The Swap Matching Problem Revisited

In this paper, we revisit the much studied problem of Pattern Matching with Swaps (Swap Matching problem, for short). We first present a graph-theoretic model, which opens a new and so far unexplored avenue to solve the problem. Then, using the model, we devise two efficient algorithms to solve the swap matching problem. The resulting algorithms are adaptations of the classic shift-and algorithm. For patterns having length similar to the word-size of the target machine, both the algorithms run in linear time considering a fixed alphabet.Comment: 23 pages, 3 Figures and 17 Table

Cartesian Tree Matching and Indexing

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

A Simple Algorithm for Approximating the Text-To-Pattern Hamming Distance

The algorithmic task of computing the Hamming distance between a given pattern of length m and each location in a text of length n, both over a general alphabet Sigma, is one of the most fundamental algorithmic tasks in string algorithms. The fastest known runtime for exact computation is tilde O(nsqrt m). We recently introduced a complicated randomized algorithm for obtaining a (1 +/- eps) approximation for each location in the text in O( (n/eps) log(1/eps) log n log m log |Sigma|) total time, breaking a barrier that stood for 22 years. In this paper, we introduce an elementary and simple randomized algorithm that takes O((n/eps) log n log m) time

Cartesian 트리에 기반한 문자열 매칭 및 인덱싱

학위논문 (석사) -- 서울대학교 대학원 : 공과대학 컴퓨터공학부, 2020. 8. 박근수.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.본 논문에서는 Cartesian 트리에 기반한 새로운 매칭 기준인 Cartesian 트리 매칭을 제안한다. 이는 두 문자열의 Cartesian 트리가 서로 같을 때, 두 문자열을 매칭된 것으로 정의하는 문제이다. Cartesian 트리 매칭의 기준 하에서, 본 연구에서는 길이 n인 텍스트와 길이 m인 패턴 사이의 단일패턴매칭 문제와 길이 n인 텍스트와 길이의 합이 m인 여러 개의 패턴 사이의 다중패턴매칭 문제를 정의하고, 단일패턴매칭 문제를 해결하는 O(n+m) 시간 알고리즘과 다중패턴매칭 문제를 해결하는 O((n+m) log k) 시간 결정론적 알고리즘 및 O(n+m) 시간 무작위 알고리즘을 제시한다. 또한, Cartesian 트리 매칭에 대한 인덱스 자료구조인 Cartesian 접미사트리를 정의하고, 이를 구축하는 O(n) 시간 무작위 알고리즘을 제시한다. 본 논문에서는 Cartesian tree를 표현하는 방식인 부모거리표현 (parent-distance representation)을 정의하고, 이를 이용하여 위 문제들을 해결하는 효율적인 알고리즘들을 제시한다.Chapter 1 Introduction 1 Chapter 2 Problem Definition 4 2.1 Basic notations 4 2.2 Cartesian tree matching 4 Chapter 3 Single Pattern Matching in O(n + m) Time 7 3.1 Parent-distance representation 7 3.2 Computing parent-distance representation 9 3.3 Failure function 11 3.4 Text search 13 3.5 Computing failure function 13 3.6 Correctness and time complexity 14 3.7 Cartesian tree signature 15 Chapter 4 Multiple Pattern Matching in O((n + m) log k) Time 17 4.1 Constructing the Aho-Corasick automaton 17 4.2 Multiple pattern matching 21 Chapter 5 Cartesian Suffix Tree in Randomized O(n) Time 22 5.1 Defining Cartesian suffix tree 22 5.2 Constructing Cartesian suffix tree 23 Chapter 6 Conclusion 26 Bibliography 27 요약 31Maste

Towards Optimal Approximate Streaming Pattern Matching by Matching Multiple Patterns in Multiple Streams

Recently, there has been a growing focus in solving approximate pattern matching problems in the streaming model. Of particular interest are the pattern matching with k-mismatches (KMM) problem and the pattern matching with w-wildcards (PMWC) problem. Motivated by reductions from these problems in the streaming model to the dictionary matching problem, this paper focuses on designing algorithms for the dictionary matching problem in the multi-stream model where there are several independent streams of data (as opposed to just one in the streaming model), and the memory complexity of an algorithm is expressed using two quantities: (1) a read-only shared memory storage area which is shared among all the streams, and (2) local stream memory that each stream stores separately. In the dictionary matching problem in the multi-stream model the goal is to preprocess a dictionary D={P_1,P_2,...,P_d} of d=|D| patterns (strings with maximum length m over alphabet Sigma) into a data structure stored in shared memory, so that given multiple independent streaming texts (where characters arrive one at a time) the algorithm reports occurrences of patterns from D in each one of the texts as soon as they appear. We design two efficient algorithms for the dictionary matching problem in the multi-stream model. The first algorithm works when all the patterns in D have the same length m and costs O(d log m) words in shared memory, O(log m log d) words in stream memory, and O(log m) time per character. The second algorithm works for general D, but the time cost per character becomes O(log m+log d log log d). We also demonstrate the usefulness of our first algorithm in solving both the KMM problem and PMWC problem in the streaming model. In particular, we obtain the first almost optimal (up to poly-log factors) algorithm for the PMWC problem in the streaming model. We also design a new algorithm for the KMM problem in the streaming model that, up to poly-log factors, has the same bounds as the most recent results that use different techniques. Moreover, for most inputs, our algorithm for KMM is significantly faster on average