Pub Date : 2024-07-12DOI: 10.1142/s012905412450014x
Guan-Zhi Chen, Chang-Biau Yang, Yu-Cheng Chang
The longest increasing subsequence (LIS) problem aims to find the subsequence exhibiting an increasing trend in a numeric sequence with the maximum length. In this paper, we generalize the LIS problem to the longest wave subsequence (LWS) problem, which encompasses two versions: LWSt and LWSr. Given a numeric sequence [Formula: see text] of distinct values and a target trend sequence [Formula: see text], the LWSt problem aims to identify the longest subsequence of [Formula: see text] that preserves the trend of the prefix of [Formula: see text]. And, the LWSr problem aims to find the longest subsequence of [Formula: see text] within [Formula: see text] segments, alternating increasing and decreasing subsequences. We propose two efficient algorithms for solving the two versions of the LWS problem. For the LWSt problem, the time complexity of our algorithm is O[Formula: see text], where [Formula: see text] represents the length of the given numeric sequence [Formula: see text]. Additionally, we propose an O[Formula: see text]-time algorithm for solving the LWSr problem. In both algorithms, we utilize the priority queues for the insertion, deletion, and successor operations.
{"title":"The Longest Wave Subsequence Problem: Generalizations of the Longest Increasing Subsequence Problem","authors":"Guan-Zhi Chen, Chang-Biau Yang, Yu-Cheng Chang","doi":"10.1142/s012905412450014x","DOIUrl":"https://doi.org/10.1142/s012905412450014x","url":null,"abstract":"The longest increasing subsequence (LIS) problem aims to find the subsequence exhibiting an increasing trend in a numeric sequence with the maximum length. In this paper, we generalize the LIS problem to the longest wave subsequence (LWS) problem, which encompasses two versions: LWSt and LWSr. Given a numeric sequence [Formula: see text] of distinct values and a target trend sequence [Formula: see text], the LWSt problem aims to identify the longest subsequence of [Formula: see text] that preserves the trend of the prefix of [Formula: see text]. And, the LWSr problem aims to find the longest subsequence of [Formula: see text] within [Formula: see text] segments, alternating increasing and decreasing subsequences. We propose two efficient algorithms for solving the two versions of the LWS problem. For the LWSt problem, the time complexity of our algorithm is O[Formula: see text], where [Formula: see text] represents the length of the given numeric sequence [Formula: see text]. Additionally, we propose an O[Formula: see text]-time algorithm for solving the LWSr problem. In both algorithms, we utilize the priority queues for the insertion, deletion, and successor operations.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141653493","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 : 2024-07-12DOI: 10.1142/s0129054124500163
Junzhen Wang, Shumin Zhang, Bo Zhu
The [Formula: see text]-set tree connectivity, as a natural extension of classical connectivity, is a very important index to evaluate the fault-tolerance of interconnection networks. Let [Formula: see text] be a connected graph and a subset [Formula: see text], an [Formula: see text]-tree of graph [Formula: see text] is a tree [Formula: see text] that contains all the vertices of [Formula: see text] and [Formula: see text]. Two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are internally disjoint if and only if [Formula: see text] and [Formula: see text]. The cardinality of maximum internally disjoint [Formula: see text]-trees is defined as [Formula: see text], and the [Formula: see text]-set tree connectivity is defined by [Formula: see text]. In this paper, we show that the [Formula: see text]-set tree connectivity of folded hypercube when [Formula: see text], that is, [Formula: see text], where [Formula: see text] is folded hypercube for [Formula: see text].
{"title":"The 4-Set Tree Connectivity of Folded Hypercube","authors":"Junzhen Wang, Shumin Zhang, Bo Zhu","doi":"10.1142/s0129054124500163","DOIUrl":"https://doi.org/10.1142/s0129054124500163","url":null,"abstract":"The [Formula: see text]-set tree connectivity, as a natural extension of classical connectivity, is a very important index to evaluate the fault-tolerance of interconnection networks. Let [Formula: see text] be a connected graph and a subset [Formula: see text], an [Formula: see text]-tree of graph [Formula: see text] is a tree [Formula: see text] that contains all the vertices of [Formula: see text] and [Formula: see text]. Two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are internally disjoint if and only if [Formula: see text] and [Formula: see text]. The cardinality of maximum internally disjoint [Formula: see text]-trees is defined as [Formula: see text], and the [Formula: see text]-set tree connectivity is defined by [Formula: see text]. In this paper, we show that the [Formula: see text]-set tree connectivity of folded hypercube when [Formula: see text], that is, [Formula: see text], where [Formula: see text] is folded hypercube for [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141652452","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 : 2024-07-12DOI: 10.1142/s0129054124500151
Mahin Bahrami, Dariush Kiani, Zahed Rahmati
A graph [Formula: see text] is called a [Formula: see text]-dot product graph if there is a function [Formula: see text] such that for any two distinct vertices [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]. The minimum value [Formula: see text] such that [Formula: see text] is a [Formula: see text]-dot product graph, is called the dot product dimension [Formula: see text] of [Formula: see text]. In this paper, we give an efficient algorithm for computing the dot product dimension of outerplanar graphs of at most two edge-disjoint cycles. If the graph has two cycles, we only consider those outerplanar graphs if both cycles have exactly one vertex in common and the length of one of the cycles is greater than or equal to six.
{"title":"An Efficient Algorithm to Compute Dot Product Dimension of Some Outerplanar Graphs","authors":"Mahin Bahrami, Dariush Kiani, Zahed Rahmati","doi":"10.1142/s0129054124500151","DOIUrl":"https://doi.org/10.1142/s0129054124500151","url":null,"abstract":"A graph [Formula: see text] is called a [Formula: see text]-dot product graph if there is a function [Formula: see text] such that for any two distinct vertices [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]. The minimum value [Formula: see text] such that [Formula: see text] is a [Formula: see text]-dot product graph, is called the dot product dimension [Formula: see text] of [Formula: see text]. In this paper, we give an efficient algorithm for computing the dot product dimension of outerplanar graphs of at most two edge-disjoint cycles. If the graph has two cycles, we only consider those outerplanar graphs if both cycles have exactly one vertex in common and the length of one of the cycles is greater than or equal to six.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141653087","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 : 2024-07-12DOI: 10.1142/s0129054124420012
O. Martynova, Alexander Okhotin
Finite automata that traverse graphs by moving along their edges are known as graph-walking automata (GWA). This paper investigates the state complexity of Boolean operations for this model. It is proved that the union of GWA with [Formula: see text] and [Formula: see text] states, with [Formula: see text], operating on graphs with [Formula: see text] labels of edge end-points, is representable by a GWA with [Formula: see text] states, and at least [Formula: see text] states are necessary in the worst case. For the intersection, the upper bound is [Formula: see text] and the lower bound is [Formula: see text]. The upper bound for the complementation is [Formula: see text], and the lower bound is [Formula: see text].
{"title":"State Complexity of Boolean Operations on Graph-Walking Automata","authors":"O. Martynova, Alexander Okhotin","doi":"10.1142/s0129054124420012","DOIUrl":"https://doi.org/10.1142/s0129054124420012","url":null,"abstract":"Finite automata that traverse graphs by moving along their edges are known as graph-walking automata (GWA). This paper investigates the state complexity of Boolean operations for this model. It is proved that the union of GWA with [Formula: see text] and [Formula: see text] states, with [Formula: see text], operating on graphs with [Formula: see text] labels of edge end-points, is representable by a GWA with [Formula: see text] states, and at least [Formula: see text] states are necessary in the worst case. For the intersection, the upper bound is [Formula: see text] and the lower bound is [Formula: see text]. The upper bound for the complementation is [Formula: see text], and the lower bound is [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141654651","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 : 2024-07-10DOI: 10.1142/s0129054124420024
V. Geffert, Alexander Okhotin
It is shown that a two-way deterministic finite automaton (2DFA) with [Formula: see text] states over an alphabet [Formula: see text] can be transformed to an equivalent one-way automaton (1DFA) with [Formula: see text] states, where [Formula: see text]. This reflects the fact that, by keeping the last processed symbol [Formula: see text] in memory, the simulating 1DFA can remember one of [Formula: see text] states in which the automaton moves by [Formula: see text] to the right, and a function that maps [Formula: see text] states moving to the left to [Formula: see text] states moving to the right; cf. ca. [Formula: see text] functions in the classical construction. A close lower bound of [Formula: see text] states is established using a 2-symbol alphabet, with witness languages defined by direction-determinate 2DFA. The same lower bound is also achieved with witness languages defined by sweeping 2DFA, at the expense of using a 5-symbol alphabet. In addition, the complexity of transforming a sweeping or a direction-determinate 2DFA to a 1DFA is shown to be exactly [Formula: see text].
{"title":"Deterministic One-Way Simulation of Two-Way Deterministic Finite Automata Over Small Alphabets","authors":"V. Geffert, Alexander Okhotin","doi":"10.1142/s0129054124420024","DOIUrl":"https://doi.org/10.1142/s0129054124420024","url":null,"abstract":"It is shown that a two-way deterministic finite automaton (2DFA) with [Formula: see text] states over an alphabet [Formula: see text] can be transformed to an equivalent one-way automaton (1DFA) with [Formula: see text] states, where [Formula: see text]. This reflects the fact that, by keeping the last processed symbol [Formula: see text] in memory, the simulating 1DFA can remember one of [Formula: see text] states in which the automaton moves by [Formula: see text] to the right, and a function that maps [Formula: see text] states moving to the left to [Formula: see text] states moving to the right; cf. ca. [Formula: see text] functions in the classical construction. A close lower bound of [Formula: see text] states is established using a 2-symbol alphabet, with witness languages defined by direction-determinate 2DFA. The same lower bound is also achieved with witness languages defined by sweeping 2DFA, at the expense of using a 5-symbol alphabet. In addition, the complexity of transforming a sweeping or a direction-determinate 2DFA to a 1DFA is shown to be exactly [Formula: see text].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141661647","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 : 2024-07-09DOI: 10.1142/s0129054124500084
Nirmala Bhatt, B. Gorain, Kaushik Mondal, S. Pandit
The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].
最大独立集问题在图论和相关领域得到了广泛研究。图的独立集是图中不相邻顶点的子集。最大独立集是规模最大的独立集。本文研究的是分布式环境下某些类别几何交集图中的最大独立集问题。更确切地说，我们研究了两种几何交集图（区间图和轴平行线段交集图）上的最大独立集问题，并提出了确定性分布式算法，其模型与本地通信模型相似，但稍弱于本地通信模型。我们用[公式：见正文]轮和[公式：见正文]消息计算区间图上的最大独立集，其中[公式：见正文]是最大独立集的大小，[公式：见正文]是图中的节点数。我们提供了[公式：见正文]的回合数匹配下限，而[公式：见正文]则是信息复杂度的微不足道的下限。因此，我们的算法在时间和信息上都是最优的。我们还研究了区间数图[公式：见正文]中的最大独立集问题，这是区间图的一种特例，其中的区间长度完全[公式：见正文]不同。我们提出了一种[公式：见正文]轮运行的[公式：见正文]近似算法。对于轴平行线段相交图，我们设计了一种[公式：见正文]近似计算算法，它可以在[公式：见正文]轮中得到解。本文的结果扩展了 Molla 等人 [J. Parallel Distrib.］
{"title":"Distributed Independent Sets in Interval and Segment Intersection Graphs","authors":"Nirmala Bhatt, B. Gorain, Kaushik Mondal, S. Pandit","doi":"10.1142/s0129054124500084","DOIUrl":"https://doi.org/10.1142/s0129054124500084","url":null,"abstract":"The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141836056","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 : 2024-07-09DOI: 10.1142/s0129054124500102
Rui Xiao, Qunying Liao
The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code [Formula: see text] with minimum distance [Formula: see text], if it has an [Formula: see text]-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases.
{"title":"An Improvement for Error-Correcting Pairs of Some Special MDS Codes","authors":"Rui Xiao, Qunying Liao","doi":"10.1142/s0129054124500102","DOIUrl":"https://doi.org/10.1142/s0129054124500102","url":null,"abstract":"The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code [Formula: see text] with minimum distance [Formula: see text], if it has an [Formula: see text]-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code [Formula: see text] with minimum distance [Formula: see text] to have an [Formula: see text]-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141665847","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 : 2024-07-09DOI: 10.1142/s0129054124500138
Bobin George, Jinta Jose, Rajesh K. Thumbakara
Soft set theory is a mathematical approach to address the challenges of handling vague or uncertain information. It is a more advanced version of classical set theory that deals with imprecise elements and enables the flexible representation of uncertain data. It involves categorizing the elements of the universe based on specific parameters. Semigraph is a generalization of a graph which is different from a hypergraph. A hypergraph extends the concept of a graph by allowing any subset of vertices to form an edge. Semigraphs, on the other hand, distinguish themselves from hypergraphs by imposing a specific order on the vertices within each edge. Soft semigraphs were developed using the principles of soft set theory applied to semigraphs. This study introduces Eulerian and Hamiltonian soft semigraphs. We establish a necessary and sufficient condition for a soft semigraph to be Eulerian, relying on parameters such as [Formula: see text]-part consecutive adjacent degree, [Formula: see text]-part end degree, and the [Formula: see text]-part consecutive adjacency graph. Additionally, we provide the conditions for a soft semigraph to be Hamiltonian. We introduce the concept of maximal non-Hamiltonian [Formula: see text]-part. Finally, we define the closure of a soft semigraph and demonstrate the relationship between a Hamiltonian soft semigraph and its closure.
{"title":"Eulerian and Hamiltonian Soft Semigraphs","authors":"Bobin George, Jinta Jose, Rajesh K. Thumbakara","doi":"10.1142/s0129054124500138","DOIUrl":"https://doi.org/10.1142/s0129054124500138","url":null,"abstract":"Soft set theory is a mathematical approach to address the challenges of handling vague or uncertain information. It is a more advanced version of classical set theory that deals with imprecise elements and enables the flexible representation of uncertain data. It involves categorizing the elements of the universe based on specific parameters. Semigraph is a generalization of a graph which is different from a hypergraph. A hypergraph extends the concept of a graph by allowing any subset of vertices to form an edge. Semigraphs, on the other hand, distinguish themselves from hypergraphs by imposing a specific order on the vertices within each edge. Soft semigraphs were developed using the principles of soft set theory applied to semigraphs. This study introduces Eulerian and Hamiltonian soft semigraphs. We establish a necessary and sufficient condition for a soft semigraph to be Eulerian, relying on parameters such as [Formula: see text]-part consecutive adjacent degree, [Formula: see text]-part end degree, and the [Formula: see text]-part consecutive adjacency graph. Additionally, we provide the conditions for a soft semigraph to be Hamiltonian. We introduce the concept of maximal non-Hamiltonian [Formula: see text]-part. Finally, we define the closure of a soft semigraph and demonstrate the relationship between a Hamiltonian soft semigraph and its closure.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141665908","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 : 2024-06-08DOI: 10.1142/s0129054124500060
Ali Q. M. Al-Saedi, R. I. Nabiyyi, M. Javanian
The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes.
{"title":"Limit Law for Zagreb and Wiener Indices of Random Exponential Recursive Trees","authors":"Ali Q. M. Al-Saedi, R. I. Nabiyyi, M. Javanian","doi":"10.1142/s0129054124500060","DOIUrl":"https://doi.org/10.1142/s0129054124500060","url":null,"abstract":"The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes.","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141369420","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 : 2024-04-18DOI: 10.1142/s0129054124500047
Shuai Liu, Yan Wang, Jianxi Fan, Baolei Cheng
The existence of multiple edge-disjoint Hamiltonian cycles (EDHCs for short) is a desirable property of interconnection networks. These parallel cycles can provide an advantage for algorithms that require a ring structure. Additionally, EDHCs can enhance all-to-all data broadcasting and edge fault tolerance in network communications. In this paper, we investigate the construction of EDHCs in the balanced hypercube, which is a variant of the hypercube with many attractive properties, such as strong connectivity, regularity, and symmetry. In particular, each processor in the balanced hypercube has a backup processor that shares the common neighbors, enabling fault tolerance and efficient system reconfiguration. In 2019, Lü et al. provided an algorithm to construct two EDHCs in an -dimensional balanced hypercube for . We further study this topic and give some construction schemes to construct EDHCs in for . Since is -regular, our result is optimal for (). In addition, we simulate the fault-tolerant data broadcasting through these parallel cycles as transmission channels.
{"title":"Edge-Disjoint Hamiltonian Cycles in Balanced Hypercubes with Applications to Fault-Tolerant Data Broadcasting","authors":"Shuai Liu, Yan Wang, Jianxi Fan, Baolei Cheng","doi":"10.1142/s0129054124500047","DOIUrl":"https://doi.org/10.1142/s0129054124500047","url":null,"abstract":"<p>The existence of multiple edge-disjoint Hamiltonian cycles (EDHCs for short) is a desirable property of interconnection networks. These parallel cycles can provide an advantage for algorithms that require a ring structure. Additionally, EDHCs can enhance all-to-all data broadcasting and edge fault tolerance in network communications. In this paper, we investigate the construction of EDHCs in the balanced hypercube, which is a variant of the hypercube with many attractive properties, such as strong connectivity, regularity, and symmetry. In particular, each processor in the balanced hypercube has a backup processor that shares the common neighbors, enabling fault tolerance and efficient system reconfiguration. In 2019, Lü <i>et al.</i> provided an algorithm to construct two EDHCs in an <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>-dimensional balanced hypercube <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>B</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> for <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. We further study this topic and give some construction schemes to construct <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mn>2</mn></mrow><mrow><mo stretchy=\"false\">⌊</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi><mo stretchy=\"false\">⌋</mo></mrow></msup></math></span><span></span> EDHCs in <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>B</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> for <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>2</mn></math></span><span></span>. Since <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>B</mi><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> is <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn><mi>n</mi></math></span><span></span>-regular, our result is optimal for <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup></math></span><span></span> (<span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>r</mi><mo>≥</mo><mn>1</mn></math></span><span></span>). In addition, we simulate the fault-tolerant data broadcasting through these parallel cycles as transmission channels.</p>","PeriodicalId":50323,"journal":{"name":"International Journal of Foundations of Computer Science","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630649","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}
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