix arrays and suffix trees. It achieves a fast $\smash{O(m /\lg_{|\Sigma|} n + \lg_{|\Sigma|}^\epsilon n)}$ search time in the worst case, for any constant $0 < \epsilon \leq 1$, using at most $\smash{\bigl(\epsilon^{-1} + O(1)\bigr) \, n \lg |\Sigma|}$ bits of storage. Our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case, and for many scenarios, the space is actually sublinear in practice. As a concrete example, the compressed suffix array for a typical 100 MB \textsc{ascii} file can require 30--40 MB or less, while the raw suffix array requires 500 MB. Our theoretical bounds improve \emph{both} time and space of previous indexing schemes. Listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output-sensitive cost, giving $O(\mathit{occ} \, \smash{\lg_{|\Sigma|}^\epsilon n})$ time as a result. When the patterns are sufficiently long, we can use auxiliary data structures in $O(n \lg |\Sigma|)$ bits to obtain a total search bound of $O(m /\lg_{|\Sigma|} n + \mathit{occ})$ time, which is optimal.

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Compressed Suffix Arrays and Suffix Trees with Applications to Text Indexing and String Matching

Expression.- Spectral Clustering Gene Ontology Terms to Group Genes by Function.- Dynamic De-Novo Prediction of microRNAs Associated with Cell Conditions: A Search Pruned by Expression.- Clustering Gene Expression Series with Prior Knowledge.- A Linear Time Biclustering Algorithm for Time Series Gene Expression Data.- Time-Window Analysis of Developmental Gene Expression Data with Multiple Genetic Backgrounds.- Phylogeny.- A Lookahead Branch-and-Bound Algorithm for the Maximum Quartet Consistency Problem.- Computing the Quartet Distance Between Trees of Arbitrary Degree.- Using Semi-definite Programming to Enhance Supertree Resolvability.- An Efficient Reduction from Constrained to Unconstrained Maximum Agreement Subtree.- Pattern Identification in Biogeography.- On the Complexity of Several Haplotyping Problems.- A Hidden Markov Technique for Haplotype Reconstruction.- Algorithms for Imperfect Phylogeny Haplotyping (IPPH) with a Single Homoplasy or Recombination Event.- Networks.- A Faster Algorithm for Detecting Network Motifs.- Reaction Motifs in Metabolic Networks.- Reconstructing Metabolic Networks Using Interval Analysis.- Genome Rearrangements.- A 1.375-Approximation Algorithm for Sorting by Transpositions.- A New Tight Upper Bound on the Transposition Distance.- Perfect Sorting by Reversals Is Not Always Difficult.- Minimum Recombination Histories by Branch and Bound.- Sequences.- A Unifying Framework for Seed Sensitivity and Its Application to Subset Seeds.- Generalized Planted (l,d)-Motif Problem with Negative Set.- Alignment of Tandem Repeats with Excision, Duplication, Substitution and Indels (EDSI).- The Peres-Shields Order Estimator for Fixed and Variable Length Markov Models with Applications to DNA Sequence Similarity.- Multiple Structural RNA Alignment with Lagrangian Relaxation.- Faster Algorithms for Optimal Multiple Sequence Alignment Based on Pairwise Comparisons.- Ortholog Clustering on a Multipartite Graph.- Linear Time Algorithm for Parsing RNA Secondary Structure.- A Compressed Format for Collections of Phylogenetic Trees and Improved Consensus Performance.- Structure.- Optimal Protein Threading by Cost-Splitting.- Efficient Parameterized Algorithm for Biopolymer Structure-Sequence Alignment.- Rotamer-Pair Energy Calculations Using a Trie Data Structure.- Improved Maintenance of Molecular Surfaces Using Dynamic Graph Connectivity.- The Main Structural Regularities of the Sandwich Proteins.- Discovery of Protein Substructures in EM Maps.

Algorithms in Bioinformatics: 5th International Workshop, WABI 2005, Mallorca, Spain, October 3-6, 2005, Proceedings (Lecture Notes in Computer Science / Lecture Notes in Bioinformatics)

This book presents a practical approach to string matching problems, focusing on the algorithms and implementations that perform best in practice. It covers searching for simple, multiple, and extended strings, as well as regular expressions, exactly and approximately. It includes all of the most significant new developments in complex pattern searching. The clear explanations, step-by-step examples, algorithms pseudo-code, and implementation efficiency maps will enable researchers, professionals, and students in bioinformatics, computer science, and software engineering to choose the most appropriate algorithms for their applications.

Flexible Pattern Matching in Strings: Practical On-Line Search Algorithms for Texts and Biological Sequences

This paper considers various flavors of the following online problem: preprocess a text or collection of strings, so that given a query string p, all matches of p with the text can be reported quickly. In this paper we consider matches in which a bounded number of mismatches are allowed, or in which a bounded number of "don't care" characters are allowed. The specific problems we look at are: indexing, in which there is a single text t, and we seek locations where p matches a substring of t; dictionary queries, in which a collection of strings is given upfront, and we seek those strings which match p in their entirety; and dictionary matching, in which a collection of strings is given upfront, and we seek those substrings of a (long) p which match an original string in its entirety. These are all instances of an all-to-all matching problem, for which we provide a single solution.The performance bounds all have a similar character. For example, for the indexing problem with n=|t| and m=|p|, the query time for k substitutions is O(m + (c1 log n)k⁄k! + # matches), with a data structure of size O(n (c2 log n)k⁄k!) and a preprocessing time of O(n (c2 log n)k⁄k!), where c1,c2 > 1 are constants. The deterministic preprocessing assumes a weakly nonuniform RAM model; this assumption is not needed if randomization is used in the preprocessing.

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Dictionary matching and indexing with errors and don't cares

The string matching with mismatches problem is that of finding the number of mismatches between a pattern P of length m and every length m substring of the text T. Currently, the fastest algorithms for this problem are the following. The Galil-Giancarlo algorithm finds all locations where the pattern has at most k errors (where k is part of the input) in time O(nk). The Abrahamson algorithm finds the number of mismatches at every location in time O(n√ m log m). We present an algorithm that is faster than both. Our algorithm finds all locations where the pattern has at most k errors in time O(n√k log k). We also show an algorithm that solves the above problem in time O((n + (nk3)/m) log k).

Faster algorithms for string matching with k mismatches

A directed multigraph is said to be d-regular if the indegree and outdegree of every vertex is exactly d. By Hall's theorem, one can represent such a multigraph as a combination of at most n2 cycle covers, each taken with an appropriate multiplicity. We prove that if the d-regular multigraph does not contain more than ⌊d/2⌋ copies of any 2-cycle then we can find a similar decomposition into n2pairs of cycle covers where each 2-cycle occurs in at most one component of each pair. Our proof is constructive and gives a polynomial algorithm to find such a decomposition. Since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs, we also give an alternative, simpler algorithm to extract a single such pair.This combinatorial theorem then comes handy in rounding a fractional solution of an LP relaxation of the maximum Traveling Salesman Problem (TSP) problem. The first stage of the rounding procedure obtains two cycle covers that do not share a 2-cycle with weight at least twice the weight of the optimal solution. Then we show how to extract a tour from the 2 cycle covers, whose weight is at least 2/3 of the weight of the longest tour. This improves upon the previous 5/8 approximation with a simpler algorithm. Utilizing a reduction from maximum TSP to the shortest superstring problem, we obtain a 2.5-approximation algorithm for the latter problem, which is again much simpler than the previous one.For minimum asymmetric TSP, the same technique gives two cycle covers, not sharing a 2-cycle, with weight at most twice the weight of the optimum. Assuming triangle inequality, we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0.842 log2n larger than optimal. This improves upon a previous approximation algorithm with approximation guarantee of 0.999 log2n. Other applications of the rounding procedure are approximation algorithms for maximum 3-cycle cover (factor 2/3, previously 3/5) and maximum asymmetric TSP with triangle inequality (factor 10/13, previously 3/4).

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Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs

We present a collection of new results on problems related to 3SUM, including: The first truly subquadratic algorithm for computing the (min,+) convolution for monotone increasing sequences with integer values bounded by O(n), solving 3SUM for monotone sets in 2D with integer coordinates bounded by O(n), and preprocessing a binary string for histogram indexing (also called jumbled indexing).The running time is O(n(9+√177)/12, polylog,n)=O(n1.859) with randomization, or O(n1.864) deterministically. This greatly improves the previous n2/2Ω(√log n) time bound obtained from Williams' recent result on all-pairs shortest paths [STOC'14], and answers an open question raised by several researchers studying the histogram indexing problem. The first algorithm for histogram indexing for any constant alphabet size that achieves truly subquadratic preprocessing time and truly sublinear query time. A truly subquadratic algorithm for integer 3SUM in the case when the given set can be partitioned into n1-δ clusters each covered by an interval of length n, for any constant δ>0. An algorithm to preprocess any set of n integers so that subsequently 3SUM on any given subset can be solved in O(n13/7, polylog,n) time.All these results are obtained by a surprising new technique, based on the Balog--Szemeredi--Gowers Theorem from additive combinatorics.

Moshe Lewenstein

Papers

Dictionary matching and indexing with errors and don't cares

Faster algorithms for string matching with k mismatches

Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs

Clustered Integer 3SUM via Additive Combinatorics

New results on induced matchings