Let G be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in O(|V |W /k(|V |, W /N))-time by employing an algorithm designed by Feder and Motwani as a subroutine, where |V | and N denote the number of nodes and the maximum edge weight of G, respectively and k(x, y) = log x/ log(x 2 /y). The parameter W is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case W = O(|E|) where |E| be the number of edges of G and in worst case W = W, that is, |E| ≤ W ≤ W. In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter W. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.
{"title":"A Modified Decomposition Algorithm for Maximum Weight Bipartite Matching and Its Experimental Evaluation","authors":"Shibsankar Das","doi":"10.7561/SACS.2020.1.39","DOIUrl":"https://doi.org/10.7561/SACS.2020.1.39","url":null,"abstract":"Let G be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in O(|V |W /k(|V |, W /N))-time by employing an algorithm designed by Feder and Motwani as a subroutine, where |V | and N denote the number of nodes and the maximum edge weight of G, respectively and k(x, y) = log x/ log(x 2 /y). The parameter W is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case W = O(|E|) where |E| be the number of edges of G and in worst case W = W, that is, |E| ≤ W ≤ W. In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter W. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125413687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide an axiomatisation for =pTr , a variant of probabilistic trace equivalence as formulated by Bernardo et al., 2014, in the setting of the alternating model of Hansson. The equivalence considers traces individually instead of trace distributions. We show that our axiomatisation is sound and also complete for recursion-free sequential processes. Due to the nature of the trace equivalence, the axiomatisation is particularly complex.
{"title":"A Complete Axiomatisation for Probabilistic Trace Equivalence","authors":"Ferry Timmers, J. F. Groote","doi":"10.7561/SACS.2020.1.69","DOIUrl":"https://doi.org/10.7561/SACS.2020.1.69","url":null,"abstract":"We provide an axiomatisation for =pTr , a variant of probabilistic trace equivalence as formulated by Bernardo et al., 2014, in the setting of the alternating model of Hansson. The equivalence considers traces individually instead of trace distributions. We show that our axiomatisation is sound and also complete for recursion-free sequential processes. Due to the nature of the trace equivalence, the axiomatisation is particularly complex.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127767607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of computing a maximum weight matching in a bipartite graph is one of the fundamental algorithmic problems that has played an important role in the development of combinatorial optimization and algorithmics. Let Gw,σ is a collection of all weighted bipartite graphs, each having σ and w as the size of each of the non-empty subset of the vertex partition and the total weight of the graph, respectively. We give a tight lower bound dw−σ σ e + 1 for the set {Wt(mwm(G)) | G ∈ Gw,σ} which denotes the collection of weights of maximum weight bipartite matchings of all the graphs in Gw,σ.
二部图中最大权值匹配的计算问题是组合优化和算法发展中重要的基本算法问题之一。设Gw，σ是所有加权二部图的集合，每个图的顶点划分的非空子集和图的总权值分别为σ和w。我们给出了集合{Wt(mwm(G)) | G∈Gw，σ}的一个紧下界dw−σ σ e + 1，它表示Gw，σ中所有图的最大权值二部匹配的权值集合。
{"title":"An Optimum Lower Bound for the Weights of Maximum Weight Matching in Bipartite Graphs","authors":"Shibsankar Das","doi":"10.7561/SACS.2020.1.25","DOIUrl":"https://doi.org/10.7561/SACS.2020.1.25","url":null,"abstract":"The problem of computing a maximum weight matching in a bipartite graph is one of the fundamental algorithmic problems that has played an important role in the development of combinatorial optimization and algorithmics. Let Gw,σ is a collection of all weighted bipartite graphs, each having σ and w as the size of each of the non-empty subset of the vertex partition and the total weight of the graph, respectively. We give a tight lower bound dw−σ σ e + 1 for the set {Wt(mwm(G)) | G ∈ Gw,σ} which denotes the collection of weights of maximum weight bipartite matchings of all the graphs in Gw,σ.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114984493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced.Three examples of such specifications are given. A preference is formulated for a specification by means of infinitely many equations which can be presented via a finite number of so-called schematic equations phrased in terms of an infinite signature. On the basis of the latter specification three topics are discussed: (i) fracterm decomposition operators and the numerator paradox, (ii) foundational specifications of arithmetical datatypes, and (iii) poly-infix operations.
{"title":"Most General Algebraic Specifications for an Abstract Datatype of Rational Numbers","authors":"J. Bergstra","doi":"10.7561/SACS.2020.1.1","DOIUrl":"https://doi.org/10.7561/SACS.2020.1.1","url":null,"abstract":"The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced.Three examples of such specifications are given. A preference is formulated for a specification by means of infinitely many equations which can be presented via a finite number of so-called schematic equations phrased in terms of an infinite signature. On the basis of the latter specification three topics are discussed: (i) fracterm decomposition operators and the numerator paradox, (ii) foundational specifications of arithmetical datatypes, and (iii) poly-infix operations.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133089670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of identifying tandem scattered subsequences within a string. Our algorithm identifies a longest subsequence which occurs twice without overlap in a string. This algorithm is based on the Hunt-Szymanski algorithm, therefore its performance improves if the string is not self similar, which occurs naturally on strings over large alphabets. Our algorithm relies on new results for data structures that support dynamic longest increasing sub-sequences. In the process we also obtain improved algorithms for the decremental string comparison problem.
{"title":"Small Longest Tandem Scattered Subsequences","authors":"L. Russo, Alexandre P. Francisco","doi":"10.7561/SACS.2021.1.79","DOIUrl":"https://doi.org/10.7561/SACS.2021.1.79","url":null,"abstract":"We consider the problem of identifying tandem scattered subsequences within a string. Our algorithm identifies a longest subsequence which occurs twice without overlap in a string. This algorithm is based on the Hunt-Szymanski algorithm, therefore its performance improves if the string is not self similar, which occurs naturally on strings over large alphabets. Our algorithm relies on new results for data structures that support dynamic longest increasing sub-sequences. In the process we also obtain improved algorithms for the decremental string comparison problem.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"140 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125760391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A stay point of a moving entity is a region in which it spends a significant amount of time. In this paper, we identify all stay points of an entity in a certain time interval, where the entity is allowed to leave the region but it should return within a given time limit. This definition of stay points seems more natural in many applications of trajectory analysis than those that do not limit the time of entity’s absence from the region. We present an O(n log n) algorithm for trajectories in R with n vertices and a (1 + )-approximation algorithm for trajectories in R to identify all such stay points. Our algorithm runs in O(kn), where k depends on and the ratio of the duration of the trajectory to the allowed gap time. We also present an algorithm to answer stay point queries in logarithmic time, after an O(kn log n) time preprocessing.
{"title":"Identifying and Querying Regularly Visited Places","authors":"A. Rudi","doi":"10.7561/sacs.2019.2.185","DOIUrl":"https://doi.org/10.7561/sacs.2019.2.185","url":null,"abstract":"A stay point of a moving entity is a region in which it spends a significant amount of time. In this paper, we identify all stay points of an entity in a certain time interval, where the entity is allowed to leave the region but it should return within a given time limit. This definition of stay points seems more natural in many applications of trajectory analysis than those that do not limit the time of entity’s absence from the region. We present an O(n log n) algorithm for trajectories in R with n vertices and a (1 + )-approximation algorithm for trajectories in R to identify all such stay points. Our algorithm runs in O(kn), where k depends on and the ratio of the duration of the trajectory to the allowed gap time. We also present an algorithm to answer stay point queries in logarithmic time, after an O(kn log n) time preprocessing.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122398046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We first present a probabilistic version of ACP that rests on the principle that probabilistic choices are always resolved before choices involved in alternative composition and parallel composition are resolved and then extend this probabilistic version of ACP with a form of interleaving in which parallel processes are interleaved according to what is known as a process-scheduling policy in the field of operating systems. We use the term strategic interleaving for this more constrained form of interleaving. The extension covers probabilistic process-scheduling policies.
{"title":"Probabilistic process algebra and strategic interleaving","authors":"K. Middelburg","doi":"10.7561/sacs.2020.2.205","DOIUrl":"https://doi.org/10.7561/sacs.2020.2.205","url":null,"abstract":"We first present a probabilistic version of ACP that rests on the principle that probabilistic choices are always resolved before choices involved in alternative composition and parallel composition are resolved and then extend this probabilistic version of ACP with a form of interleaving in which parallel processes are interleaved according to what is known as a process-scheduling policy in the field of operating systems. We use the term strategic interleaving for this more constrained form of interleaving. The extension covers probabilistic process-scheduling policies.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127464957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kleene algebra with tests (KAT) was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete transitions guarded by Boolean tests. This paper introduces two generalisations of this structure able to express programs as weighted transitions and tests with outcomes in non necessarily bivalent truth spaces: graded Kleene algebra with tests (GKAT) and a variant where tests are also idempotent (I-GKAT). On this context, and in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT [22], we discuss the encoding of a graded PHL in I-GKAT and of its while-free fragment in GKAT. Moreover, to establish semantics for these structures four new algebras are defined: FSET(T), FREL(K,T) and FLANG(K,T) over complete residuated lattices K and T, and M(n,A) over a GKAT or I-GKAT A. As a final exercise, the paper discusses some program equivalence proofs in a graded context.
{"title":"Generalising KAT to verify weighted computations","authors":"Leandro Gomes, A. Madeira, L. Barbosa","doi":"10.7561/sacs.2019.2.141","DOIUrl":"https://doi.org/10.7561/sacs.2019.2.141","url":null,"abstract":"Kleene algebra with tests (KAT) was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete transitions guarded by Boolean tests. This paper introduces two generalisations of this structure able to express programs as weighted transitions and tests with outcomes in non necessarily bivalent truth spaces: graded Kleene algebra with tests (GKAT) and a variant where tests are also idempotent (I-GKAT). On this context, and in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT [22], we discuss the encoding of a graded PHL in I-GKAT and of its while-free fragment in GKAT. Moreover, to establish semantics for these structures four new algebras are defined: FSET(T), FREL(K,T) and FLANG(K,T) over complete residuated lattices K and T, and M(n,A) over a GKAT or I-GKAT A. As a final exercise, the paper discusses some program equivalence proofs in a graded context.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127558564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the problem of reporting all maximal collinear subsets of a point set S in R for d ≥ 3. An algorithm for this problem can be used to detect if any three of the points are collinear or find the line that intersects the most points in S. Besides, obtaining such maximal subsets is necessary for some problems about the collinearity relation among points, such as when covering them with the fewest lines. We present practical algorithms to find all maximal collinear subsets of a set of n points, including one with space complexity O(n) and time complexity O(dn log n), and one with space complexity O(n) and time complexity O(dn).
{"title":"Enumerating Collinear Points in Higher Dimensions","authors":"A. Rudi, R. A. Rufai","doi":"10.7561/SACS.2019.1.81","DOIUrl":"https://doi.org/10.7561/SACS.2019.1.81","url":null,"abstract":"In this paper, we study the problem of reporting all maximal collinear subsets of a point set S in R for d ≥ 3. An algorithm for this problem can be used to detect if any three of the points are collinear or find the line that intersects the most points in S. Besides, obtaining such maximal subsets is necessary for some problems about the collinearity relation among points, such as when covering them with the fewest lines. We present practical algorithms to find all maximal collinear subsets of a set of n points, including one with space complexity O(n) and time complexity O(dn log n), and one with space complexity O(n) and time complexity O(dn).","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124072527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce and investigate weighted context-free grammars over an arbitrary bimonoid K. Thus, we do not assume that the operations of K are commutative or idempotent or they distribute over each other. We prove a Chomsky-Schützenberger type theorem for the series generated by our grammars. Moreover, we show that the class of series generated by weighted right-linear grammars over a linearly ordered alphabet Σ and K coincides with that of recognizable series over Σ and K.
{"title":"Weighted Context-Free Grammars Over Bimonoids","authors":"George Rahonis, Faidra Torpari","doi":"10.7561/SACS.2019.1.59","DOIUrl":"https://doi.org/10.7561/SACS.2019.1.59","url":null,"abstract":"We introduce and investigate weighted context-free grammars over an arbitrary bimonoid K. Thus, we do not assume that the operations of K are commutative or idempotent or they distribute over each other. We prove a Chomsky-Schützenberger type theorem for the series generated by our grammars. Moreover, we show that the class of series generated by weighted right-linear grammars over a linearly ordered alphabet Σ and K coincides with that of recognizable series over Σ and K.","PeriodicalId":394919,"journal":{"name":"Sci. Ann. Comput. Sci.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122984270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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