In recent years, significant advances have been made in the design and analysis of fully dynamic maximal matching algorithms. However, these theoretical results have received very little attention from the practical perspective. Few of the algorithms are implemented and tested on real datasets, and their practical potential is far from understood. In this paper, we attempt to bridge the gap between theory and practice that is currently observed for the fully dynamic maximal matching problem. We engineer several algorithms and empirically study those algorithms on an extensive set of dynamic instances.
Finding large or heavy matchings in graphs is a ubiquitous combinatorial optimization problem. In this paper, we engineer the first non-trivial implementations for approximating the dynamic weighted matching problem. Our first algorithm is based on random walks/paths combined with dynamic programming. The second algorithm has been introduced by Stubbs and Williams without an implementation. Roughly speaking, their algorithm uses dynamic unweighted matching algorithms as a subroutine (within a multilevel approach); this allows us to use previous work on dynamic unweighted matching algorithms as…
We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic $(2+ε)$-approximate integral matching algorithm with small polylog worst-case update time. For the $(2+ε)$-approximation regime only a \emph{fractional} fully-dynamic $(2+ε)$-matching algorithm with worst…
The algorithm of Micali and Vazirani \cite{MV} finds a maximum cardinality matching in time $O(\sqrt n m)$ if an efficient set-merging algorithm is used. The latter is provided by the incremental-tree set-merging algorithm of \cite{GabTar}. Details of this application to matching were omitted from \cite{GabTar} and are presented in this note.
Enumerating matchings is a classical problem in the field of enumeration algorithms. There are polynomial-delay enumeration algorithms for several settings, such as enumerating perfect matchings, maximal matchings, and (weighted) matchings in specific orders. In this paper, we present polynomial-delay enumeration algorithms for maximal matchings with cardinality at least given threshold $t$. Our algorithm enumerates all such matchings in $O(nm)$ delay with exponential space, where $n$ and $m$ are the number of vertices and edges of an input graph, respectively. We also present a polynomial-del…
For all practical purposes, the Micali-Vazirani general graph maximum matching algorithm is still the most efficient known algorithm for the problem. The purpose of this paper is to provide a complete proof of correctness of the algorithm in the simplest possible terms; graph-theoretic machinery developed for this purpose also helps simplify the algorithm.
String matching is the problem of finding all the occurrences of a pattern in a text. We propose improved versions of the fast family of string matching algorithms based on hashing $q$-grams. The improvement consists of considering minimal values $q$ such that each $q$-grams of the pattern has a unique hash value. The new algorithms are fastest than algorithm of the HASH family for short patterns on large size alphabets.
We present a simple proof that the competitive ratio of any randomized online matching algorithm for the line is at least $\sqrt{\log_2(n\!+\!1)}/12$ for all $n=2^i\!-\!1: i\in\mathbb{N}$.
Fully dynamic graph algorithms that achieve polylogarithmic or better time per operation use either a hierarchical graph decomposition or random-rank based approach. There are so far two graph properties for which efficient algorithms for both types of data structures exist, namely fully dynamic (Delta + 1) coloring and fully dynamic maximal matching. In this paper we present an extensive experimental study of these two types of algorithms for these two problems together with very simple baseline algorithms to determine which of these algorithms are the fastest. Our results indicate that the d…
We put forth a new string matching algorithm which matches the pattern from neither the left nor the right end, instead a special position. Comparing with the Knuth-Morris-Pratt algorithm and the Boyer-Moore algorithm, the new algorithm is more flexible to pick the position for starting comparisons. The option really brings it a saving in cost.
Huang et al.~(STOC 2018) introduced the fully online matching problem, a generalization of the classic online bipartite matching problem in that it allows all vertices to arrive online and considers general graphs. They showed that the ranking algorithm by Karp et al.~(STOC 1990) is strictly better than $0.5$-competitive and the problem is strictly harder than the online bipartite matching problem in that no algorithms can be $(1-1/e)$-competitive. This paper pins down two tight competitive ratios of classic algorithms for the fully online matching problem. For the fractional version of the pr…
In this paper we analyze the expected time complexity of the auction algorithm for the matching problem on random bipartite graphs. We prove that the expected time complexity of the auction algorithm for bipartite matching is $O\left(\frac{N\log^2(N)}{\log\left(Np\right)}\right)$ on sequential machines. This is equivalent to other augmenting path algorithms such as the HK algorithm. Furthermore, we show that the algorithm can be implemented on parallel machines with $O(\log(N))$ processors and shared memory with an expected time complexity of $O(N\log(N))$.
Given a text $T$ and a pattern $P$, the order-preserving matching problem is to find all substrings in $T$ which have the same relative orders as $P$. Order-preserving matching has been an active research area since it was introduced by Kubica et al. \cite{kubica2013linear} and Kim et al. \cite{kim2014order}. In this paper we present two algorithms for the multiple order-preserving matching problem, one of which runs in sublinear time on average and the other in linear time on average. Both algorithms run much faster than the previous algorithms.
Cartesian tree matching is the problem of finding all substrings in a given text which have the same Cartesian trees as that of a given pattern. In this paper, we deal with Cartesian tree matching for the case of multiple patterns. We present two fingerprinting methods, i.e., the parent-distance encoding and the binary encoding. By combining an efficient fingerprinting method and a conventional multiple string matching algorithm, we can efficiently solve multiple pattern Cartesian tree matching. We propose three practical algorithms for multiple pattern Cartesian tree matching based on the Wu-…
The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1/2-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned …
Given a pattern x of length m and a text y of length n, both over an ordered alphabet, the order-preserving pattern matching problem consists in finding all substrings of the text with the same relative order as the pattern. It is an approximate variant of the well known exact pattern matching problem which has gained attention in recent years. This interesting problem finds applications in a lot of fields as time series analysis, like share prices on stock markets, weather data analysis or to musical melody matching. In this paper we present two new filtering approaches which turn out to be m…
An $(f,g)$-semi-matching in a bipartite graph $G=(U \cup V,E)$ is a set of edges $M \subseteq E$ such that each vertex $u\in U$ is incident with at most $f(u)$ edges of $M$, and each vertex $v\in V$ is incident with at most $g(v)$ edges of $M$. In this paper we give an algorithm that for a graph with $n$ vertices and $m$ edges, $n\le m$, constructs a maximum $(f,g)$-semi-matching in running time $O(m\cdot \min (\sqrt{\sum_{u\in U}f(u)}, \sqrt{\sum_{v\in V}g(v)}))$. Using the reduction of [5], our result on maximum $(f,g)$-semi-matching problem directly implies an algorithm for the optimal semi…
We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in $O(m\sqrt{n}\log(nN))$ time, $O(m\sqrt{n})$ per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs. Here $m,n,$ and $N$ bound the number of edges, vertices, and magnitude of any edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in $O(m\sqrt{n\log nα(m,n)} \log(nN))$ time.
The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-Ω(1))$-approximate algorithm with $O(m^{3/8})=O(n^…
We study the problem of finding "fair" stable matchings in the Stable Marriage problem with Incomplete lists (SMI). For an instance $I$ of SMI there may be many stable matchings, providing significantly different outcomes for the sets of men and women. We introduce two new notions of fairness in SMI. Firstly, a regret-equal stable matching minimises the difference in ranks of a worst-off man and a worst-off woman, among all stable matchings. Secondly, a min-regret sum stable matching minimises the sum of ranks of a worst-off man and a worst-off woman, among all stable matchings. We present two…
Motivated by applications in online dating and kidney exchange, the stochastic matching problem was introduced by Chen, Immorlica, Karlin, Mahdian and Rudra (2009). They have proven a 4-approximation of a simple greedy strategy, but conjectured that it is in fact a 2-approximation. In this paper we confirm this hypothesis.
We present a new dynamic matching sparsification scheme. From this scheme we derive a framework for dynamically rounding fractional matchings against \emph{adaptive adversaries}. Plugging in known dynamic fractional matching algorithms into our framework, we obtain numerous randomized dynamic matching algorithms which work against adaptive adversaries (the first such algorithms, as all previous randomized algorithms for this problem assumed an \emph{oblivious} adversary). In particular, for any constant $ε>0$, our framework yields $(2+ε)$-approximate algorithms with constant update time or pol…
An algorithm is presented which produces the minimum cost bipartite matching between two sets of M points each, where the cost of matching two points is proportional to the minimum distance by which a particle could reach one point from the other while constrained to travel on a connected set of curves, or roads. Given any such roadmap, the algorithm obtains O(M log M) total runtime in terms of M, which is the best possible bound in the sense that any algorithm for minimal matching has runtime Omega(M log M). The algorithm is strongly polynomial and is based on a capacity-scaling approach to t…
We study a natural generalization of stable matching to the maximum weight stable matching problem and we obtain a combinatorial polynomial time algorithm for it by reducing it to the problem of finding a maximum weight ideal cut in a DAG. We give the first polynomial time algorithm for the latter problem; this algorithm is also combinatorial. The combinatorial nature of our algorithms not only means that they are efficient but also that they enable us to obtain additional structural and algorithmic results: - We show that the set, $\cal M'$, of maximum weight stable matchings forms a sublatti…
Given a text $T$ and a pattern $P$ over alphabet $Σ$, the classic exact matching problem searches for all occurrences of pattern $P$ in text $T$. Unlike exact matching problem, order-preserving pattern matching (OPPM) considers the relative order of elements, rather than their real values. In this paper, we propose an efficient algorithm for OPPM problem using the "duel-and-sweep" paradigm. Our algorithm runs in $O(n + m\log m)$ time in general and $O(n + m)$ time under an assumption that the characters in a string can be sorted in linear time with respect to the string size. We also perform e…
Finding optimal matchings in dense graphs is of general interest and of particular importance in social, transportation and biological networks. While developing optimal solutions for various matching problems is important, the running times of the fastest available optimal matching algorithms are too costly. However, when the vertices of the graphs are point-sets in $R^d$ and edge weights correspond to the euclidean distances, the available optimal matching algorithms are substantially faster. In this paper, we propose a novel network embedding based heuristic algorithm to solve various match…
We show that the pseudoflow algorithm for maximum flow is particularly efficient for the bipartite matching problem both in theory and in practice. We develop several implementations of the pseudoflow algorithm for bipartite matching, and compare them over a wide set of benchmark instances to state-of-the-art implementations of push-relabel and augmenting path algorithms that are specifically designed to solve these problems. The experiments show that the pseudoflow variants are in most cases faster than the other algorithms. We also show that one particular implementation---the matching pseud…
In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algori…
The pattern matching problem with swaps is to find all occurrences of a pattern in a text while allowing the pattern to swap adjacent symbols. The goal is to design fast matching algorithm that takes advantage of the bit parallelism of bitwise machine instructions and has only streaming access to the input. We introduce a new approach to solve this problem based on the graph theoretic model and compare its performance to previously known algorithms. We also show that an approach using deterministic finite automata cannot achieve similarly efficient algorithms. Furthermore, we describe a fatal …
Karp, Vazirani, and Vazirani (STOC 1990) initiated the study of online bipartite matching, which has held a central role in online algorithms ever since. Of particular importance are the Ranking algorithm for integral matching and the Water-filling algorithm for fractional matching. Most algorithms in the literature can be viewed as adaptations of these two in the corresponding models. Recently, Huang et al.~(STOC 2018, SODA 2019) introduced a more general model called \emph{fully online matching}, which considers general graphs and allows all vertices to arrive online. They also generalized R…
In this work, we study longest common substring, pattern matching, and wildcard pattern matching in the asymmetric streaming model. In this streaming model, we have random access to one string and streaming access to the other one. We present streaming algorithms with provable guarantees for these three fundamental problems. In particular, our algorithms for pattern matching improve the upper bound and beat the unconditional lower bounds on the memory of randomized and deterministic streaming algorithms. In addition to this, we present algorithms for wildcard pattern matching in the asymmetric…
We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a $(2-δ)$-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a $(1+δ)$-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintai…
Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph problems. For general m-edge and n-vertex graphs, it is well-known to be solvable in $O(m \sqrt{n})$ time. We develop a linear-time algorithm to find maximum-cardinality matchings on cocomparability graphs, a prominent subclass of perfect graphs that contains interval graphs as well as permutation graphs. Our algorithm is based on the recently discovered Lexicographic Depth First Search (LDFS).
Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P denotes a set of posts and ranks on the edges denote preferences of the agents over posts. A matching M in G is rank-maximal if it matches the maximum number of applicants to their top-rank post, subject to this, the maximum number of applicants to their second rank post and so on. In this paper, we develop a switching graph characterization of rank-maximal matchings, which is a useful tool that encodes all rank-maximal matchings in an instance. The characterization leads to simple and efficient algorithms for several i…
We investigate the gap between theory and practice for exact branching algorithms. In theory, branch-and-reduce algorithms currently have the best time complexity for numerous important problems. On the other hand, in practice, state-of-the-art methods are based on different approaches, and the empirical efficiency of such theoretical algorithms have seldom been investigated probably because they are seemingly inefficient because of the plethora of complex reduction rules. In this paper, we design a branch-and-reduce algorithm for the vertex cover problem using the techniques developed for the…
We consider the maximum vertex-weighted matching problem (MVM), in which non-negative weights are assigned to the vertices of a graph, the weight of a matching is the sum of the weights of the matched vertices, and we are required to compute a matching of maximum weight. We describe an exact algorithm for MVM with $O(|V|\, |E|)$ time complexity, and then we design a $2/3$-approximation algorithm for MVM on bipartite graphs by restricting the length of augmenting paths to at most three. The latter algorithm has time complexity $O(|E| + |V| \log |V|)$. The approximation algorithm solves two MVM …
Cartesian tree matching is the problem of finding all substrings of a given text which have the same Cartesian trees as that of a given pattern. So far there is one linear-time solution for Cartesian tree matching, which is based on the KMP algorithm. We improve the running time of the previous solution by introducing new representations. We present the framework of a binary filtration method and an efficient verification technique for Cartesian tree matching. Any exact string matching algorithm can be used as a filtration for Cartesian tree matching on our framework. We also present a SIMD so…
Several papers have achieved time $O(\sqrt n m)$ for cardinality matching, starting from first principles. This results in a long derivation. We simplify the task by employing well-known concepts for maximum weight matching. We use Edmonds' algorithm to derive the structure of shortest augmenting paths. We extend this to a complete algorithm for maximum cardinality matching in time $O(\sqrt n m)$.
Online bipartite matching has been extensively studied. In the unweighted setting, Karp et al. gave an optimal $(1 - 1/e)$-competitive randomized algorithm. In the weighted setting, optimal algorithms have been achieved only under assumptions on the edge weights. For the general case, little was known beyond the trivial $1/2$-competitive greedy algorithm. Recently, Fahrbach et al. have presented an 0.5086-competitive algorithm (for the problem in a model, namely free-disposal), overcoming the long-standing barrier of 1/2. Besides, in designing competitive algorithms for the online matching pro…
Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in bipartite graphs and in $O( (\frac{1}ε)^{\frac{1}ε})$ or $O(\text{poly} (\frac{1}ε) \cdot \log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for ex…
Given a pattern $w$ and a text $t$, the speed of a pattern matching algorithm over $t$ with regard to $w$, is the ratio of the length of $t$ to the number of text accesses performed to search $w$ into $t$. We first propose a general method for computing the limit of the expected speed of pattern matching algorithms, with regard to $w$, over iid texts. Next, we show how to determine the greatest speed which can be achieved among a large class of algorithms, altogether with an algorithm running this speed. Since the complexity of this determination make it impossible to deal with patterns of len…
Estimating the size of the maximum matching is a canonical problem in graph algorithms, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. * Insert-Only Streams: We present a one-pass algorithm that takes O(c log^2 n) space and approximates the size of the maximum matching in graphs with arboricity c within a factor of O(c). This improves significantly on the state-of-the-art O~(cn^{2/3})-space streaming algorithms. * Dynamic S…
Given a set of patterns called a dictionary and a text, the dictionary matching problem is a task to find all occurrence positions of all patterns in the text. The dictionary matching problem can be solved efficiently by using the Aho-Corasick algorithm. Recently, Matsuoka et al. [TCS, 2016] proposed a generalization of pattern matching problem under substring consistent equivalence relations and presented a generalization of the Knuth-Morris-Pratt algorithm to solve this problem. An equivalence relation $\approx$ is a substring consistent equivalence relation (SCER) if for two strings $X,Y$, …
The popular matching problem is of matching a set of applicants to a set of posts, where each applicant has a preference list, ranking a non-empty subset of posts in the order of preference, possibly with ties. A matching M is popular if there is no other matching M' such that more applicants prefer M' to M. We give the first NC algorithm to solve the popular matching problem without ties. We also give an NC algorithm that solves the maximum-cardinality popular matching problem. No NC or RNC algorithms were known for the matching problem in preference systems prior to this work. Moreover, we g…
This article identifies a key algorithmic ingredient in the edge-weighted online matching algorithm by Zadimoghaddam (2017) and presents a simplified algorithm and its analysis to demonstrate how it works in the unweighted case.
Maximum weight matching is one of the most fundamental combinatorial optimization problems with a wide range of applications in data mining and bioinformatics. Developing distributed weighted matching algorithms is challenging due to the sequential nature of efficient algorithms for this problem. In this paper, we develop a simple distributed algorithm for the problem on general graphs with approximation guarantee of $2+\varepsilon$ that (nearly) matches that of the sequential greedy algorithm. A key advantage of this algorithm is that it can be easily implemented in only two rounds of computa…
String matching is a fundamental problem in algorithm. This study examines the development and construction of two reversible string-matching algorithms: a naive string-matching algorithm and the Rabin-Karp algorithm. The algorithms are used to introduce reversible computing concepts, beginning from basic reversible programming techniques to more advanced considerations about the injectivization of the polynomial hash-update function employed by the Rabin-Karp algorithm. The results are two clean input-preserving reversible algorithms that require no additional space and have the same asymptot…
Pattern matching is a fundamental process in almost every scientific domain. The problem involves finding the positions of a given pattern (usually of short length) in a reference stream of data (usually of large length). The matching can be an exact or as an approximate (inexact). Exact matching is to search for the pattern without allowing for mismatches (or insertions and deletions) of one or more characters in the pattern), while approximate matching is the opposite. For exact matching, several data structures that can be built in linear time and space are used and in practice nowadays. Fo…
We design quantum algorithms for maximum matching. Working in the query model, in both adjacency matrix and adjacency list settings, we improve on the best known algorithms for general graphs, matching previously obtained results for bipartite graphs. In particular, for a graph with $n$ nodes and $m$ edges, our algorithm makes $O(n^{7/4})$ queries in the matrix model and $O(n^{3/4}(m+n)^{1/2})$ queries in the list model. Our approach combines Gabow's classical maximum matching algorithm [Gabow, Fundamenta Informaticae, '17] with the guessing tree method of Beigi and Taghavi [Beigi and Taghavi,…
We present a new algorithm which detects the maximal possible number of matched disjoint pairs satisfying a given caliper when a bipartite matching is done with respect to a scalar index (e.g., propensity score), and constructs a corresponding matching. Variable width calipers are compatible with the technique, provided that the width of the caliper is a Lipschitz function of the index. If the observations are ordered with respect to the index then the matching needs $O(N)$ operations, where $N$ is the total number of subjects to be matched. The case of 1-to-$n$ matching is also considered. We…
We present a $\frac53$-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A $\frac74$-approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a $\frac{7}{4}$-approximation algorithm," {\em Math. Program.}, 182(1):315--354, 2020; arXiv:1810.07816. Our improvement is based on new algorithmic techniques, and some of these may lead to ad…
Graphs are widely used to model complicated data semantics in many application domains. In this paper, two novel and efficient algorithms Fast-ON and Fast-P are proposed for solving the subgraph isomorphism problem. The two algorithms are based on Ullman algorithm [Ullmann 1976], apply vertex-at-a-time matching manner and path-at-a-time matching manner respectively, and use effective heuristics to cut the search space. Comparing to the well-known algorithms, Fast-ON and Fast-P achieve up to 1-4 orders of magnitude speed-up for both dense and sparse graph data.
This paper presents an $O(\log\log \bar{d})$ round massively parallel algorithm for $1+ε$ approximation of maximum weighted $b$-matchings, using near-linear memory per machine. Here $\bar{d}$ denotes the average degree in the graph and $ε$ is an arbitrarily small positive constant. Recall that $b$-matching is the natural and well-studied generalization of the matching problem where different vertices are allowed to have multiple (and differing number of) incident edges in the matching. Concretely, each vertex $v$ is given a positive integer budget $b_v$ and it can have up to $b_v$ incident edg…
We propose a framework for the exact probabilistic analysis of window-based pattern matching algorithms, such as Boyer-Moore, Horspool, Backward DAWG Matching, Backward Oracle Matching, and more. In particular, we show how to efficiently obtain the distribution of such an algorithm's running time cost for any given pattern in a random text model, which can be quite general, from simple uniform models to higher-order Markov models or hidden Markov models (HMMs). Furthermore, we provide a technique to compute the exact distribution of \emph{differences} in running time cost of two algorithms. In…
