Theoretical Computer Science最新文献

{"title":"Towards strong regret minimization sets: Balancing freshness and diversity in data selection","authors":"Hongjie Guo ,&nbsp;Jianzhong Li ,&nbsp;Hong Gao","doi":"10.1016/j.tcs.2024.114986","DOIUrl":"10.1016/j.tcs.2024.114986","url":null,"abstract":"<div><div>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-<em>k</em> 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 <em>k</em> 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.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1026 ","pages":"Article 114986"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720861","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}

{"title":"Adding direction constraints to the 1-2-3 Conjecture","authors":"Julien Bensmail ,&nbsp;Hervé Hocquard ,&nbsp;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}

{"title":"Dichotomies for tree minor containment with structural parameters","authors":"Tatsuya Gima ,&nbsp;Soh Kumabe ,&nbsp;Kazuhiro Kurita ,&nbsp;Yuto Okada ,&nbsp;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}

{"title":"Chemical mass-action systems as analog computers: Implementing arithmetic computations at specified speed","authors":"David F. Anderson ,&nbsp;Badal Joshi","doi":"10.1016/j.tcs.2024.114983","DOIUrl":"10.1016/j.tcs.2024.114983","url":null,"abstract":"<div><div>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, <em>m</em>th roots (for <span><math><mi>m</mi><mo>≥</mo><mn>2</mn></math></span>), 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.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1025 ","pages":"Article 114983"},"PeriodicalIF":0.9,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142719766","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}

{"title":"Finer-grained reductions in fine-grained hardness of approximation","authors":"Elie Abboud ,&nbsp;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}

{"title":"Gathering on a circle with limited visibility by anonymous oblivious robots","authors":"Giuseppe Antonio Di Luna ,&nbsp;Ryuhei Uehara ,&nbsp;Giovanni Viglietta ,&nbsp;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>&lt;</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}

{"title":"On central placements of new vertices in a planar point set","authors":"Peter Damaschke ,&nbsp;Fredrik Ekstedt ,&nbsp;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}

{"title":"Density of k-ary words with 0, 1, 2 - error overlaps","authors":"Marcella Anselmo ,&nbsp;Manuela Flores ,&nbsp;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}

{"title":"Vertex-independent spanning trees in complete Josephus cubes","authors":"Qi He,&nbsp;Yan Wang,&nbsp;Jianxi Fan,&nbsp;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}

{"title":"Fault-tolerant mutual visibility without any axis agreement in presence of mobility failure","authors":"Subhajit Pramanick,&nbsp;Saswata Jana,&nbsp;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}