DOCSLIB.ORG

Topics in Combinatorial Pattern Matching

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topics in Combinatorial Pattern Matching Downloaded from orbit.dtu.dk on: Sep 27, 2021 Topics in combinatorial pattern matching Vildhøj, Hjalte Wedel Publication date: 2015 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Vildhøj, H. W. (2015). Topics in combinatorial pattern matching. Technical University of Denmark. DTU Compute PHD-2014 No. 348 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. TOPICS IN COMBINATORIAL PATTERN MATCHING Hjalte Wedel Vildhøj PHD-2014-348 Technical University of Denmark Department of Applied Mathematics and Computer Science Richard Petersens Plads, Building 324, 2800 Kongens Lyngby, Denmark Phone +45 4525 3031 [email protected] www.compute.dtu.dk PHD-2014-348 ISSN: 0909-3192 PREFACE This doctoral dissertation was prepared at the Department of Applied Mathe- matics and Computer Science at the Technical University of Denmark in partial fulfilment of the requirements for acquiring a doctoral degree. The dissertation presents selected results within the general topic of theoretical computer science and more specifically combinatorial pattern matching. All included results were obtained and published in peer-reviewed conference proceedings or journals during my enrollment as a PhD student from September 2011 to September 2014. Acknowledgements I am grateful to my advisors Inge Li Gørtz and Philip Bille who have provided excellent guidance and support during my studies and have introduced me to countless of interesting problems, papers and people. I also thank my colleagues, office mates and friends Frederik Rye Skjoldjensen, Patrick Hagge Cording, and Søren Vind for making the past three years anything but lonely. I am further grateful to our section head Paul Fischer and the rest of our section for creating a fantastic atmosphere for research. I also owe my thanks to Gadi Landau and Oren Weimann for hosting me during my external stay at Haifa University in Israel, and for broadening my mind about their beautiful country. I also thank Raphaël Clifford and Benjamin Sach for hosting me during a memorable research visit at Bristol University. Moreover, I wish to thank all my co-authors Philip Bille, Søren Bøg, Patrick Hagge Cording, Johannes Fischer, Inge Li Gørtz, Tomasz Kociumaka, Tsvi Kopelowitz, Benjamin Sach, Tatiana Starikovskaya, Morten Stöckel, Søren Vind and David Kofoed Wind. Last but not least, I thank Maja for her love and support. Hjalte Wedel Vildhøj Copenhagen, September 2014 i ABSTRACT This dissertation studies problems in the general theme of combinatorial pattern matching. More specifically, we study the following topics: Longest Common Extensions. We revisit the longest common extension (LCE) problem, that is, preprocess a string T into a compact data structure that sup- ports fast LCE queries. An LCE query takes a pair (i; j) of indices in T and returns the length of the longest common prefix of the suffixes of T starting at positions i and j. Such queries are also commonly known as longest common prefix (LCP) queries. We study the time-space trade-offs for the problem, that is, the space used for the data structure vs. the worst-case time for answering an LCE query. Let n be the length of T . Given a parameter τ, 1 τ n, we show how to achieve either (n=pτ) space and (τ) query time,≤ or≤ (n/τ) space and (τ log( LCE(i; j)O/τ)) query time, whereO LCE(i; j) denotesO the length of theO LCE returnedj byj the query. These bounds providej thej first smooth trade-offs for the LCE problem and almost match the previously known bounds at the extremes when τ = 1 or τ = n. We apply the result to obtain improved bounds for several applications where the LCE problem is the computational bottleneck, including approximate string matching and computing palindromes. We also present an efficient technique to reduce LCE queries on two strings to one string. Finally, we give a lower bound on the time-space product for LCE data structures in the non-uniform cell probe model showing that our second trade-off is nearly optimal. Fingerprints in Compressed Strings. The Karp-Rabin fingerprint of a string is a type of hash value that due to its strong properties has been used in many string algorithms. We show how to construct a data structure for a string S of size N compressed by a context-free grammar of size n that supports fin- gerprint queries. That is, given indices i and j, the answer to a query is the iii iv TOPICS IN COMBINATORIAL PATTERN MATCHING fingerprint of the substring S[i; j]. We present the first (n) space data struc- tures that answer fingerprint queries without decompressingO any characters. For Straight Line Programs (SLP) we get (log N) query time, and for Linear SLPs (an SLP derivative that captures LZ78O compression and its variations) we get (log log N) query time. Hence, our data structures has the same time and spaceO complexity as for random access in SLPs. We utilize the fingerprint data structures to solve the longest common extension problem in query time (log N log `) and (log ` log log ` + log log N) for SLPs and Linear SLPs, respec- tively.O Here, ` = LCE(O i; j) denotes the length of the LCE. j j Sparse Text Indexing. We present efficient algorithms for constructing sparse suffix trees, sparse suffix arrays and sparse positions heaps for b arbitrary posi- tions of a text T of length n while using only (b) words of space during the construction. Our main contribution is to showO that the sparse suffix tree (and array) can be constructed in (n log2 b) time. To achieve this we develop a technique, that allows to efficientlyO answer b longest common prefix queries on suffixes of T , using only (b) space. Our first solution is Monte-Carlo and outputs the correct tree withO high probability. We then give a Las-Vegas algo- rithm which also uses (b) space and runs in the same time bounds with high probability when b = O(pn). Furthermore, additional tradeoffs between the space usage and the constructionO time for the Monte-Carlo algorithm are given. Finally, we show that at the expense of slower pattern queries, it is possible to construct sparse position heaps in (n + b log b) time and (b) space. O O The Longest Common Substring Problem. Given m documents of total length n, we consider the problem of finding a longest string common to at least d 2 of the documents. This problem is known as the longest common substring (LCS)≥ problem and has a classic (n) space and (n) time solution (Weiner [FOCS’73], Hui [CPM’92]). However,O the use of linearO space is impractical in many applica- tions. We show several time-space trade-offs for this problem. Our main result is that for any trade-off parameter 1 τ n, the LCS problem can be solved in (τ) space and (n2/τ) time, thus≤ providing≤ the first smooth deterministic time-spaceO trade-offO from constant to linear space. The result uses a new and very simple algorithm, which computes a τ-additive approximation to the LCS in (n2/τ) time and (1) space. We also show a time-space trade-off lower bound forO deterministic branchingO programs, which implies that any deterministic RAM algorithm solving the LCS problem on documents from a sufficiently large al- p phabet in (τ) space must use Ω(n log(n=(τ log n))= log log(n=(τ log n)) time. O ABSTRACT v Structural Properties of Suffix Trees. We study structural and combinatorial properties of suffix trees. Given an unlabeled tree on n nodes and suffix links of its internal nodes, we ask the question “Is aT suffix tree?", i.e., is there a string S whose suffix tree has the same topologicalT structure as ? We place no restrictions on S, in particular we do not require that S ends withT a unique symbol. This corresponds to considering the more general definition of implicit or extended suffix trees. Such general suffix trees have many applications and are for example needed to allow efficient updates when suffix trees are built online. We prove that is a suffix tree if and only if it is realized by a string S of length n 1, and weT give a linear-time algorithm for inferring S when the first letter on− each edge is known. DANISH ABSTRACT Denne afhandling studerer problemer inden for det generelle omrade kombina- torisk mønstergenkendelse. Vi studerer følgende emner: Længste fælles præfiks. Vi vender tilbage til længste-fælles-præfiks-problemet, det vil sige præprocesser en streng T til en kompakt datastruktur, der under- støtter hurtige LCE-forespørgsler. En LCE-forespørgsel tager et par (i; j) af positioner i T og returnerer det længste fælles præfiks af de to suffikser, der starter pa position i og j i T .Sadanne forespørgsler er ogsa kendt som LCP- forespørgsler. Vi studerer mulige afvejninger af tid og plads for problemet – det vil sige den plads, som datastrukturen anvender versus den tid, den skal bruge til at svare pa en LCE-forespørgsel.
Load more

Recommended publications
  • Pattern Matching Using Similarity Measures
    Pattern matching using similarity measures Patroonvergelijking met behulp van gelijkenismaten (met een samenvatting in het Nederlands) PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van Rector Magnificus, Prof. Dr. H. O. Voorma, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op maandag 18 september 2000 des morgens te 10:30 uur door Michiel Hagedoorn geboren op 13 juli 1972, te Renkum promotor: Prof. Dr. M. H. Overmars Faculteit Wiskunde & Informatica co-promotor: Dr. R. C. Veltkamp Faculteit Wiskunde & Informatica ISBN 90-393-2460-3 PHILIPS '$ The&&% research% described in this thesis has been made possible by financial support from Philips Research Laboratories. The work in this thesis has been carried out in the graduate school ASCI. Contents 1 Introduction 1 1.1Patternmatching.......................... 1 1.2Applications............................. 4 1.3Obtaininggeometricpatterns................... 7 1.4 Paradigms in geometric pattern matching . 8 1.5Similaritymeasurebasedpatternmatching........... 11 1.6Overviewofthisthesis....................... 16 2 A theory of similarity measures 21 2.1Pseudometricspaces........................ 22 2.2Pseudometricpatternspaces................... 30 2.3Embeddingpatternsinafunctionspace............. 40 2.4TheHausdorffmetric........................ 46 2.5Thevolumeofsymmetricdifference............... 54 2.6 Reflection visibility based distances . 60 2.7Summary.............................. 71 2.8Experimentalresults.......................
    [Show full text]
  • Pattern Matching Using Fuzzy Methods David Bell and Lynn Palmer, State of California: Genetic Disease Branch
    Pattern Matching Using Fuzzy Methods David Bell and Lynn Palmer, State of California: Genetic Disease Branch ABSTRACT The formula is : Two major methods of detecting similarities Euclidean Distance and 2 a "fuzzy Hamming distance" presented in a paper entitled "F %%y Dij : ; ( <3/i - yj 4 Hamming Distance: A New Dissimilarity Measure" by Bookstein, Klein, and Raita, will be compared using entropy calculations to Euclidean distance is especially useful for comparing determine their abilities to detect patterns in data and match data matching whole word object elements of 2 vectors. The records. .e find that both means of measuring distance are useful code in Java is: depending on the context. .hile fuzzy Hamming distance outperforms in supervised learning situations such as searches, the Listing 1: Euclidean Distance Euclidean distance measure is more useful in unsupervised pattern matching and clustering of data. /** * EuclideanDistance.java INTRODUCTION * * Pattern matching is becoming more of a necessity given the needs for * Created: Fri Oct 07 08:46:40 2002 such methods for detecting similarities in records, epidemiological * * @author David Bell: DHS-GENETICS patterns, genetics and even in the emerging fields of criminal * @version 1.0 behavioral pattern analysis and disease outbreak analysis due to */ possible terrorist activity. 0nfortunately, traditional methods for /** Abstact Distance class linking or matching records, data fields, etc. rely on exact data * @param None matches rather than looking for close matches or patterns. 1f course * @return Distance Template proximity pattern matches are often necessary when dealing with **/ messy data, data that has inexact values and/or data with missing key abstract class distance{ values.
    [Show full text]
  • CSCI 2041: Pattern Matching Basics
    CSCI 2041: Pattern Matching Basics Chris Kauffman Last Updated: Fri Sep 28 08:52:58 CDT 2018 1 Logistics Reading Assignment 2 I OCaml System Manual: Ch I Demo in lecture 1.4 - 1.5 I Post today/tomorrow I Practical OCaml: Ch 4 Next Week Goals I Mon: Review I Code patterns I Wed: Exam 1 I Pattern Matching I Fri: Lecture 2 Consider: Summing Adjacent Elements 1 (* match_basics.ml: basic demo of pattern matching *) 2 3 (* Create a list comprised of the sum of adjacent pairs of 4 elements in list. The last element in an odd-length list is 5 part of the return as is. *) 6 let rec sum_adj_ie list = 7 if list = [] then (* CASE of empty list *) 8 [] (* base case *) 9 else 10 let a = List.hd list in (* DESTRUCTURE list *) 11 let atail = List.tl list in (* bind names *) 12 if atail = [] then (* CASE of 1 elem left *) 13 [a] (* base case *) 14 else (* CASE of 2 or more elems left *) 15 let b = List.hd atail in (* destructure list *) 16 let tail = List.tl atail in (* bind names *) 17 (a+b) :: (sum_adj_ie tail) (* recursive case *) The above function follows a common paradigm: I Select between Cases during a computation I Cases are based on structure of data I Data is Destructured to bind names to parts of it 3 Pattern Matching in Programming Languages I Pattern Matching as a programming language feature checks that data matches a certain structure the executes if so I Can take many forms such as processing lines of input files that match a regular expression I Pattern Matching in OCaml/ML combines I Case analysis: does the data match a certain structure I Destructure Binding: bind names to parts of the data I Pattern Matching gives OCaml/ML a certain "cool" factor I Associated with the match/with syntax as follows match something with | pattern1 -> result1 (* pattern1 gives result1 *) | pattern2 -> (* pattern 2..
    [Show full text]
  • Compiling Pattern Matching to Good Decision Trees
    Submitted to ML’08 Compiling Pattern Matching to good Decision Trees Luc Maranget INRIA Luc.marangetinria.fr Abstract In this paper we study compilation to decision tree, whose We address the issue of compiling ML pattern matching to efficient primary advantage is never testing a given subterm of the subject decisions trees. Traditionally, compilation to decision trees is op- value more than once (and whose primary drawback is potential timized by (1) implementing decision trees as dags with maximal code size explosion). Our aim is to refine naive compilation to sharing; (2) guiding a simple compiler with heuristics. We first de- decision trees, and to compare the output of such an optimizing sign new heuristics that are inspired by necessity, a notion from compiler with optimized backtracking automata. lazy pattern matching that we rephrase in terms of decision tree se- Compilation to decision can be very sensitive to the testing mantics. Thereby, we simplify previous semantical frameworks and order of subject value subterms. The situation can be explained demonstrate a direct connection between necessity and decision by the example of an human programmer attempting to translate a ML program into a lower-level language without pattern matching. tree runtime efficiency. We complete our study by experiments, 1 showing that optimized compilation to decision trees is competi- Let f be the following function defined on triples of booleans : tive. We also suggest some heuristics precisely. l e t f x y z = match x,y,z with | _,F,T -> 1 Categories and Subject Descriptors D 3. 3 [Programming Lan- | F,T,_ -> 2 guages]: Language Constructs and Features—Patterns | _,_,F -> 3 | _,_,T -> 4 General Terms Design, Performance, Sequentiality.
    [Show full text]
  • Combinatorial Pattern Matching
    Combinatorial Pattern Matching 1 A Recurring Problem Finding patterns within sequences Variants on this idea Finding repeated motifs amoungst a set of strings What are the most frequent k-mers How many time does a specific k-mer appear Fundamental problem: Pattern Matching Find all positions of a particular substring in given sequence? 2 Pattern Matching Goal: Find all occurrences of a pattern in a text Input: Pattern p = p1, p2, … pn and text t = t1, t2, … tm Output: All positions 1 < i < (m – n + 1) such that the n-letter substring of t starting at i matches p def bruteForcePatternMatching(p, t): locations = [] for i in xrange(0, len(t)-len(p)+1): if t[i:i+len(p)] == p: locations.append(i) return locations print bruteForcePatternMatching("ssi", "imissmissmississippi") [11, 14] 3 Pattern Matching Performance Performance: m - length of the text t n - the length of the pattern p Search Loop - executed O(m) times Comparison - O(n) symbols compared Total cost - O(mn) per pattern In practice, most comparisons terminate early Worst-case: p = "AAAT" t = "AAAAAAAAAAAAAAAAAAAAAAAT" 4 We can do better! If we preprocess our pattern we can search more effciently (O(n)) Example: imissmissmississippi 1. s 2. s 3. s 4. SSi 5. s 6. SSi 7. s 8. SSI - match at 11 9. SSI - match at 14 10. s 11. s 12. s At steps 4 and 6 after finding the mismatch i ≠ m we can skip over all positions tested because we know that the suffix "sm" is not a prefix of our pattern "ssi" Even works for our worst-case example "AAAAT" in "AAAAAAAAAAAAAAT" by recognizing the shared prefixes ("AAA" in "AAAA").
    [Show full text]
  • Integrating Pattern Matching Within String Scanning a Thesis
    Integrating Pattern Matching Within String Scanning A Thesis Presented in Partial Fulfillment of the Requirements for the Degree of Master of Science with a Major in Computer Science in the College of Graduate Studies University of Idaho by John H. Goettsche Major Professor: Clinton Jeffery, Ph.D. Committee Members: Robert Heckendorn, Ph.D.; Robert Rinker, Ph.D. Department Administrator: Frederick Sheldon, Ph.D. July 2015 ii Authorization to Submit Thesis This Thesis of John H. Goettsche, submitted for the degree of Master of Science with a Major in Computer Science and titled \Integrating Pattern Matching Within String Scanning," has been reviewed in final form. Permission, as indicated by the signatures and dates below, is now granted to submit final copies to the College of Graduate Studies for approval. Major Professor: Date: Clinton Jeffery, Ph.D. Committee Members: Date: Robert Heckendorn, Ph.D. Date: Robert Rinker, Ph.D. Department Administrator: Date: Frederick Sheldon, Ph.D. iii Abstract A SNOBOL4 like pattern data type and pattern matching operation were introduced to the Unicon language in 2005, but patterns were not integrated with the Unicon string scanning control structure and hence, the SNOBOL style patterns were not adopted as part of the language at that time. The goal of this project is to make the pattern data type accessible to the Unicon string scanning control structure and vice versa; and also make the pattern operators and functions lexically consistent with Unicon. To accomplish these goals, a Unicon string matching operator was changed to allow the execution of a pattern match in the anchored mode, pattern matching unevaluated expressions were revised to handle complex string scanning functions, and the pattern matching lexemes were revised to be more consistent with the Unicon language.
    [Show full text]
  • Practical Authenticated Pattern Matching with Optimal Proof Size
    Practical Authenticated Pattern Matching with Optimal Proof Size Dimitrios Papadopoulos Charalampos Papamanthou Boston University University of Maryland [email protected] [email protected] Roberto Tamassia Nikos Triandopoulos Brown University RSA Laboratories & Boston University [email protected] [email protected] ABSTRACT otherwise. More elaborate models for pattern matching involve We address the problem of authenticating pattern matching queries queries expressed as regular expressions over Σ or returning multi- over textual data that is outsourced to an untrusted cloud server. By ple occurrences of p, and databases allowing search over multiple employing cryptographic accumulators in a novel optimal integrity- texts or other (semi-)structured data (e.g., XML data). This core checking tool built directly over a suffix tree, we design the first data-processing problem has numerous applications in a wide range authenticated data structure for verifiable answers to pattern match- of topics including intrusion detection, spam filtering, web search ing queries featuring fast generation of constant-size proofs. We engines, molecular biology and natural language processing. present two main applications of our new construction to authen- Previous works on authenticated pattern matching include the ticate: (i) pattern matching queries over text documents, and (ii) schemes by Martel et al. [28] for text pattern matching, and by De- exact path queries over XML documents. Answers to queries are vanbu et al. [16] and Bertino et al. [10] for XML search. In essence, verified by proofs of size at most 500 bytes for text pattern match- these works adopt the same general framework: First, by hierarchi- ing, and at most 243 bytes for exact path XML search, indepen- cally applying a cryptographic hash function (e.g., SHA-2) over the dently of the document or answer size.
    [Show full text]
  • Pattern Matching
    Pattern Matching Document #: P1371R1 Date: 2019-06-17 Project: Programming Language C++ Evolution Reply-to: Sergei Murzin <[email protected]> Michael Park <[email protected]> David Sankel <[email protected]> Dan Sarginson <[email protected]> Contents 1 Revision History 3 2 Introduction 3 3 Motivation and Scope 3 4 Before/After Comparisons4 4.1 Matching Integrals..........................................4 4.2 Matching Strings...........................................4 4.3 Matching Tuples...........................................4 4.4 Matching Variants..........................................5 4.5 Matching Polymorphic Types....................................5 4.6 Evaluating Expression Trees.....................................6 4.7 Patterns In Declarations.......................................8 5 Design Overview 9 5.1 Basic Syntax.............................................9 5.2 Basic Model..............................................9 5.3 Types of Patterns........................................... 10 5.3.1 Primary Patterns....................................... 10 5.3.1.1 Wildcard Pattern................................. 10 5.3.1.2 Identifier Pattern................................. 10 5.3.1.3 Expression Pattern................................ 10 5.3.2 Compound Patterns..................................... 11 5.3.2.1 Structured Binding Pattern............................ 11 5.3.2.2 Alternative Pattern................................ 12 5.3.2.3 Parenthesized Pattern............................... 15 5.3.2.4 Case Pattern...................................
    [Show full text]
  • Pattern Matching
    Pattern Matching Gonzalo Navarro Department of Computer Science University of Chile Abstract An important subtask of the pattern discovery process is pattern matching, where the pattern sought is already known and we want to determine how often and where it occurs in a sequence. In this paper we review the most practical techniques to find patterns of different kinds. We show how regular expressions can be searched for with general techniques, and how simpler patterns can be dealt with more simply and efficiently. We consider exact as well as approximate pattern matching. Also, we cover both sequential searching, where the sequence cannot be preprocessed, and indexed searching, where we have a data structure built over the sequence to speed up the search. 1 Introduction A central subproblem in pattern discovery is to determine how often a candidate pattern occurs, as well as possibly some information on its frequency distribution across the text. In general, a pattern will be a description of a (finite or infinite) set of strings, each string being a sequence of symbols. Usually, a good pattern must be as specific (that is, denote a small set of strings) and as frequent as possible. Hence, given a candidate pattern, it is usual to ask for its frequency, as well as to examine its occurrences looking for more specific instances of the pattern that are frequent enough. In this paper we focus on pattern matching. Given a known pattern (which can be more general or more specific), we wish to count how many times it occurs in the text, and to point out its occurrence positions.
    [Show full text]
  • Pattern Matching
    Pattern Matching An Introduction to File Globs and Regular Expressions Copyright 20062009 Stewart Weiss The danger that lies ahead Much to your disadvantage, there are two different forms of patterns in UNIX, one used when representing file names, and another used by commands, such as grep, sed, awk, and vi. You need to remember that the two types of patterns are different. Still worse, the textbook covers both of these in the same chapter, and I will do the same, so as not to digress from the order in the book. This will make it a little harder for you, but with practice you will get the hang of it. 2 CSci 132 Practical UNIX with Perl File globs In card games, a wildcard is a playing card that can be used as if it were any other card, such as the Joker. Computer science has borrowed the idea of a wildcard, and taken it several steps further. All shells give you the ability to write patterns that represent sets of filenames, using special characters called wildcards. (These patterns are not regular expressions, but they look like them.) The patterns are called file globs. The name "glob" comes from the name of the original UNIX program that expanded the pattern into a set of matching filenames. A string is a wildcard pattern if it contains one of the characters '?', '*' or '['. 3 CSci 132 Practical UNIX with Perl File glob rules Rule 1: a character always matches itself, except for the wildcards. So a matches 'a' and 'b' matches 'b' and so on.
    [Show full text]
  • Regular Expressions and Pattern Matching [email protected]
    Regular Expressions and Pattern Matching [email protected] Regular Expression (regex): a separate language, allowing the construction of patterns. used in most programming languages. very powerful in Perl. Pattern Match: using regex to search data and look for a match. Overview: how to create regular expressions how to use them to match and extract data biological context So Why Regex? Parse files of data and information: fasta embl / genbank format html (web-pages) user input to programs Check format Find illegal characters (validation) Search for sequences motifs Simple Patterns place regex between pair of forward slashes (/ /). try: #!/usr/bin/perl while (<STDIN>) { if (/abc/) { print “1 >> $_”; } } Run the script. Type in something that contains abc: abcfoobar Type in something that doesn't: fgh cba foobar ab c foobar print statement is returned if abc is matched within the typed input. Simple Patterns (2) Can also match strings from files. genomes_desc.txt contains a few text lines containing information about three genomes. try: #!/usr/bin/perl open IN, “<genomes_desc.txt”; while (<IN>) { if (/elegans/) { #match lines with this regex print; #print lines with match } } Parses each line in turn. Looks for elegans anywhere in line $_ Flexible matching There are many characters with special meanings – metacharacters. star (*) matches any number of instances /ab*c/ => 'a' followed by zero or more 'b' followed by 'c' => abc or abbbbbbbc or ac plus (+) matches at least one instance /ab+c/ => 'a' followed by one or more 'b' followed by 'c' => abc or abbc or abbbbbbbbbbbbbbc NOT ac question mark (?) matches zero or one instance /ab?c/ => 'a' followed by 0 or 1 'b' followed by 'c' => abc or ac More General Quantifiers Match a character a specific number or range of instances {x} will match x number of instances.
    [Show full text]
  • Cartesian Tree Matching and Indexing
    Cartesian Tree Matching and Indexing Sung Gwan Park Seoul National University, Korea [email protected] Amihood Amir Bar-Ilan University, Israel [email protected] Gad M. Landau University of Haifa, Israel and New York University, USA [email protected] Kunsoo Park1 Seoul National University, Korea [email protected] Abstract We introduce a new metric of match, called Cartesian tree matching, which means that two strings match if they have the same Cartesian trees. Based on Cartesian tree matching, we define single pattern matching for a text of length n and a pattern of length m, and multiple pattern matching for a text of length n and k patterns of total length m. We present an O(n + m) time algorithm for single pattern matching, and an O((n + m) log k) deterministic time or O(n + m) randomized time algorithm for multiple pattern matching. We also define an index data structure called Cartesian suffix tree, and present an O(n) randomized time algorithm to build the Cartesian suffix tree. Our efficient algorithms for Cartesian tree matching use a representation of the Cartesian tree, called the parent-distance representation. 2012 ACM Subject Classification Theory of computation → Pattern matching Keywords and phrases Cartesian tree matching, Pattern matching, Indexing, Parent-distance representation Digital Object Identifier 10.4230/LIPIcs.CPM.2019.13 1 Introduction String matching is one of fundamental problems in computer science, and it can be applied to many practical problems. In many applications string matching has variants derived from exact matching (which can be collectively called generalized matching), such as order- preserving matching [19, 20, 22], parameterized matching [4, 7, 8], jumbled matching [9], overlap matching [3], pattern matching with swaps [2], and so on.
    [Show full text]