Pub Date : 2024-11-26DOI: 10.1016/j.tcs.2024.114986
Hongjie Guo , Jianzhong Li , Hong Gao
Multi-criteria decision-making typically requires selecting a concise, representative set from large databases. Regret minimization set (RMS) queries have emerged as a solution to circumvent the necessity of a utility function in top-k queries and to address the expansive result sets produced by skyline queries. However, traditional RMS formulations only ensure one result under any utility function and do not account for the diversity and freshness of results. This study introduces the concept of strong regret minimization set (SRMS), ensuring the utility value accuracy of selected k data points under any utility function while incorporating result diversity and freshness. We explore two new computational challenges: the Minimum Size problem, focusing on reducing the result set size with bounded utility error, and the Max-sum Diversity and Freshness problem, aiming to optimize the diversity and freshness of the selected set. Both problems are proved to be NP-hard, and we develop approximation algorithms for them. Experimental results on both real-world and synthetic data show high efficiency and scalability of proposed algorithms.
多标准决策通常需要从大型数据库中选择具有代表性的简明集合。遗憾最小化集（RMS）查询作为一种解决方案应运而生，它规避了 top-k 查询中效用函数的必要性，并解决了天际线查询产生的庞大结果集问题。然而，传统的 RMS 公式只能确保在任何效用函数下都有一个结果，而且不考虑结果的多样性和新鲜度。本研究引入了强遗憾最小化集（SRMS）的概念，在任意效用函数下确保所选 k 个数据点的效用值准确性，同时考虑结果的多样性和新鲜度。我们探索了两个新的计算挑战：最小尺寸问题（侧重于在效用误差受限的情况下缩小结果集尺寸）和最大总和多样性和新鲜度问题（旨在优化所选结果集的多样性和新鲜度）。这两个问题都被证明是 NP 难问题，我们为它们开发了近似算法。在真实世界和合成数据上的实验结果表明，所提出的算法具有很高的效率和可扩展性。
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Pub Date : 2024-11-22DOI: 10.1016/j.tcs.2024.114985
Julien Bensmail , Hervé Hocquard , Clara Marcille
In connection with the so-called 1-2-3 Conjecture, we introduce and study a new variant of proper labellings, obtained when aiming at designing, for an oriented graph, an oriented colouring through the sums of labels incident to its vertices. Formally, for an oriented graph and a k-labelling of its arcs, for every vertex , one can compute the sum of labels assigned by ℓ to its incident arcs. We call ℓ an oriented labelling if the sum function σ indeed forms an oriented colouring of . That is, for any two arcs and of , if , then we must have . We denote by the smallest k such that oriented k-labellings of exist (if any).
We study this new parameter in general and in particular contexts. In particular, we observe that there is no constant bound on in general, contrarily to the undirected case. Still, we establish connections between this parameter and others, such as the oriented chromatic number, from which we deduce other types of bounds, some of which we improve upon for some classes of oriented graphs. We also investigate other aspects of this parameter, such as the complexity of determining for a given oriented graph , or the possible relationships between and the underlying graph G of .
关于所谓的 1-2-3 猜想，我们介绍并研究了适当标记的一种新变体，其目的是通过其顶点的标记之和为定向图设计定向着色。形式上，对于一个定向图及其弧的 k 标签，对于每个顶点 ，我们可以计算 ℓ 分配给其入射弧的标签之和σ(v)。如果和函数 σ 确实构成了 ℓ 的定向着色，我们称 ℓ 为定向标签。 也就是说，对于 ℓ 的任意两条弧，如果 σ(a)=σ(d), 那么我们必须有 σ(b)≠σ(c).我们用最小的 k 表示，这样就存在定向 k 标签（如果有的话）。我们将研究这个新参数的一般情况和特殊情况。特别是，我们观察到，与不定向情况相反，一般情况下没有常数约束。尽管如此，我们还是在这个参数和其他参数（如有向色度数）之间建立了联系，并由此推导出其他类型的约束，其中一些约束在某些有向图类中得到了改进。我们还研究了这一参数的其他方面，例如确定给定定向图 ，或其底层图 G 之间可能存在的关系的复杂性。
{"title":"Adding direction constraints to the 1-2-3 Conjecture","authors":"Julien Bensmail , Hervé Hocquard , Clara Marcille","doi":"10.1016/j.tcs.2024.114985","DOIUrl":"10.1016/j.tcs.2024.114985","url":null,"abstract":"<div><div>In connection with the so-called 1-2-3 Conjecture, we introduce and study a new variant of proper labellings, obtained when aiming at designing, for an oriented graph, an oriented colouring through the sums of labels incident to its vertices. Formally, for an oriented graph <figure><img></figure> and a <em>k</em>-labelling <figure><img></figure> of its arcs, for every vertex <figure><img></figure>, one can compute the sum <span><math><mi>σ</mi><mo>(</mo><mi>v</mi><mo>)</mo></math></span> of labels assigned by <em>ℓ</em> to its incident arcs. We call <em>ℓ</em> an oriented labelling if the sum function <em>σ</em> indeed forms an oriented colouring of <figure><img></figure>. That is, for any two arcs <figure><img></figure> and <figure><img></figure> of <figure><img></figure>, if <span><math><mi>σ</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><mi>σ</mi><mo>(</mo><mi>d</mi><mo>)</mo></math></span>, then we must have <span><math><mi>σ</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>≠</mo><mi>σ</mi><mo>(</mo><mi>c</mi><mo>)</mo></math></span>. We denote by <figure><img></figure> the smallest <em>k</em> such that oriented <em>k</em>-labellings of <figure><img></figure> exist (if any).</div><div>We study this new parameter in general and in particular contexts. In particular, we observe that there is no constant bound on <figure><img></figure> in general, contrarily to the undirected case. Still, we establish connections between this parameter and others, such as the oriented chromatic number, from which we deduce other types of bounds, some of which we improve upon for some classes of oriented graphs. We also investigate other aspects of this parameter, such as the complexity of determining <figure><img></figure> for a given oriented graph <figure><img></figure>, or the possible relationships between <figure><img></figure> and the underlying graph <em>G</em> of <figure><img></figure>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114985"},"PeriodicalIF":0.9,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142719633","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}
The problem of determining whether a graph G contains another graph H as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While the problem is -complete in general, it can be tractable on some restricted graph classes. This study focuses on the case where both G and H are trees, known as the tree minor containment problem. Even in this case, the problem is known to be -complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of H is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide complexity dichotomies for the problem based on three structural parameters: diameter, pathwidth, and path eccentricity.
确定一个图 G 是否包含另一个图 H 作为次要图的问题，称为次要图包含问题，是图算法领域的一个基本问题。虽然这个问题在一般情况下是 NP-完全的，但在某些受限的图类中是可以解决的。本研究侧重于 G 和 H 都是树的情况，即所谓的树次要包含问题。即使在这种情况下，该问题也是已知的 NP-完全问题。相比之下，已知的多项式时间算法适用于两棵树都是毛毛虫或 H 的最大度是常数的情况。我们的研究旨在澄清树小包含问题的可处理性和不可处理性的界限。具体来说，我们根据三个结构参数：直径、路径宽度和路径偏心率，为该问题提供了复杂性二分法。
{"title":"Dichotomies for tree minor containment with structural parameters","authors":"Tatsuya Gima , Soh Kumabe , Kazuhiro Kurita , Yuto Okada , Yota Otachi","doi":"10.1016/j.tcs.2024.114984","DOIUrl":"10.1016/j.tcs.2024.114984","url":null,"abstract":"<div><div>The problem of determining whether a graph <em>G</em> contains another graph <em>H</em> as a minor, referred to as the <em>minor containment problem</em>, is a fundamental problem in the field of graph algorithms. While the problem is <span><math><mi>NP</mi></math></span>-complete in general, it can be tractable on some restricted graph classes. This study focuses on the case where both <em>G</em> and <em>H</em> are trees, known as the <em>tree minor containment problem</em>. Even in this case, the problem is known to be <span><math><mi>NP</mi></math></span>-complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of <em>H</em> is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide complexity dichotomies for the problem based on three structural parameters: diameter, pathwidth, and path eccentricity.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 114984"},"PeriodicalIF":0.9,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720862","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 : 2024-11-22DOI: 10.1016/j.tcs.2024.114983
David F. Anderson , Badal Joshi
Recent technological advances allow us to view chemical mass-action systems as analog computers. In this context, the inputs to a computation are encoded as initial values of certain chemical species while the outputs are the limiting values of other chemical species. In this paper, we design chemical systems that carry out the elementary arithmetic computations of: identification, inversion, mth roots (for ), addition, multiplication, absolute difference, rectified subtraction over non-negative real numbers, and partial real inversion over real numbers. We prove that these “elementary modules” have a speed of computation that is independent of the inputs to the computation. Moreover, we prove that finite sequences of such elementary modules, running in parallel, can carry out composite arithmetic over real numbers, also at a rate that is independent of inputs. Furthermore, we show that the speed of a composite computation is precisely the speed of the slowest elementary step. Specifically, the scale of the composite computation, i.e. the number of elementary steps involved in the composite, does not affect the overall asymptotic speed – a feature of the parallel computing nature of our algorithm. Our proofs require the careful mathematical analysis of certain non-autonomous systems, and we believe this analysis will be useful in different areas of applied mathematics, dynamical systems, and the theory of computation. We close with a discussion on future research directions, including numerous important open theoretical questions pertaining to the field of computation with reaction networks.
最近的技术进步使我们能够将化学物质作用系统视为模拟计算机。在这种情况下，计算的输入被编码为某些化学物质的初始值，而输出则是其他化学物质的极限值。在本文中，我们设计的化学系统可进行以下基本算术计算：识别、反演、第 m 次根（m≥2 时）、加法、乘法、绝对差、非负实数的整除减法和实数的部分实数反演。我们证明，这些 "基本模块 "的计算速度与计算的输入无关。此外，我们还证明，并行运行的此类基本模块的有限序列可以对实数进行复合运算，而且运算速度与输入无关。此外，我们还证明，复合计算的速度正是最慢基本步骤的速度。具体来说，复合计算的规模，即复合计算所涉及的基本步骤的数量，不会影响整体渐近速度--这是我们算法并行计算特性的一个特征。我们的证明需要对某些非自治系统进行仔细的数学分析，我们相信这种分析将有助于应用数学、动力系统和计算理论等不同领域的研究。最后，我们讨论了未来的研究方向，包括与反应网络计算领域相关的许多重要的开放性理论问题。
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Pub Date : 2024-11-22DOI: 10.1016/j.tcs.2024.114976
Elie Abboud , Noga Ron-Zewi
We investigate the relation between δ and ϵ required for obtaining a -approximation in time for closest pair problems under various distance metrics, and for other related problems in fine-grained complexity.
Specifically, our main result shows that if it is impossible to (exactly) solve the (bichromatic) inner product (IP) problem for vectors of dimension in time , then there is no -approximation algorithm for (bichromatic) Euclidean Closest Pair running in time , where (where ≈ hides polylog factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of δ on ϵ, on the order of . Our result implies in turn that no -approximation algorithm exists for Euclidean closest pair for , unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of for Euclidean closest pair, given by the best known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions, a similar result follows for a host of other related problems in fine-grained hardness of approximation.
Our reduction combines the hardness of approximation framework of Chen and Williams, together with an MA communication protocol for IP over a small alphabet, that is inspired by the MA protocol of Chen (Theory of Computing, 2020).
我们研究了在时间 N2-ϵ 内获得 (1+δ)-approximation 所需的δ 和 ϵ 之间的关系，以解决各种距离度量下的最近对问题，以及细粒度复杂性中的其他相关问题。具体来说，我们的主要结果表明，如果不可能在 N2-ϵ时间内（精确）求解维数为 clogN 的向量的（双色）内积（IP）问题，那么在 N2-2ϵ（其中 δ≈(ϵ/c)2（其中 ≈ 隐藏了多对数因子）时间内运行的（双色）欧氏最接近对算法就不存在（1+δ）-逼近算法。这改进了陈和威廉姆斯（SODA 2019）的先前结果，该结果给出了δ对ϵ的较小多项式依赖性，其数量级为δ≈(ϵ/c)6。我们的结果反过来意味着，对于δ≈ϵ4 的欧氏最邻近对，不存在 (1+δ)- 近似算法，除非对 IP 进行算法改进。这反过来又非常接近δ≈ϵ3 对欧氏最邻近对的近似保证，由 Almam、Chan 和 Williams（FOCS 2016）的已知最佳算法给出。我们的还原法结合了陈和威廉姆斯的近似硬度框架，以及受陈的 MA 协议（《计算理论》，2020 年）启发的小字母表上 IP 的 MA 通信协议。
{"title":"Finer-grained reductions in fine-grained hardness of approximation","authors":"Elie Abboud , Noga Ron-Zewi","doi":"10.1016/j.tcs.2024.114976","DOIUrl":"10.1016/j.tcs.2024.114976","url":null,"abstract":"<div><div>We investigate the relation between <em>δ</em> and <em>ϵ</em> required for obtaining a <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>δ</mi><mo>)</mo></math></span>-approximation in time <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span> for closest pair problems under various distance metrics, and for other related problems in fine-grained complexity.</div><div>Specifically, our main result shows that if it is impossible to (exactly) solve the (bichromatic) inner product (IP) problem for vectors of dimension <span><math><mi>c</mi><mi>log</mi><mo></mo><mi>N</mi></math></span> in time <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow></msup></math></span>, then there is no <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>δ</mi><mo>)</mo></math></span>-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>2</mn><mi>ϵ</mi></mrow></msup></math></span>, where <span><math><mi>δ</mi><mo>≈</mo><msup><mrow><mo>(</mo><mi>ϵ</mi><mo>/</mo><mi>c</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> (where ≈ hides polylog factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of <em>δ</em> on <em>ϵ</em>, on the order of <span><math><mi>δ</mi><mo>≈</mo><msup><mrow><mo>(</mo><mi>ϵ</mi><mo>/</mo><mi>c</mi><mo>)</mo></mrow><mrow><mn>6</mn></mrow></msup></math></span>. Our result implies in turn that no <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>δ</mi><mo>)</mo></math></span>-approximation algorithm exists for Euclidean closest pair for <span><math><mi>δ</mi><mo>≈</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of <span><math><mi>δ</mi><mo>≈</mo><msup><mrow><mi>ϵ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> for Euclidean closest pair, given by the best known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions, a similar result follows for a host of other related problems in fine-grained hardness of approximation.</div><div>Our reduction combines the hardness of approximation framework of Chen and Williams, together with an MA communication protocol for IP over a small alphabet, that is inspired by the MA protocol of Chen (Theory of Computing, 2020).</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 114976"},"PeriodicalIF":0.9,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720863","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 : 2024-11-19DOI: 10.1016/j.tcs.2024.114974
Giuseppe Antonio Di Luna , Ryuhei Uehara , Giovanni Viglietta , Yukiko Yamauchi
A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant ϑ from the robot's current location, where (angles are expressed in radians).
We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric.
We prove that, if (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if , no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected.
The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.
{"title":"Gathering on a circle with limited visibility by anonymous oblivious robots","authors":"Giuseppe Antonio Di Luna , Ryuhei Uehara , Giovanni Viglietta , Yukiko Yamauchi","doi":"10.1016/j.tcs.2024.114974","DOIUrl":"10.1016/j.tcs.2024.114974","url":null,"abstract":"<div><div>A swarm of anonymous oblivious mobile robots, operating in deterministic Look-Compute-Move cycles, is confined within a circular track. All robots agree on the clockwise direction (chirality), they are activated by an adversarial semi-synchronous scheduler (SSYNCH), and an active robot always reaches the destination point it computes (rigidity). Robots have limited visibility: each robot can see only the points on the circle that have an angular distance strictly smaller than a constant <em>ϑ</em> from the robot's current location, where <span><math><mn>0</mn><mo><</mo><mi>ϑ</mi><mo>≤</mo><mi>π</mi></math></span> (angles are expressed in radians).</div><div>We study the Gathering problem for such a swarm of robots: that is, all robots are initially in distinct locations on the circle, and their task is to reach the same point on the circle in a finite number of turns, regardless of the way they are activated by the scheduler. Note that, due to the anonymity of the robots, this task is impossible if the initial configuration is rotationally symmetric; hence, we have to make the assumption that the initial configuration be rotationally asymmetric.</div><div>We prove that, if <span><math><mi>ϑ</mi><mo>=</mo><mi>π</mi></math></span> (i.e., each robot can see the entire circle except its antipodal point), there is a distributed algorithm that solves the Gathering problem for swarms of any size. By contrast, we also prove that, if <span><math><mi>ϑ</mi><mo>≤</mo><mi>π</mi><mo>/</mo><mn>2</mn></math></span>, no distributed algorithm solves the Gathering problem, regardless of the size of the swarm, even under the assumption that the initial configuration is rotationally asymmetric and the visibility graph of the robots is connected.</div><div>The latter impossibility result relies on a probabilistic technique based on random perturbations, which is novel in the context of anonymous mobile robots. Such a technique is of independent interest, and immediately applies to other Pattern-Formation problems.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114974"},"PeriodicalIF":0.9,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703607","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 : 2024-11-17DOI: 10.1016/j.tcs.2024.114973
Peter Damaschke , Fredrik Ekstedt , Raad Salman
The vertices of an edge-weighted clique shall be placed in the plane so as to minimize the sum of all weighted distances, called the spread. Driven by practical applications in factory layout planning, we consider this problem under several constraints. First we show, in the Manhattan metric, the NP-completeness of the version where some vertices are already placed, and some minimum distance is prescribed between any two vertices. However, we can optimally append one new vertex to n placed vertices in time. For the problem without minimum distance requirements but with many unplaced vertices, we give some structural properties of optimal solutions.
边缘加权小块的顶点应放置在平面上，以最小化所有加权距离的总和，即扩散。在工厂布局规划实际应用的驱动下，我们在几个约束条件下考虑了这个问题。首先，在曼哈顿度量中，我们展示了一些顶点已被放置，且任意两个顶点之间规定了最小距离的版本的 NP 完备性。然而，我们可以在 O(n2) 时间内优化地将一个新顶点添加到 n 个已放置的顶点上。对于没有最小距离要求但有许多未放置顶点的问题，我们给出了最优解的一些结构特性。
{"title":"On central placements of new vertices in a planar point set","authors":"Peter Damaschke , Fredrik Ekstedt , Raad Salman","doi":"10.1016/j.tcs.2024.114973","DOIUrl":"10.1016/j.tcs.2024.114973","url":null,"abstract":"<div><div>The vertices of an edge-weighted clique shall be placed in the plane so as to minimize the sum of all weighted distances, called the spread. Driven by practical applications in factory layout planning, we consider this problem under several constraints. First we show, in the Manhattan metric, the NP-completeness of the version where some vertices are already placed, and some minimum distance is prescribed between any two vertices. However, we can optimally append one new vertex to <em>n</em> placed vertices in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> time. For the problem without minimum distance requirements but with many unplaced vertices, we give some structural properties of optimal solutions.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114973"},"PeriodicalIF":0.9,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703604","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 : 2024-11-15DOI: 10.1016/j.tcs.2024.114958
Marcella Anselmo , Manuela Flores , Maria Madonia
An overlap, or border, of a word is a prefix that is equal to the suffix of the same length. An overlap with q errors is a prefix which has distance q from the suffix of the same length; here, 0-error overlaps are classic ones. Unbordered, or bifix-free, words are a central notion in combinatorics on words and have a prominent role in many related areas, such as pattern matching or frame synchronization. On the other hand, words with 2-error overlaps arose as a characterization of isometric words, a notion recently introduced in the framework of hypercubes and their isometric subgraphs. This paper investigates the density of words with 0, 1, 2-error overlaps, where the words are taken over a generic k-ary alphabet, , and the distance they refer to is the Hamming or the Lee distance. Estimates on the limit density values are provided and compared in the case of binary and quaternary alphabets.
{"title":"Density of k-ary words with 0, 1, 2 - error overlaps","authors":"Marcella Anselmo , Manuela Flores , Maria Madonia","doi":"10.1016/j.tcs.2024.114958","DOIUrl":"10.1016/j.tcs.2024.114958","url":null,"abstract":"<div><div>An overlap, or border, of a word is a prefix that is equal to the suffix of the same length. An overlap with <em>q</em> errors is a prefix which has distance <em>q</em> from the suffix of the same length; here, 0-error overlaps are classic ones. Unbordered, or bifix-free, words are a central notion in combinatorics on words and have a prominent role in many related areas, such as pattern matching or frame synchronization. On the other hand, words with 2-error overlaps arose as a characterization of isometric words, a notion recently introduced in the framework of hypercubes and their isometric subgraphs. This paper investigates the density of words with 0, 1, 2-error overlaps, where the words are taken over a generic <em>k</em>-ary alphabet, <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, and the distance they refer to is the Hamming or the Lee distance. Estimates on the limit density values are provided and compared in the case of binary and quaternary alphabets.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114958"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703606","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 : 2024-11-15DOI: 10.1016/j.tcs.2024.114969
Qi He, Yan Wang, Jianxi Fan, Baolei Cheng
Vertex-independent spanning trees (short for VISTs) serve as pivotal constructs in numerous network algorithms and have been the subject of extensive research for three decades. The n-dimensional complete Josephus cube , derived from the Josephus cube, was first proposed to achieve better fault tolerance while maximizing routing efficiency (no sacrificing routing efficiency). Compared to the Josephus cube, it exhibits enhanced symmetry, improved connectivity, and better fault tolerance while maintaining efficient embedding, incremental scalability, and short diameter (). This paper studies the existence and construction of VISTs in rooted at an arbitrary vertex. To determine the specific connection edge between vertex v and its parent in the spanning tree , three algorithms were first proposed to calculate the values of , , and , respectively, where and . Based on these algorithms, a parallel algorithm is proposed to construct () VISTs in using processors. As is -connected, our algorithm is designed to yield the optimal number of resulting VISTs for . Finally, we present the theoretical proof of the parallel algorithm and demonstrate that its time complexity is .
顶点无关生成树（简称 VIST）是众多网络算法中的关键结构，三十年来一直是广泛研究的主题。由约瑟夫立方体衍生而来的 n 维完整约瑟夫立方体 CJCn 最早被提出，旨在实现更好的容错性，同时最大限度地提高路由效率（不牺牲路由效率）。与约瑟夫立方体相比，它在保持高效嵌入、增量可扩展性和短直径（⌈n2⌉）的同时，表现出更强的对称性、更好的连通性和更好的容错性。本文研究了 CJCn 中以任意顶点为根的 n+2 VIST 的存在和构造。为了确定顶点 v 与其父节点在生成树 Ti 中的具体连接边，首先提出了三种算法，分别计算 Fv,i、Mv,i 和 Hv,i 的值，其中 v∈V(CJCn)，i∈{0,1,⋯,n+1}。在这些算法的基础上，提出了一种并行算法，使用 2n 个处理器在 CJCn 中构建 n+2 (n≥4) 个 VIST。由于 CJCn 是 (n+2)-connected 的，因此我们设计的算法能产生 n≥4 的最佳 VIST 数量。最后，我们给出了并行算法的理论证明，并证明其时间复杂度为 O(n)。
{"title":"Vertex-independent spanning trees in complete Josephus cubes","authors":"Qi He, Yan Wang, Jianxi Fan, Baolei Cheng","doi":"10.1016/j.tcs.2024.114969","DOIUrl":"10.1016/j.tcs.2024.114969","url":null,"abstract":"<div><div>Vertex-independent spanning trees (short for VISTs) serve as pivotal constructs in numerous network algorithms and have been the subject of extensive research for three decades. The <em>n</em>-dimensional 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>, derived from the Josephus cube, was first proposed to achieve better fault tolerance while maximizing routing efficiency (no sacrificing routing efficiency). Compared to the Josephus cube, it exhibits enhanced symmetry, improved connectivity, and better fault tolerance while maintaining efficient embedding, incremental scalability, and short diameter (<span><math><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></math></span>). This paper studies the existence and construction of <span><math><mi>n</mi><mo>+</mo><mn>2</mn></math></span> VISTs in <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> rooted at an arbitrary vertex. To determine the specific connection edge between vertex <em>v</em> and its parent in the spanning tree <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, three algorithms were first proposed to calculate the values of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span>, and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>v</mi><mo>,</mo><mi>i</mi></mrow></msub></math></span>, respectively, where <span><math><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mi>i</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>}</mo></math></span>. Based on these algorithms, a parallel algorithm is proposed to construct <span><math><mi>n</mi><mo>+</mo><mn>2</mn></math></span> (<span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>) VISTs in <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> using <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> processors. As <span><math><mi>C</mi><mi>J</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-connected, our algorithm is designed to yield the optimal number of resulting VISTs for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. Finally, we present the theoretical proof of the parallel algorithm and demonstrate that its time complexity is <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114969"},"PeriodicalIF":0.9,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703605","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}
We aim to solve the mutual visibility problem using N autonomous, indistinguishable, homogeneous, oblivious and opaque point robots in the presence of mobility failure. The faulty robot cannot move when it becomes faulty, but the light remains working. Initially, from any arbitrary configuration, the problem of mutual visibility using robots aims to reach a configuration where any two robots can see each other. The challenge is to reach to such a configuration in the presence of faulty robots along with obstructed visibility under which two robots see each other only if the line segment joining them does not have any robots. Every robot operates in the conventional Look-Compute-Move cycles. Robots neither have any agreement in their coordinate system nor have the knowledge of N. The problem is not solvable for a specific symmetric initial configuration of the robots. We propose an algorithm that tolerates number of faulty robots and uses a constant number of colors in the FSYNC setting. To be specific, the algorithm requires 21 colors and runs in synchronous rounds. We present another algorithm much simpler than the prior one but can tolerate a single faulty robot. This algorithm needs only 2 colors in the SSYNC and 5 colors in the ASYNC setting.
{"title":"Fault-tolerant mutual visibility without any axis agreement in presence of mobility failure","authors":"Subhajit Pramanick, Saswata Jana, Partha Sarathi Mandal","doi":"10.1016/j.tcs.2024.114970","DOIUrl":"10.1016/j.tcs.2024.114970","url":null,"abstract":"<div><div>We aim to solve the mutual visibility problem using <em>N</em> autonomous, indistinguishable, homogeneous, oblivious and opaque point robots in the presence of mobility failure. The faulty robot cannot move when it becomes faulty, but the light remains working. Initially, from any arbitrary configuration, the problem of mutual visibility using robots aims to reach a configuration where any two robots can see each other. The challenge is to reach to such a configuration in the presence of faulty robots along with obstructed visibility under which two robots see each other only if the line segment joining them does not have any robots. Every robot operates in the conventional <em>Look-Compute-Move</em> cycles. Robots neither have any agreement in their coordinate system nor have the knowledge of <em>N</em>. The problem is not solvable for a specific symmetric initial configuration of the robots. We propose an algorithm that tolerates <span><math><mi>f</mi><mspace></mspace><mo>(</mo><mo>≤</mo><mi>N</mi><mo>)</mo></math></span> number of faulty robots and uses a constant number of colors in the FSYNC setting. To be specific, the algorithm requires 21 colors and runs in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> synchronous rounds. We present another algorithm much simpler than the prior one but can tolerate a single faulty robot. This algorithm needs only 2 colors in the SSYNC and 5 colors in the ASYNC setting.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114970"},"PeriodicalIF":0.9,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142702874","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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