Cartesian Tree Matching and Indexing
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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....................... -
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. -
Balanced Trees Part One
Balanced Trees Part One Balanced Trees ● Balanced search trees are among the most useful and versatile data structures. ● Many programming languages ship with a balanced tree library. ● C++: std::map / std::set ● Java: TreeMap / TreeSet ● Many advanced data structures are layered on top of balanced trees. ● We’ll see several later in the quarter! Where We're Going ● B-Trees (Today) ● A simple type of balanced tree developed for block storage. ● Red/Black Trees (Today/Thursday) ● The canonical balanced binary search tree. ● Augmented Search Trees (Thursday) ● Adding extra information to balanced trees to supercharge the data structure. Outline for Today ● BST Review ● Refresher on basic BST concepts and runtimes. ● Overview of Red/Black Trees ● What we're building toward. ● B-Trees and 2-3-4 Trees ● Simple balanced trees, in depth. ● Intuiting Red/Black Trees ● A much better feel for red/black trees. A Quick BST Review Binary Search Trees ● A binary search tree is a binary tree with 9 the following properties: 5 13 ● Each node in the BST stores a key, and 1 6 10 14 optionally, some auxiliary information. 3 7 11 15 ● The key of every node in a BST is strictly greater than all keys 2 4 8 12 to its left and strictly smaller than all keys to its right. Binary Search Trees ● The height of a binary search tree is the 9 length of the longest path from the root to a 5 13 leaf, measured in the number of edges. 1 6 10 14 ● A tree with one node has height 0. -
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.. -
Position Heaps for Cartesian-Tree Matching on Strings and Tries
Position Heaps for Cartesian-tree Matching on Strings and Tries Akio Nishimoto1, Noriki Fujisato1, Yuto Nakashima1, and Shunsuke Inenaga1;2 1Department of Informatics, Kyushu University, Japan fnishimoto.akio, noriki.fujisato, yuto.nakashima, [email protected] 2PRESTO, Japan Science and Technology Agency, Japan Abstract The Cartesian-tree pattern matching is a recently introduced scheme of pattern matching that detects fragments in a sequential data stream which have a similar structure as a query pattern. Formally, Cartesian-tree pattern matching seeks all substrings S0 of the text string S such that the Cartesian tree of S0 and that of a query pattern P coincide. In this paper, we present a new indexing structure for this problem, called the Cartesian-tree Position Heap (CPH ). Let n be the length of the input text string S, m the length of a query pattern P , and σ the alphabet size. We show that the CPH of S, denoted CPH(S), supports pattern matching queries in O(m(σ + log(minfh; mg)) + occ) time with O(n) space, where h is the height of the CPH and occ is the number of pattern occurrences. We show how to build CPH(S) in O(n log σ) time with O(n) working space. Further, we extend the problem to the case where the text is a labeled tree (i.e. a trie). Given a trie T with N nodes, we show that the CPH of T , denoted CPH(T ), supports pattern matching queries on the trie in O(m(σ2 +log(minfh; mg))+occ) time with O(Nσ) space. -
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. -
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"). -
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. -
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. -
Lecture Notes of CSCI5610 Advanced Data Structures
Lecture Notes of CSCI5610 Advanced Data Structures Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong July 17, 2020 Contents 1 Course Overview and Computation Models 4 2 The Binary Search Tree and the 2-3 Tree 7 2.1 The binary search tree . .7 2.2 The 2-3 tree . .9 2.3 Remarks . 13 3 Structures for Intervals 15 3.1 The interval tree . 15 3.2 The segment tree . 17 3.3 Remarks . 18 4 Structures for Points 20 4.1 The kd-tree . 20 4.2 A bootstrapping lemma . 22 4.3 The priority search tree . 24 4.4 The range tree . 27 4.5 Another range tree with better query time . 29 4.6 Pointer-machine structures . 30 4.7 Remarks . 31 5 Logarithmic Method and Global Rebuilding 33 5.1 Amortized update cost . 33 5.2 Decomposable problems . 34 5.3 The logarithmic method . 34 5.4 Fully dynamic kd-trees with global rebuilding . 37 5.5 Remarks . 39 6 Weight Balancing 41 6.1 BB[α]-trees . 41 6.2 Insertion . 42 6.3 Deletion . 42 6.4 Amortized analysis . 42 6.5 Dynamization with weight balancing . 43 6.6 Remarks . 44 1 CONTENTS 2 7 Partial Persistence 47 7.1 The potential method . 47 7.2 Partially persistent BST . 48 7.3 General pointer-machine structures . 52 7.4 Remarks . 52 8 Dynamic Perfect Hashing 54 8.1 Two random graph results . 54 8.2 Cuckoo hashing . 55 8.3 Analysis . 58 8.4 Remarks . 59 9 Binomial and Fibonacci Heaps 61 9.1 The binomial heap . -
CMSC 420: Lecture 7 Randomized Search Structures: Treaps and Skip Lists
CMSC 420 Dave Mount CMSC 420: Lecture 7 Randomized Search Structures: Treaps and Skip Lists Randomized Data Structures: A common design techlque in the field of algorithm design in- volves the notion of using randomization. A randomized algorithm employs a pseudo-random number generator to inform some of its decisions. Randomization has proved to be a re- markably useful technique, and randomized algorithms are often the fastest and simplest algorithms for a given application. This may seem perplexing at first. Shouldn't an intelligent, clever algorithm designer be able to make better decisions than a simple random number generator? The issue is that a deterministic decision-making process may be susceptible to systematic biases, which in turn can result in unbalanced data structures. Randomness creates a layer of \independence," which can alleviate these systematic biases. In this lecture, we will consider two famous randomized data structures, which were invented at nearly the same time. The first is a randomized version of a binary tree, called a treap. This data structure's name is a portmanteau (combination) of \tree" and \heap." It was developed by Raimund Seidel and Cecilia Aragon in 1989. (Remarkably, this 1-dimensional data structure is closely related to two 2-dimensional data structures, the Cartesian tree by Jean Vuillemin and the priority search tree of Edward McCreight, both discovered in 1980.) The other data structure is the skip list, which is a randomized version of a linked list where links can point to entries that are separated by a significant distance. This was invented by Bill Pugh (a professor at UMD!). -
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...................................