Pub Date : 2023-07-17DOI: 10.1142/s0129054123480039
Zoltán Fülöp, H. Vogler
We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.
{"title":"A Comparison of Sets of Recognizable Weighted Tree Languages Over Specific Sets of Bounded Lattices","authors":"Zoltán Fülöp, H. Vogler","doi":"10.1142/s0129054123480039","DOIUrl":"https://doi.org/10.1142/s0129054123480039","url":null,"abstract":"We consider weighted tree automata (wta) over bounded lattices, locally finite bounded lattices, distributive bounded lattices, and finite chains and we consider the weighted tree languages defined by their run semantics and by their initial algebra semantics. Due to these eight combinations, there are eight sets of weighted tree languages. Moreover, we consider the four sets of weighted tree languages which are recognized by crisp-deterministic wta over the above mentioned sets of weight algebras. We give all inclusion relations among all these twelve sets of weighted tree languages. More precisely, we prove that eleven of the twelve sets are equal, and this set is a proper subset of the set of weighted tree languages recognized by wta over bounded lattices with initial algebra semantics.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49289430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-04DOI: 10.1142/s0129054123460024
Lucie Ciencialová, Ludek Cienciala, E. Csuhaj-Varjú
Reaction systems are well-known formal models of interactions between biochemical reactions. A reaction system is a finite set of triples (reactants, inhibitors, products) that represent chemical reactions, where the reactants, the inhibitors, and the products are objects corresponding to the chemicals. The reactions may facilitate or inhibit each other. A distributed reaction system consists of a finite set of reaction systems that interact with their environment (function in a given context). The environment is a finite set of reactants provided by a context automaton. In the preceding paper, we studied distributed reaction systems where in each step, the context automaton provided a separate set of reactants to the component reaction systems. We assigned languages to these distributed reaction systems and provided representations of some well-known language classes by these constructs. In this paper, the context is provided for the whole distributed reaction system and the component reaction systems distribute the context among each other in different ways (the same context is valid for each component, or the context is split among the components). As in the preceding paper, we assign languages to these new types of distributed reaction systems and provide representations of well-known language classes (the class of right-linear simple matrix languages, the recursively enumerable language class).
{"title":"Language Classes of Extended Distributed Reaction Systems","authors":"Lucie Ciencialová, Ludek Cienciala, E. Csuhaj-Varjú","doi":"10.1142/s0129054123460024","DOIUrl":"https://doi.org/10.1142/s0129054123460024","url":null,"abstract":"Reaction systems are well-known formal models of interactions between biochemical reactions. A reaction system is a finite set of triples (reactants, inhibitors, products) that represent chemical reactions, where the reactants, the inhibitors, and the products are objects corresponding to the chemicals. The reactions may facilitate or inhibit each other. A distributed reaction system consists of a finite set of reaction systems that interact with their environment (function in a given context). The environment is a finite set of reactants provided by a context automaton. In the preceding paper, we studied distributed reaction systems where in each step, the context automaton provided a separate set of reactants to the component reaction systems. We assigned languages to these distributed reaction systems and provided representations of some well-known language classes by these constructs. In this paper, the context is provided for the whole distributed reaction system and the component reaction systems distribute the context among each other in different ways (the same context is valid for each component, or the context is split among the components). As in the preceding paper, we assign languages to these new types of distributed reaction systems and provide representations of well-known language classes (the class of right-linear simple matrix languages, the recursively enumerable language class).","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49394867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-30DOI: 10.1142/s0129054123420066
Kazuo Iwama, Shuichi Miyazaki
This paper has two objectives. One is to give a linear time algorithm that solves the stable roommates problem (i.e., obtains one stable matching) using the stable marriage problem. The idea is that a stable matching of a roommate instance [Formula: see text] is a stable matching (that however must satisfy a certain condition) of some marriage instance [Formula: see text]. [Formula: see text] is obtained just by making two copies of [Formula: see text], one for the men’s table and the other for the women’s table. The second objective is to investigate the possibility of reducing the roommate problem to the marriage problem (with one-to-one correspondence between their stable matchings) in polynomial time. For a given [Formula: see text], we construct the rotation POSET [Formula: see text] of [Formula: see text] and then we “halve” it to obtain [Formula: see text], by which we can forget the above condition and can use all the closed subsets of [Formula: see text] for all the stable matchings of [Formula: see text]. Unfortunately this approach works (runs in polynomial time) only for restricted instances.
本文有两个目的。一是给出一个线性时间算法，利用稳定婚姻问题求解稳定室友问题(即得到一个稳定匹配)。其思想是，室友实例的稳定匹配[公式:见文本]是某些婚姻实例的稳定匹配(但必须满足特定条件)[公式:见文本]。只需将[公式:见文]复制两份即可得到[公式:见文]，一份用于男子牌桌，另一份用于女子牌桌。第二个目标是研究在多项式时间内将室友问题简化为婚姻问题的可能性(他们的稳定匹配之间有一对一的对应关系)。对于给定的[Formula: see text]，我们构造[Formula: see text]的旋转POSET [Formula: see text]，然后将其“对半”得到[Formula: see text]，这样我们就可以忽略上述条件，并且可以使用[Formula: see text]的所有闭子集来进行[Formula: see text]的所有稳定匹配。不幸的是，这种方法只适用于有限的实例(在多项式时间内运行)。
{"title":"Marriage and Roommate","authors":"Kazuo Iwama, Shuichi Miyazaki","doi":"10.1142/s0129054123420066","DOIUrl":"https://doi.org/10.1142/s0129054123420066","url":null,"abstract":"This paper has two objectives. One is to give a linear time algorithm that solves the stable roommates problem (i.e., obtains one stable matching) using the stable marriage problem. The idea is that a stable matching of a roommate instance [Formula: see text] is a stable matching (that however must satisfy a certain condition) of some marriage instance [Formula: see text]. [Formula: see text] is obtained just by making two copies of [Formula: see text], one for the men’s table and the other for the women’s table. The second objective is to investigate the possibility of reducing the roommate problem to the marriage problem (with one-to-one correspondence between their stable matchings) in polynomial time. For a given [Formula: see text], we construct the rotation POSET [Formula: see text] of [Formula: see text] and then we “halve” it to obtain [Formula: see text], by which we can forget the above condition and can use all the closed subsets of [Formula: see text] for all the stable matchings of [Formula: see text]. Unfortunately this approach works (runs in polynomial time) only for restricted instances.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136016632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-28DOI: 10.1142/s0129054123480015
Christof Löding, W. Thomas
The class of Boolean combinations of tree languages recognized by deterministic top-down tree automata (also known as deterministic root-to-frontier automata) is studied. The problem of determining for a given regular tree language whether it belongs to this class is open. We provide some progress by two results: First, a characterization of this class by a natural extension of deterministic top-down tree automata is presented, and as an application we obtain a convenient method to show that certain regular tree languages are outside this class. In the second result, it is shown that, for fixed [Formula: see text], it is decidable whether a regular tree language is a Boolean combination of [Formula: see text] tree languages recognized by deterministic top-down tree automata.
{"title":"On the Boolean Closure of Deterministic Top-Down Tree Automata","authors":"Christof Löding, W. Thomas","doi":"10.1142/s0129054123480015","DOIUrl":"https://doi.org/10.1142/s0129054123480015","url":null,"abstract":"The class of Boolean combinations of tree languages recognized by deterministic top-down tree automata (also known as deterministic root-to-frontier automata) is studied. The problem of determining for a given regular tree language whether it belongs to this class is open. We provide some progress by two results: First, a characterization of this class by a natural extension of deterministic top-down tree automata is presented, and as an application we obtain a convenient method to show that certain regular tree languages are outside this class. In the second result, it is shown that, for fixed [Formula: see text], it is decidable whether a regular tree language is a Boolean combination of [Formula: see text] tree languages recognized by deterministic top-down tree automata.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44301054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-28DOI: 10.1142/s0129054123410071
Pooja Goyal, B. S. Panda
A set [Formula: see text] of a graph [Formula: see text] is called a connected power dominating set of [Formula: see text] if [Formula: see text], the subgraph induced by [Formula: see text], is connected and every vertex in the graph can be observed from [Formula: see text], following the two observation rules for power system monitoring: Rule [Formula: see text]: if [Formula: see text], then [Formula: see text] can observe itself and all its neighbors, and Rule [Formula: see text]: for an already observed vertex whose all neighbors except one are observed, then the only unobserved neighbor becomes observed as well. Given a graph [Formula: see text], Minimum Connected Power Domination is to find a connected power dominating set of minimum cardinality of [Formula: see text] and Decide Connected Power Domination is the decision version of Minimum Connected Power Domination. Decide Connected Power Domination is known to be NP -complete for general graphs. In this paper, we prove that Decide Connected Power Domination remains NP -complete for star-convex bipartite graphs, perfect elimination bipartite graphs and split graphs. This answers some open problems posed in [B. Brimkov, D. Mikesell and L. Smith, Connected power domination in graphs, J. Comb. Optim. 38(1) (2019) 292–315]. On the positive side, we show that Minimum Connected Power Domination is polynomial-time solvable for chain graphs, a proper subclass of perfect elimination bipartite graph, and for threshold graphs, a proper subclass of split graphs. Further, we show that Minimum Connected Power Domination cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text], for bipartite graphs as well as for chordal graphs. Finally, we show that Minimum Connected Power Domination is APX -hard for bounded degree graphs.
{"title":"Hardness Results of Connected Power Domination for Bipartite Graphs and Chordal Graphs","authors":"Pooja Goyal, B. S. Panda","doi":"10.1142/s0129054123410071","DOIUrl":"https://doi.org/10.1142/s0129054123410071","url":null,"abstract":"A set [Formula: see text] of a graph [Formula: see text] is called a connected power dominating set of [Formula: see text] if [Formula: see text], the subgraph induced by [Formula: see text], is connected and every vertex in the graph can be observed from [Formula: see text], following the two observation rules for power system monitoring: Rule [Formula: see text]: if [Formula: see text], then [Formula: see text] can observe itself and all its neighbors, and Rule [Formula: see text]: for an already observed vertex whose all neighbors except one are observed, then the only unobserved neighbor becomes observed as well. Given a graph [Formula: see text], Minimum Connected Power Domination is to find a connected power dominating set of minimum cardinality of [Formula: see text] and Decide Connected Power Domination is the decision version of Minimum Connected Power Domination. Decide Connected Power Domination is known to be NP -complete for general graphs. In this paper, we prove that Decide Connected Power Domination remains NP -complete for star-convex bipartite graphs, perfect elimination bipartite graphs and split graphs. This answers some open problems posed in [B. Brimkov, D. Mikesell and L. Smith, Connected power domination in graphs, J. Comb. Optim. 38(1) (2019) 292–315]. On the positive side, we show that Minimum Connected Power Domination is polynomial-time solvable for chain graphs, a proper subclass of perfect elimination bipartite graph, and for threshold graphs, a proper subclass of split graphs. Further, we show that Minimum Connected Power Domination cannot be approximated within [Formula: see text] for any [Formula: see text] unless [Formula: see text], for bipartite graphs as well as for chordal graphs. Finally, we show that Minimum Connected Power Domination is APX -hard for bounded degree graphs.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135260004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-28DOI: 10.1142/s0129054123500089
V. Mkrtchyan, Ojas D. Parekh, K. Subramani
This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.
{"title":"Approximation Algorithms for Partial Vertex Covers in Trees","authors":"V. Mkrtchyan, Ojas D. Parekh, K. Subramani","doi":"10.1142/s0129054123500089","DOIUrl":"https://doi.org/10.1142/s0129054123500089","url":null,"abstract":"This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44076317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-28DOI: 10.1142/s0129054123500144
Zhiwei Wang, Chen Tian, Zhanlin Wang, Yuhang Wang
Robust subgroup multisignature allows any subgroup of signers from a global set to sign a given message on behalf of the whole group, and the individual signatures should be verified before the combination process, which resists poison signature attacks. An emerging application of robust subgroup multisignatures in blockchain is that a qualified subgroup of a global set of users has reached agreement. In the integrated blockchain and edge computing system, the edge server can naturally act as a combiner in multisignatures and help other end devices produce the final aggregate signature. In this paper, we propose a robust subgroup multisignature with one-time public keys in order that has two advantages for solving the signers ordering problem and one-time public key problem simultaneously. Our scheme is a nontrivial extension of Galindo et al.’s robust subgroup multisignature scheme and can be proven unforgeable, robust and chronological in random oracles. Our scheme can also be suitable for the consortium blockchain by adding a noninteractive zero-knowledge (NIZK) proof system for certifying the one-time public keys.
{"title":"Robust Subgroup Multisignature with One-Time Public Keys in Order","authors":"Zhiwei Wang, Chen Tian, Zhanlin Wang, Yuhang Wang","doi":"10.1142/s0129054123500144","DOIUrl":"https://doi.org/10.1142/s0129054123500144","url":null,"abstract":"Robust subgroup multisignature allows any subgroup of signers from a global set to sign a given message on behalf of the whole group, and the individual signatures should be verified before the combination process, which resists poison signature attacks. An emerging application of robust subgroup multisignatures in blockchain is that a qualified subgroup of a global set of users has reached agreement. In the integrated blockchain and edge computing system, the edge server can naturally act as a combiner in multisignatures and help other end devices produce the final aggregate signature. In this paper, we propose a robust subgroup multisignature with one-time public keys in order that has two advantages for solving the signers ordering problem and one-time public key problem simultaneously. Our scheme is a nontrivial extension of Galindo et al.’s robust subgroup multisignature scheme and can be proven unforgeable, robust and chronological in random oracles. Our scheme can also be suitable for the consortium blockchain by adding a noninteractive zero-knowledge (NIZK) proof system for certifying the one-time public keys.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41719676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-23DOI: 10.1142/s0129054123500120
Huifen Ge, Shumin Zhang, Chengfu Ye
The alternating group network [Formula: see text] can be used to model the topology structure of a large-scale parallel network system. In this work, the structure fault tolerance of alternating group networks based on the star and path structures is investigated. For a graph [Formula: see text] and its a connected subgraph [Formula: see text], the [Formula: see text]-structure connectivity [Formula: see text] (resp. [Formula: see text]-substructure connectivity [Formula: see text]) of [Formula: see text] is the cardinality of a minimum family [Formula: see text] whose every element is isomorphic to [Formula: see text] (resp. isomorphic to a subgraph of [Formula: see text]) such that [Formula: see text] is disconnected. Specifically, we determine [Formula: see text] and [Formula: see text] for [Formula: see text].
{"title":"The Structure Fault Tolerance of Alternating Group Networks","authors":"Huifen Ge, Shumin Zhang, Chengfu Ye","doi":"10.1142/s0129054123500120","DOIUrl":"https://doi.org/10.1142/s0129054123500120","url":null,"abstract":"The alternating group network [Formula: see text] can be used to model the topology structure of a large-scale parallel network system. In this work, the structure fault tolerance of alternating group networks based on the star and path structures is investigated. For a graph [Formula: see text] and its a connected subgraph [Formula: see text], the [Formula: see text]-structure connectivity [Formula: see text] (resp. [Formula: see text]-substructure connectivity [Formula: see text]) of [Formula: see text] is the cardinality of a minimum family [Formula: see text] whose every element is isomorphic to [Formula: see text] (resp. isomorphic to a subgraph of [Formula: see text]) such that [Formula: see text] is disconnected. Specifically, we determine [Formula: see text] and [Formula: see text] for [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45661814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-23DOI: 10.1142/s0129054123500107
Baohua Niu, Shuming Zhou, Tao Tian, Qifan Zhang
The fault diameter and wide diameter are commonly used to measure the fault tolerance and transmission delay of interconnection networks beyond traditional diameter. The [Formula: see text]-wide diameter of graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there exist at least [Formula: see text] internally vertex disjoint paths of length at most [Formula: see text] for any two distinct vertices in [Formula: see text]. The [Formula: see text]-fault diameter of graph [Formula: see text], denoted by [Formula: see text], is the maximum diameter of the survival graph obtained by deleting at most [Formula: see text] vertices in [Formula: see text]. The exchanged crossed cube, as a compounded interconnection network denoted by [Formula: see text], holds the desirable properties of both crossed cube and exchanged hypercube, while achieving a better balanced between cost and performance of the parallel computing systems. In this paper, we construct [Formula: see text] internally vertex disjoint paths between any two distinct vertices of [Formula: see text]. Moreover, we determine the upper and lower bounds of [Formula: see text]-wide diameter and [Formula: see text]-fault diameter of [Formula: see text], i.e., [Formula: see text], which shows that the exchanged crossed cube has better efficiency and reliability than that of the exchanged hypercube.
{"title":"The Wide Diameter and Fault Diameter of Exchanged Crossed Cube","authors":"Baohua Niu, Shuming Zhou, Tao Tian, Qifan Zhang","doi":"10.1142/s0129054123500107","DOIUrl":"https://doi.org/10.1142/s0129054123500107","url":null,"abstract":"The fault diameter and wide diameter are commonly used to measure the fault tolerance and transmission delay of interconnection networks beyond traditional diameter. The [Formula: see text]-wide diameter of graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there exist at least [Formula: see text] internally vertex disjoint paths of length at most [Formula: see text] for any two distinct vertices in [Formula: see text]. The [Formula: see text]-fault diameter of graph [Formula: see text], denoted by [Formula: see text], is the maximum diameter of the survival graph obtained by deleting at most [Formula: see text] vertices in [Formula: see text]. The exchanged crossed cube, as a compounded interconnection network denoted by [Formula: see text], holds the desirable properties of both crossed cube and exchanged hypercube, while achieving a better balanced between cost and performance of the parallel computing systems. In this paper, we construct [Formula: see text] internally vertex disjoint paths between any two distinct vertices of [Formula: see text]. Moreover, we determine the upper and lower bounds of [Formula: see text]-wide diameter and [Formula: see text]-fault diameter of [Formula: see text], i.e., [Formula: see text], which shows that the exchanged crossed cube has better efficiency and reliability than that of the exchanged hypercube.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45306627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-23DOI: 10.1142/s0129054123420054
T. Asano
Suppose we are given a graph with nodes characterized by amounts of supplies and demands of multiple commodities. The amounts of commodities stored at nodes (supplies) are given by positive numbers while those of demands at nodes are given by negative numbers. To meet demands we send commodities from nodes to neighbors by using vehicles, one at each node, with some loading capacity moving to and from neighbors. In this paper we adopt a one-way transportation model in which we just send commodities from a node to one of its neighbors along an edge. When we choose one neighbor at each node, we have a set of trips which naturally define a graph such that each connected component has at most one cycle, which is known as a pseudoforest. We present a linear-time algorithm for deciding whether there is a set of trips that meet all demands using one-way multi-commodity transportations on a pseudoforest with node degrees bounded by a constant. Using the algorithm, we first present an efficient algorithm for finding an optimal set of one-way one-commodity trips that minimize the maximum unmet demand on a pseudoforest, and then extend the idea to a multi-commodity problem on a pseudoforest with node degrees bounded by a constant.
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