We propose two step-by-step approaches to the analysis of robustness in biochemical networks. Our aim is to measure the ability of the network to exhibit step-by-step limited variations on the concentration of a species of interest at varying of the initial concentration of other species. The first approach we propose is reaction-by-reaction, i.e. we compare the states reached by nominal and perturbed networks after they have performed the same number of reactions. We provide a statistical technique allowing for estimating robustness, we implement it in a tool called spebnr (a Simple Python Environment for statistical estimation of Biochemical Network Robustness) and showcase it on three case studies: the EnvZ/OmpR osmoregulatory signaling system of Escherichia Coli, the mechanism of bacterial chemotaxis of Escherichia Coli, and enzyme activity at saturation. Then, we consider a time-by-time approach, in which networks are compared on the basis of the states they reached at the same time point, regardless of how many reactions occurred. This approach is implemented in Stark, and we apply it to the study the robustness of the EnvZ/OmpR osmoregulatory signaling system and the Lotka-Volterra equations.
{"title":"Robustness for biochemical networks: Step-by-step approach","authors":"Valentina Castiglioni , Ruggero Lanotte , Michele Loreti , Desiree Manicardi , Simone Tini","doi":"10.1016/j.tcs.2024.114934","DOIUrl":"10.1016/j.tcs.2024.114934","url":null,"abstract":"<div><div>We propose two step-by-step approaches to the analysis of robustness in biochemical networks. Our aim is to measure the ability of the network to exhibit step-by-step limited variations on the concentration of a species of interest at varying of the initial concentration of other species. The first approach we propose is reaction-by-reaction, i.e. we compare the states reached by nominal and perturbed networks after they have performed the same number of reactions. We provide a statistical technique allowing for estimating robustness, we implement it in a tool called <span>spebnr</span> (<em>a Simple Python Environment for statistical estimation of Biochemical Network Robustness</em>) and showcase it on three case studies: the EnvZ/OmpR osmoregulatory signaling system of Escherichia Coli, the mechanism of bacterial chemotaxis of Escherichia Coli, and enzyme activity at saturation. Then, we consider a time-by-time approach, in which networks are compared on the basis of the states they reached at the same time point, regardless of how many reactions occurred. This approach is implemented in <span>Stark</span>, and we apply it to the study the robustness of the EnvZ/OmpR osmoregulatory signaling system and the Lotka-Volterra equations.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1022 ","pages":"Article 114934"},"PeriodicalIF":0.9,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142551927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-24DOI: 10.1016/j.tcs.2024.114935
Alessandro De Luca, Gabriele Fici
We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We give a closed formula for counting the exact number of balanced words with a zeroes and b ones. We also study minimal non-balanced words.
我们展示了 Christoffel 词和二进制平衡词的组合结果，这些结果是由它们作为数字段近似值的几何解释所激发的。我们给出了一个封闭公式，用于计算有 a 个 0 和 b 个 1 的平衡词的精确数量。我们还研究了最小非平衡词。
{"title":"Some results on digital segments and balanced words","authors":"Alessandro De Luca, Gabriele Fici","doi":"10.1016/j.tcs.2024.114935","DOIUrl":"10.1016/j.tcs.2024.114935","url":null,"abstract":"<div><div>We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We give a closed formula for counting the exact number of balanced words with <em>a</em> zeroes and <em>b</em> ones. We also study minimal non-balanced words.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1021 ","pages":"Article 114935"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554149","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}
This work deals with the problem of gathering n oblivious mobile entities, called robots, at a point (not known beforehand) placed on an infinite triangular grid. The robots are considered to be myopic, i.e., robots have limited visibility. Earlier works of gathering mostly considered the robots either on a plane or on a circle or on a rectangular grid under both full and limited visibility. In the triangular grid, there are two works to the best of our knowledge. The first one is by Cicerone et al. on arbitrary pattern formation where full visibility is considered. The other one by Shibata et al. which considers seven robots with 2-hop visibility that form a hexagon with one robot in the center of the hexagon in a collision-less environment under a fully synchronous scheduler.
In this work, we first show that gathering on a triangular grid with 1-hop vision of robots is not possible even under a fully synchronous scheduler if the robots do not agree on any axis. So one axis agreement has been considered in this work (i.e., the robots agree on a direction and its orientation). We have also shown that the lower bound for time is epochs when n number of robots are gathering on an infinite triangular grid. An algorithm is then presented where a swarm of n number of robots with 1-hop visibility can gather within epochs under a semi-synchronous scheduler. So the algorithm presented here is time optimal.
这项工作涉及的问题是，在一个无限三角形网格上的一个点（事先不知道）上，聚集 n 个被称为机器人的遗忘移动实体。机器人被认为是近视眼，即机器人的可见度有限。早期的收集工作大多将机器人置于平面、圆形或矩形网格上，既考虑了完全可见性，也考虑了有限可见性。在三角形网格中，据我们所知有两项研究。第一项是 Cicerone 等人关于任意模式形成的研究，其中考虑了完全可见性。另一项研究由 Shibata 等人完成，该研究考虑了在完全同步调度下，七个具有 2 跳可见度的机器人在无碰撞环境中组成一个六边形，其中一个机器人位于六边形的中心。在这项研究中，我们首先证明，如果机器人在任何轴上都不一致，那么即使在完全同步调度下，具有 1 跳可见度的机器人也不可能在三角形网格上聚集。因此，本研究考虑的是单轴一致（即机器人在一个方向及其方位上达成一致）。我们还证明，当 n 个机器人聚集在一个无限三角形网格上时，时间下限为 Ω(n) 个历时。随后，我们提出了一种算法，在半同步调度程序下，由 n 个机器人组成的具有 1 跳可见度的机器人群可以在 O(n) 个历时内聚集。因此，这里介绍的算法在时间上是最优的。
{"title":"Time optimal gathering of myopic robots on an infinite triangular grid","authors":"Pritam Goswami , Avisek Sharma , Satakshi Ghosh , Buddhadeb Sau","doi":"10.1016/j.tcs.2024.114930","DOIUrl":"10.1016/j.tcs.2024.114930","url":null,"abstract":"<div><div>This work deals with the problem of gathering <em>n</em> oblivious mobile entities, called robots, at a point (not known beforehand) placed on an infinite triangular grid. The robots are considered to be myopic, i.e., robots have limited visibility. Earlier works of gathering mostly considered the robots either on a plane or on a circle or on a rectangular grid under both full and limited visibility. In the triangular grid, there are two works to the best of our knowledge. The first one is by Cicerone et al. on arbitrary pattern formation where full visibility is considered. The other one by Shibata et al. which considers seven robots with 2-hop visibility that form a hexagon with one robot in the center of the hexagon in a collision-less environment under a fully synchronous scheduler.</div><div>In this work, we first show that gathering on a triangular grid with 1-hop vision of robots is not possible even under a fully synchronous scheduler if the robots do not agree on any axis. So one axis agreement has been considered in this work (i.e., the robots agree on a direction and its orientation). We have also shown that the lower bound for time is <span><math><mi>Ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> epochs when <em>n</em> number of robots are gathering on an infinite triangular grid. An algorithm is then presented where a swarm of <em>n</em> number of robots with 1-hop visibility can gather within <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> epochs under a semi-synchronous scheduler. So the algorithm presented here is time optimal.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1023 ","pages":"Article 114930"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553638","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 : 2024-10-24DOI: 10.1016/j.tcs.2024.114936
Emile Benoist , Guillaume Fertin , Géraldine Jean
<div><div>In this paper, we study the <span>Exact Subset MultiCover</span> problem (or <span>ESM</span>), which can be seen as an extension of the well-known <span>Set Cover</span> problem. Let <span><math><mo>(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> be a multiset built from set <span><math><mi>U</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></math></span> and function <span><math><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. <span>ESM</span> is defined as follows: given <span><math><mo>(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> and a collection <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> of <em>n</em> subsets of <span><math><mi>U</mi></math></span>, is it possible to find a multiset <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> with <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mo>{</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>}</mo></math></span> and <span><math><mi>g</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>→</mo><mi>N</mi></math></span>, such that (i) <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>⊆</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span>, and (ii) each element of <span><math><mi>U</mi></math></span> appears as many times in <span><math><mo>(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> as in <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span>? We study this problem under an algorithmic viewpoint and provide diverse complexity results such as polynomial cases, <span>NP</span>-hardness proofs and <span>FPT</span> algorithms. We also study two variants of <span>ESM</span>: (i) <span>Exclusive Exact Subset MultiCover</span> (<span>EESM</span>), which asks that each element of <span><math><mi>U</mi></math></span> appears in exactly one subset <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mro
本文研究的是精确子集多覆盖问题（或称 ESM），它可以看作是著名的集合覆盖问题的扩展。设 (U,f) 是由集合 U={e1,e2,...em} 和函数 f:U→N⁎ 构成的多集。ESM 的定义如下：给定（U,f）和由 U 的 n 个子集组成的集合 S={S1,S2,...,Sn}，是否可能找到一个多集合（S′,g），其中 S′={S1′,S2′,...,Sn′}和 g：S′→N，使得 (i) Si′⊆Si，且 (ii) U 的每个元素在 (U,f) 中出现的次数与在 (S′,g)中出现的次数一样多？我们从算法的角度研究了这个问题，并提供了多种复杂性结果，如多项式情况、NP-hardness 证明和 FPT 算法。我们还研究了 EESM 的两个变体：(i) Exclusive Exact Subset MultiCover (ESM)，要求 U 的每个元素都恰好出现在 S′ 的一个子集 Si′中；(ii) Maximum Exclusive Exact Subset MultiCover (Max-EESM)，这是 EESM 的优化版本，要求 U 的最大数量的元素恰好出现在 S′ 的一个子集 Si′中。对于这两个变体，我们都提供了一些复杂度结果；特别是，我们提出了一个最大 EESM 的 2 近似算法，并证明了它的严密性。对于这三个问题，我们还提供了整数线性规划（ILP）公式。
{"title":"The Exact Subset MultiCover problem","authors":"Emile Benoist , Guillaume Fertin , Géraldine Jean","doi":"10.1016/j.tcs.2024.114936","DOIUrl":"10.1016/j.tcs.2024.114936","url":null,"abstract":"<div><div>In this paper, we study the <span>Exact Subset MultiCover</span> problem (or <span>ESM</span>), which can be seen as an extension of the well-known <span>Set Cover</span> problem. Let <span><math><mo>(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> be a multiset built from set <span><math><mi>U</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></math></span> and function <span><math><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. <span>ESM</span> is defined as follows: given <span><math><mo>(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> and a collection <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> of <em>n</em> subsets of <span><math><mi>U</mi></math></span>, is it possible to find a multiset <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span> with <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mo>{</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>}</mo></math></span> and <span><math><mi>g</mi><mo>:</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>→</mo><mi>N</mi></math></span>, such that (i) <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>⊆</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for every <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span>, and (ii) each element of <span><math><mi>U</mi></math></span> appears as many times in <span><math><mo>(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> as in <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><mi>g</mi><mo>)</mo></math></span>? We study this problem under an algorithmic viewpoint and provide diverse complexity results such as polynomial cases, <span>NP</span>-hardness proofs and <span>FPT</span> algorithms. We also study two variants of <span>ESM</span>: (i) <span>Exclusive Exact Subset MultiCover</span> (<span>EESM</span>), which asks that each element of <span><math><mi>U</mi></math></span> appears in exactly one subset <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>′</mo></mro","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114936"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578448","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 : 2024-10-18DOI: 10.1016/j.tcs.2024.114926
Talley Amir, James Aspnes
The population protocol model [2] offers a theoretical framework for designing and analyzing distributed algorithms among limited-resource mobile agents. While the original population protocol model considers the concept of anonymity, the issue of privacy is not investigated thoroughly. However, there is a need for time- and space-efficient privacy-preserving techniques in the population protocol model if these algorithms are to be implemented in settings handling sensitive data, such as sensor networks, IoT devices, and drones. In this work, we introduce several formal definitions of privacy, ranging from assuring only plausible deniability of the population input vector to having a full information-theoretic guarantee that knowledge beyond an agent's input and output bear no influence on the probability of a particular input vector. We then apply these definitions to both existing and novel protocols. We show that the Remainder-computing protocol from [9] (which is proven to satisfy output independent privacy under adversarial scheduling) is not information-theoretically private under probabilistic scheduling. In contrast, we provide a new algorithm and demonstrate that it correctly and information-theoretically privately computes Remainder under probabilistic scheduling.
{"title":"Privacy in population protocols with probabilistic scheduling","authors":"Talley Amir, James Aspnes","doi":"10.1016/j.tcs.2024.114926","DOIUrl":"10.1016/j.tcs.2024.114926","url":null,"abstract":"<div><div>The population protocol model <span><span>[2]</span></span> offers a theoretical framework for designing and analyzing distributed algorithms among limited-resource mobile agents. While the original population protocol model considers the concept of anonymity, the issue of privacy is not investigated thoroughly. However, there is a need for time- and space-efficient privacy-preserving techniques in the population protocol model if these algorithms are to be implemented in settings handling sensitive data, such as sensor networks, IoT devices, and drones. In this work, we introduce several formal definitions of privacy, ranging from assuring only plausible deniability of the population input vector to having a full information-theoretic guarantee that knowledge beyond an agent's input and output bear no influence on the probability of a particular input vector. We then apply these definitions to both existing and novel protocols. We show that the <span>Remainder</span>-computing protocol from <span><span>[9]</span></span> (which is proven to satisfy output independent privacy under adversarial scheduling) is not information-theoretically private under probabilistic scheduling. In contrast, we provide a new algorithm and demonstrate that it correctly and information-theoretically privately computes <span>Remainder</span> under probabilistic scheduling.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114926"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535252","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 : 2024-10-18DOI: 10.1016/j.tcs.2024.114927
Xiao-Yan Li , Zhaoding Lin , Hongbin Zhuang , Jou-Ming Chang
Connectivity indicators are commonly used to evaluate system fault tolerance and reliability. However, with the high demand for multi-processor systems in high-performance computing and data center networks, the number of processors is getting larger, and the network is getting more complex. Thus, traditional connectivity and other indicators are hardly competent in assessing the reliability of complex networks. The matroidal connectivity and conditional matroidal connectivity are novel connectivity metrics that measure the actual fault-tolerant capability based on the constraints of each network dimension. In this paper, we study matroidal connectivity and conditional matroidal connectivity of the -arrangement graph network from the natural perspective of the partition of edge dimension and obtain their theoretically accurate values. Moreover, we conduct numerical analysis to compare matroidal connectivity with other conditional edge connectivities in . Additionally, we explore the distribution pattern of edge failure through simulation experiments in and attain the relation of conditional matroidal connectivity related to network scales. Our investigations include two famous network classes: alternating group graphs and star graphs.
{"title":"Enabling high reliability via matroidal connectivity and conditional matroidal connectivity on arrangement graph networks","authors":"Xiao-Yan Li , Zhaoding Lin , Hongbin Zhuang , Jou-Ming Chang","doi":"10.1016/j.tcs.2024.114927","DOIUrl":"10.1016/j.tcs.2024.114927","url":null,"abstract":"<div><div>Connectivity indicators are commonly used to evaluate system fault tolerance and reliability. However, with the high demand for multi-processor systems in high-performance computing and data center networks, the number of processors is getting larger, and the network is getting more complex. Thus, traditional connectivity and other indicators are hardly competent in assessing the reliability of complex networks. The matroidal connectivity and conditional matroidal connectivity are novel connectivity metrics that measure the actual fault-tolerant capability based on the constraints of each network dimension. In this paper, we study matroidal connectivity and conditional matroidal connectivity of the <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span>-arrangement graph network <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> from the natural perspective of the partition of edge dimension and obtain their theoretically accurate values. Moreover, we conduct numerical analysis to compare matroidal connectivity with other conditional edge connectivities in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>. Additionally, we explore the distribution pattern of edge failure through simulation experiments in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> and attain the relation of conditional matroidal connectivity related to network scales. Our investigations include two famous network classes: alternating group graphs and star graphs.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114927"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535253","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 : 2024-10-18DOI: 10.1016/j.tcs.2024.114905
Oscar H. Ibarra , Ian McQuillan
The notions of stability, anti-stability, and error-correctability of a language that is modified by making contextual insertions in the words of the language were introduced in a previous paper by Bottoni et al. in 2011, where it was shown that these properties are decidable for regular languages. The authors proposed investigating the decidability of these properties for other classes of languages. Here, we derive necessary and sufficient conditions for a class of languages to have decidable stable, anti-stable, and error-correctable properties, and use these conditions to exhibit general classes of languages (strictly greater than the regular languages) for which the properties are decidable, and also simple classes (the first such classes) for which the properties are undecidable. We obtain identical results for the case when contextual deletions (instead of insertions) are made in the words of the language, and also with mixes of insertions and deletions. Our constructions also demonstrate that certain general problems involving nondeterministic generalized sequential machines (s) applied to languages accepted by deterministic machine models are decidable, which is surprising as the deterministic language families do not need to be closed under mappings.
{"title":"On decision problems concerning contextual insertions and deletions","authors":"Oscar H. Ibarra , Ian McQuillan","doi":"10.1016/j.tcs.2024.114905","DOIUrl":"10.1016/j.tcs.2024.114905","url":null,"abstract":"<div><div>The notions of stability, anti-stability, and error-correctability of a language that is modified by making contextual insertions in the words of the language were introduced in a previous paper by Bottoni et al. in 2011, where it was shown that these properties are decidable for regular languages. The authors proposed investigating the decidability of these properties for other classes of languages. Here, we derive necessary and sufficient conditions for a class of languages to have decidable stable, anti-stable, and error-correctable properties, and use these conditions to exhibit general classes of languages (strictly greater than the regular languages) for which the properties are decidable, and also simple classes (the first such classes) for which the properties are undecidable. We obtain identical results for the case when contextual deletions (instead of insertions) are made in the words of the language, and also with mixes of insertions and deletions. Our constructions also demonstrate that certain general problems involving nondeterministic generalized sequential machines (<span><math><mtext>GSM</mtext></math></span>s) applied to languages accepted by deterministic machine models are decidable, which is surprising as the deterministic language families do not need to be closed under <span><math><mtext>GSM</mtext></math></span> mappings.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114905"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-18DOI: 10.1016/j.tcs.2024.114925
Zhaoman Huang , Mingzu Zhang , Chia-Wei Lee
The h-extra connectivity and the h-extra diagnosability are key parameters for evaluating the reliability and fault-tolerance of the interconnection networks of the multiprocessor systems, and play an important role in designing and maintaining interconnection networks. Recently, various self-diagnostic models have emerged to assess the fault-tolerance in interconnection networks. These interconnection networks are typically expressed by a connected graph . For a non-complete graph G and , the h-extra cut signifies a vertex subset R of G, whose removal results in disconnected, with each remaining component containing at least vertices. And the h-extra connectivity of G is defined as the minimum cardinality of all h-extra cuts of G. The h-extra diagnosability for a graph G denotes the maximum number of detectable faulty vertices when focusing on these h-extra faulty sets only. The complete Josephus cube , a variant of , exhibits superior properties compared to hypercube , and also boasts higher connectivity. In this study, with the help of the exact value of the h-extra connectivity of , the explicit expression of h-extra diagnosability of under both the PMC model for and and the MM* model for and are identified to share the same value .
{"title":"Connectivity and diagnosability of the complete Josephus cube networks under h-extra fault-tolerant model","authors":"Zhaoman Huang , Mingzu Zhang , Chia-Wei Lee","doi":"10.1016/j.tcs.2024.114925","DOIUrl":"10.1016/j.tcs.2024.114925","url":null,"abstract":"<div><div>The <em>h</em>-extra connectivity and the <em>h</em>-extra diagnosability are key parameters for evaluating the reliability and fault-tolerance of the interconnection networks of the multiprocessor systems, and play an important role in designing and maintaining interconnection networks. Recently, various self-diagnostic models have emerged to assess the fault-tolerance in interconnection networks. These interconnection networks are typically expressed by a connected graph <span><math><mi>G</mi><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span>. For a non-complete graph <em>G</em> and <span><math><mi>h</mi><mo>≥</mo><mn>0</mn></math></span>, the <em>h</em>-extra cut signifies a vertex subset <em>R</em> of <em>G</em>, whose removal results in <span><math><mi>G</mi><mo>−</mo><mi>R</mi></math></span> disconnected, with each remaining component containing at least <span><math><mi>h</mi><mo>+</mo><mn>1</mn></math></span> vertices. And the <em>h</em>-extra connectivity of <em>G</em> is defined as the minimum cardinality of all <em>h</em>-extra cuts of <em>G</em>. The <em>h</em>-extra diagnosability for a graph <em>G</em> denotes the maximum number of detectable faulty vertices when focusing on these <em>h</em>-extra faulty sets only. The complete Josephus cube <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, a variant of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, exhibits superior properties compared to hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and also boasts higher connectivity. In this study, with the help of the exact value of the <em>h</em>-extra connectivity of <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the explicit expression of <em>h</em>-extra diagnosability of <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> under both the PMC model for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><mn>1</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></math></span> and the MM* model for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span> and <span><math><mn>2</mn><mo>≤</mo><mi>h</mi><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></math></span> are identified to share the same value <span><math><mo>(</mo><mi>h</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>n</mi><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>h</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1020 ","pages":"Article 114925"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527261","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 : 2024-10-18DOI: 10.1016/j.tcs.2024.114914
Arya Tanmay Gupta, Sandeep S Kulkarni
In this article, we study some parallel processing algorithms for multiplication and modulo operations. We demonstrate that the state transitions that are formed under these algorithms satisfy lattice-linearity, where these algorithms induce a lattice among the global states. Lattice-linearity implies that these algorithms can be implemented in asynchronous environments, where the nodes are allowed to read old information from each other. It means that these algorithms are guaranteed to converge correctly without any synchronization overhead. These algorithms also exhibit snap-stabilizing properties, i.e., starting from an arbitrary state, the sequence of state transitions made by the system strictly follows its specification.
{"title":"Tolerance to asynchrony in algorithms for multiplication and modulo","authors":"Arya Tanmay Gupta, Sandeep S Kulkarni","doi":"10.1016/j.tcs.2024.114914","DOIUrl":"10.1016/j.tcs.2024.114914","url":null,"abstract":"<div><div>In this article, we study some parallel processing algorithms for multiplication and modulo operations. We demonstrate that the state transitions that are formed under these algorithms satisfy lattice-linearity, where these algorithms induce a lattice among the global states. Lattice-linearity implies that these algorithms can be implemented in asynchronous environments, where the nodes are allowed to read old information from each other. It means that these algorithms are guaranteed to converge correctly without any synchronization overhead. These algorithms also exhibit snap-stabilizing properties, i.e., starting from an arbitrary state, the sequence of state transitions made by the system strictly follows its specification.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1024 ","pages":"Article 114914"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535255","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}
In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a b-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online -labeling.
本文针对直径有界的磁盘交集图的经典着色问题给出了一系列在线算法。我们的算法利用了此类图的几何表示法，并受到 Fiala 等人算法的启发，但具有更好的竞争比率。这种改进来自于在着色前使用了两种分割顶点集的技术。其中一种是平面 b 折叠着色的应用。这种方法更具通用性，我们展示了如何将其应用于平面上其他形状的着色，以及如何将其调整为在线 L(2,1)- 标记。
{"title":"Online coloring of disk graphs","authors":"Joanna Chybowska-Sokół , Konstanty Junosza-Szaniawski","doi":"10.1016/j.tcs.2024.114924","DOIUrl":"10.1016/j.tcs.2024.114924","url":null,"abstract":"<div><div>In this paper, we give a family of online algorithms for the classical coloring problem of intersection graphs of disks with bounded diameter. Our algorithms make use of a geometric representation of such graphs and are inspired by an algorithm of Fiala et al., but have better competitive ratios. The improvement comes from using two techniques of partitioning the set of vertices before coloring them. One of them is an application of a <em>b</em>-fold coloring of the plane. The method is more general and we show how it can be applied to coloring other shapes on the plane as well as adjust it for online <span><math><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-labeling.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1020 ","pages":"Article 114924"},"PeriodicalIF":0.9,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527262","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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