Pub Date : 2022-02-26DOI: 10.1142/s0129054122500058
Dongqin Cheng
The 2-dimensional torus networks, denoted by [Formula: see text], is a special case of the famous [Formula: see text]-dimensional torus network. The extra connectivity, structure connectivity and sub-structure connectivity are generalized connectivity to measure the fault-tolerance of the networks. In this paper, we show the 1-extra connectivity of [Formula: see text] for [Formula: see text] and [Formula: see text] and 2-extra connectivity of [Formula: see text] for [Formula: see text]. We also determine the [Formula: see text]-structure connectivity and [Formula: see text]-substructure connectivity of 2-dimensional torus network [Formula: see text] ([Formula: see text] and [Formula: see text]) when the structure subgraph [Formula: see text], where [Formula: see text] and [Formula: see text]. These results are sharp since the connectivity of [Formula: see text] ([Formula: see text]) is 4.
用[公式:见文]表示的二维环面网络是著名的[公式:见文]一维环面网络的一种特殊情况。附加连通性、结构连通性和子结构连通性是衡量网络容错能力的广义连通性。在本文中，我们展示了[公式:见文]和[公式:见文]的[公式:见文]的1额外连通性和[公式:见文]的[公式:见文]的[公式:见文]的2额外连通性。我们还确定了二维环面网络[公式:见文]-结构连通性和[公式:见文]-子结构连通性[公式:见文]([公式:见文]和[公式:见文])当结构子图[公式:见文]时，其中[公式:见文]和[公式:见文]。这些结果很明显，因为[Formula: see text] ([Formula: see text])的连通性是4。
{"title":"Extra Connectivity and Structure Connectivity of 2-Dimensional Torus Networks","authors":"Dongqin Cheng","doi":"10.1142/s0129054122500058","DOIUrl":"https://doi.org/10.1142/s0129054122500058","url":null,"abstract":"The 2-dimensional torus networks, denoted by [Formula: see text], is a special case of the famous [Formula: see text]-dimensional torus network. The extra connectivity, structure connectivity and sub-structure connectivity are generalized connectivity to measure the fault-tolerance of the networks. In this paper, we show the 1-extra connectivity of [Formula: see text] for [Formula: see text] and [Formula: see text] and 2-extra connectivity of [Formula: see text] for [Formula: see text]. We also determine the [Formula: see text]-structure connectivity and [Formula: see text]-substructure connectivity of 2-dimensional torus network [Formula: see text] ([Formula: see text] and [Formula: see text]) when the structure subgraph [Formula: see text], where [Formula: see text] and [Formula: see text]. These results are sharp since the connectivity of [Formula: see text] ([Formula: see text]) is 4.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115219254","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}
Pub Date : 2022-02-26DOI: 10.1142/s012905412250006x
K. Mahalingam, Palak Pandoh
A two-dimensional (2D) word is a 2D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are 1D palindromes. We characterize such words and study some of their combinatorial and structural properties. We also find the number of possible palindromic conjugates of a 2D word. We investigate an upper bound on the number of distinct non-empty HV-palindromic subwords in any finite 2D word, thus, proving the conjecture given by Anisiu et al. We also identify the minimum number of HV-palindromic subwords in an infinite 2D word over a finite alphabet.
{"title":"HV-Palindromes in Two-Dimensional Words","authors":"K. Mahalingam, Palak Pandoh","doi":"10.1142/s012905412250006x","DOIUrl":"https://doi.org/10.1142/s012905412250006x","url":null,"abstract":"A two-dimensional (2D) word is a 2D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are 1D palindromes. We characterize such words and study some of their combinatorial and structural properties. We also find the number of possible palindromic conjugates of a 2D word. We investigate an upper bound on the number of distinct non-empty HV-palindromic subwords in any finite 2D word, thus, proving the conjecture given by Anisiu et al. We also identify the minimum number of HV-palindromic subwords in an infinite 2D word over a finite alphabet.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"107 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134578870","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}
Pub Date : 2022-02-23DOI: 10.1142/s0129054122500022
Tien Tran, D. Huynh
Directional Wireless Sensor Networks (DWSNs) have been a very active research topic as they have a wide range of applications in both military and civilian areas. In this paper, we study the Antenna Orientation (AO) and Antenna Orientation and Power Assignment (AOPA) problems, two important issues in DWSNs. In the AO problem, given a set [Formula: see text] of sensors equipped with directional antennas with beam-width [Formula: see text], the objective is to determine an orientation of the antennas and a minimum transmission range [Formula: see text] such that the induced symmetric communication graph is connected. We prove that the AO problem is NP-hard. We then extend our proof technique and show that the AOPA problem for DWSNs equipped with [Formula: see text] directional antennas is also NP-hard where we have to determine an orientation of the antennas as well as a power assignment [Formula: see text] to the nodes such that the resulting symmetric communication graph is connected, and (2) [Formula: see text] is minimized ([Formula: see text] is the distance-power gradient, [Formula: see text]). We propose an [Formula: see text] heuristic algorithm and demonstrate by simulation that our method is better than the Ice-cream method.
{"title":"Symmetric Connectivity in Wireless Sensor Networks with π/3 Directional Antennas","authors":"Tien Tran, D. Huynh","doi":"10.1142/s0129054122500022","DOIUrl":"https://doi.org/10.1142/s0129054122500022","url":null,"abstract":"Directional Wireless Sensor Networks (DWSNs) have been a very active research topic as they have a wide range of applications in both military and civilian areas. In this paper, we study the Antenna Orientation (AO) and Antenna Orientation and Power Assignment (AOPA) problems, two important issues in DWSNs. In the AO problem, given a set [Formula: see text] of sensors equipped with directional antennas with beam-width [Formula: see text], the objective is to determine an orientation of the antennas and a minimum transmission range [Formula: see text] such that the induced symmetric communication graph is connected. We prove that the AO problem is NP-hard. We then extend our proof technique and show that the AOPA problem for DWSNs equipped with [Formula: see text] directional antennas is also NP-hard where we have to determine an orientation of the antennas as well as a power assignment [Formula: see text] to the nodes such that the resulting symmetric communication graph is connected, and (2) [Formula: see text] is minimized ([Formula: see text] is the distance-power gradient, [Formula: see text]). We propose an [Formula: see text] heuristic algorithm and demonstrate by simulation that our method is better than the Ice-cream method.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129891982","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}
Pub Date : 2022-02-17DOI: 10.1142/s0129054122500034
Chih-Yuan Lin, Jia-Jie Liu, Yue-Li Wang, W. C. Yen, Chiun-Chieh Hsu
In this paper, we introduce a new variant of domination called the outer-paired domination. For a graph [Formula: see text], an outer-paired dominating set [Formula: see text] is a dominating set of [Formula: see text] such that the induced subgraph of [Formula: see text] contains a perfect matching. The outer-paired domination number of a graph [Formula: see text] is the cardinality of a minimum outer-paired dominating set of [Formula: see text]. We show that finding the outer-paired domination number of a graph [Formula: see text] is NP-hard on bipartite graphs, chordal graphs, and planar graphs. We also propose a linear-time algorithm for solving the outer-paired domination problem on trees.
{"title":"The Outer-Paired Domination of Graphs","authors":"Chih-Yuan Lin, Jia-Jie Liu, Yue-Li Wang, W. C. Yen, Chiun-Chieh Hsu","doi":"10.1142/s0129054122500034","DOIUrl":"https://doi.org/10.1142/s0129054122500034","url":null,"abstract":"In this paper, we introduce a new variant of domination called the outer-paired domination. For a graph [Formula: see text], an outer-paired dominating set [Formula: see text] is a dominating set of [Formula: see text] such that the induced subgraph of [Formula: see text] contains a perfect matching. The outer-paired domination number of a graph [Formula: see text] is the cardinality of a minimum outer-paired dominating set of [Formula: see text]. We show that finding the outer-paired domination number of a graph [Formula: see text] is NP-hard on bipartite graphs, chordal graphs, and planar graphs. We also propose a linear-time algorithm for solving the outer-paired domination problem on trees.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"18 5-6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134288609","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}
Pub Date : 2022-02-16DOI: 10.1142/s0129054122500010
Vasco Boavida De Brito, José Félix Costa, Diogo Poças
We consider the experimenter (e.g. the experimental physicist) as a Turing machine — the digital component — and the experiment of measurement — the analog component — as an oracle to the Turing machine. The algorithm running in the machine abstracts the experimental method of measurement (encoding the recursive structure of experimental actions) chosen by the experimenter. In this paper we prove that the central analogue-digital complexity classes [Formula: see text], [Formula: see text] and [Formula: see text] can be characterized in terms of protocols to perform measurements controlled by standard Turing machines.
{"title":"The Power of Machines That Control Experiments","authors":"Vasco Boavida De Brito, José Félix Costa, Diogo Poças","doi":"10.1142/s0129054122500010","DOIUrl":"https://doi.org/10.1142/s0129054122500010","url":null,"abstract":"We consider the experimenter (e.g. the experimental physicist) as a Turing machine — the digital component — and the experiment of measurement — the analog component — as an oracle to the Turing machine. The algorithm running in the machine abstracts the experimental method of measurement (encoding the recursive structure of experimental actions) chosen by the experimenter. In this paper we prove that the central analogue-digital complexity classes [Formula: see text], [Formula: see text] and [Formula: see text] can be characterized in terms of protocols to perform measurements controlled by standard Turing machines.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126010866","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}
Pub Date : 2022-02-07DOI: 10.1142/s0129054122500174
Subhash Bhagat, Abhinav Chakraborty, Bibhuti Das, K. Mukhopadhyaya
The gathering over meeting nodes problem requires the robots to gather at one of the pre-defined meeting nodes. This paper investigates the problem with respect to the objective function that minimizes the total number of moves made by all the robots. In other words, the sum of the distances traveled by all the robots is minimized while accomplishing the gathering task. The robots are deployed on the nodes of an anonymous two-dimensional infinite grid which has a subset of nodes marked as meeting nodes. The robots do not agree on a global coordinate system and operate under an asynchronous scheduler. A deterministic distributed algorithm has been proposed to solve the problem for all those solvable configurations, and the initial configurations for which the problem is unsolvable have been characterized. The proposed gathering algorithm is optimal with respect to the total number of moves performed by all the robots in order to finalize the gathering.
{"title":"Optimal Gathering Over Weber Meeting Nodes in Infinite Grid","authors":"Subhash Bhagat, Abhinav Chakraborty, Bibhuti Das, K. Mukhopadhyaya","doi":"10.1142/s0129054122500174","DOIUrl":"https://doi.org/10.1142/s0129054122500174","url":null,"abstract":"The gathering over meeting nodes problem requires the robots to gather at one of the pre-defined meeting nodes. This paper investigates the problem with respect to the objective function that minimizes the total number of moves made by all the robots. In other words, the sum of the distances traveled by all the robots is minimized while accomplishing the gathering task. The robots are deployed on the nodes of an anonymous two-dimensional infinite grid which has a subset of nodes marked as meeting nodes. The robots do not agree on a global coordinate system and operate under an asynchronous scheduler. A deterministic distributed algorithm has been proposed to solve the problem for all those solvable configurations, and the initial configurations for which the problem is unsolvable have been characterized. The proposed gathering algorithm is optimal with respect to the total number of moves performed by all the robots in order to finalize the gathering.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130297854","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}
Pub Date : 2021-11-30DOI: 10.1142/s0129054121500362
Yen Hung Chen
Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.
给定一个完全图[公式:见文]，具有非负边成本，两个子集[公式:见文]和[公式:见文]，[公式:见文]的[公式:见文]，[公式:见文]，[公式:见文]的分割[公式:见文]，[公式:见文]，[公式:见文]的[公式:见文]的树[公式:见文]的树[公式:见文]跨越[公式:见文]中的所有顶点，使得[公式:见文]可以被切成[公式:]子树[公式:见文本]通过移除[公式:见文本]边和每个子树[公式:见文本]跨越[公式:见文本]，[公式:见文本]中的所有顶点。聚类斯坦纳树的代价被定义为其所有边的代价之和。如果[Formula: see text]中的所有顶点都是[Formula: see text]的内部顶点，则[Formula: see text]的聚类选择内部斯坦纳树就是[Formula: see text]的聚类斯坦纳树。聚类内选斯坦纳树问题是关于在[公式:见文]和[公式:见文]中以最小代价确定聚类内选斯坦纳树[公式:见文]。在本文中，我们提出了已知的第一个具有性能比的近似算法[公式:见文本]，用于聚类选择内部Steiner树问题，其中[公式:见文本]是最著名的Steiner树问题的性能比。
{"title":"The Clustered Selected-Internal Steiner Tree Problem","authors":"Yen Hung Chen","doi":"10.1142/s0129054121500362","DOIUrl":"https://doi.org/10.1142/s0129054121500362","url":null,"abstract":"Given a complete graph [Formula: see text], with nonnegative edge costs, two subsets [Formula: see text] and [Formula: see text], a partition [Formula: see text] of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], a clustered Steiner tree is a tree [Formula: see text] of [Formula: see text] that spans all vertices in [Formula: see text] such that [Formula: see text] can be cut into [Formula: see text] subtrees [Formula: see text] by removing [Formula: see text] edges and each subtree [Formula: see text] spans all vertices in [Formula: see text], [Formula: see text]. The cost of a clustered Steiner tree is defined to be the sum of the costs of all its edges. A clustered selected-internal Steiner tree of [Formula: see text] is a clustered Steiner tree for [Formula: see text] if all vertices in [Formula: see text] are internal vertices of [Formula: see text]. The clustered selected-internal Steiner tree problem is concerned with the determination of a clustered selected-internal Steiner tree [Formula: see text] for [Formula: see text] and [Formula: see text] in [Formula: see text] with minimum cost. In this paper, we present the first known approximation algorithm with performance ratio [Formula: see text] for the clustered selected-internal Steiner tree problem, where [Formula: see text] is the best-known performance ratio for the Steiner tree problem.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115108512","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}
Pub Date : 2021-10-14DOI: 10.1142/s0129054121500349
Pingshan Li, Rong Liu, Xianglin Liu
The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.
{"title":"The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees","authors":"Pingshan Li, Rong Liu, Xianglin Liu","doi":"10.1142/s0129054121500349","DOIUrl":"https://doi.org/10.1142/s0129054121500349","url":null,"abstract":"The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.","PeriodicalId":192109,"journal":{"name":"Int. J. Found. Comput. Sci.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127313227","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}
Pub Date : 2021-09-25DOI: 10.1142/s0129054121500313
Chenli Shen, Wensong Lin
We study how a monopolist seller should price an indivisible product iteratively to the consumers who are connected by a known link-weighted directed social network. For two consumers [Formula: see text] and [Formula: see text], there is an arc directed from [Formula: see text] to [Formula: see text] if and only if [Formula: see text] is a fashion leader of [Formula: see text]. Assuming complete information about the network, the seller offers consumers a sequence of prices over time and the goal is to obtain the maximum revenue. We assume that the consumers buy the product as soon as the seller posts a price not greater than their valuations of the product. The product’s value for a consumer is determined by three factors: a fixed consumer specified intrinsic value and a variable positive (resp. negative) externality that is exerted from the consumer’s out(resp. in)-neighbours. The setting of positive externality is that the influence of fashion leaders on a consumer is the total weight of links from herself to her fashion leaders who have owned the product, and more fashion leaders of a consumer owning the product will increase the influence (external value) on the consumer. And the setting of negative externalities is that the product’s value of showing off for a consumer is the total weight of links from her followers who do not own the product to herself, and more followers of a consumer owning the product will decrease this external value for the consumer. We confirm that finding an optimal iterative pricing is NP-hard even for acyclic networks with maximum total degree [Formula: see text] and with all intrinsic values zero. We design a greedy algorithm which achieves [Formula: see text]-approximation for networks with all intrinsic values zero and show that the approximation ratio [Formula: see text] is tight. Complementary to the hardness result, we design a [Formula: see text]-approximation algorithm for Barabási–Albert networks.
我们研究了一个垄断的销售者如何对一个不可分割的产品进行迭代定价，这些消费者通过一个已知的链接加权有向社会网络连接。对于两个消费者[Formula: see text]和[Formula: see text]来说，当且仅当[Formula: see text]是[Formula: see text]的时尚领导者时，存在一条从[Formula: see text]到[Formula: see text]的弧线。假设关于网络的信息是完整的，卖方向消费者提供一段时间内的一系列价格，目标是获得最大的收益。我们假设，一旦卖家公布的价格不高于消费者对该产品的估价，消费者就会购买该产品。产品对消费者的价值是由三个因素决定的:一个固定的消费者指定的内在价值和一个可变的正价值。负的)外部性是由消费者的外部施加的。邻居。正外部性的设定是，时尚领袖对消费者的影响是她与拥有该产品的时尚领袖之间的链接的总权重，拥有该产品的消费者的时尚领袖越多，对消费者的影响(外部价值)就会增加。负外部性的设定是，产品对消费者的炫耀价值是她的追随者不拥有该产品到她自己的链接的总权重，拥有该产品的消费者的追随者越多，消费者的这种外部价值就会减少。我们证实，即使对于具有最大总度(公式:见文本)且所有内在值为零的无环网络，找到最优迭代定价也是np困难的。我们设计了一种贪心算法，实现了对所有内在值为零的网络的[公式:见文]逼近，并证明了逼近比[公式:见文]是紧的。与硬度结果相补充，我们设计了一个[公式:见文本]- Barabási-Albert网络的近似算法。
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Pub Date : 2021-09-22DOI: 10.1142/s0129054121500325
B. Ryabko
Pseudo-random number generators (PRNGs) are widely used in computer simulation, cryptography, and many other fields. In this paper, we describe a PRNG class, which, firstly, has been successfully tested using the most powerful modern test batteries, and secondly, is proved to consist of generators that generate normal sequences. The latter property means that, for any generated sequence [Formula: see text] and any binary word [Formula: see text], we have [Formula: see text] where [Formula: see text] is the number of occurrences of [Formula: see text] in the sequence [Formula: see text], [Formula: see text].
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