{"title":"Fast Uniform Dispersion of a Crash-prone Swarm","authors":"M. Amir, F. Bruckstein","doi":"10.15607/rss.2020.xvi.017","DOIUrl":"https://doi.org/10.15607/rss.2020.xvi.017","url":null,"abstract":"We consider the problem of completely covering an unknown discrete environment with a swarm of asynchronous, frequently-crashing autonomous mobile robots. We represent the environment by a discrete graph, and task the robots with occupying every vertex and with constructing an implicit distributed spanning tree of the graph. The robotic agents activate independently at random exponential waiting times of mean $1$ and enter the graph environment over time from a source location. They grow the environment's coverage by 'settling' at empty locations and aiding other robots' navigation from these locations. The robots are identical and make decisions driven by the same simple and local rule of behaviour. The local rule is based only on the presence of neighbouring robots, and on whether a settled robot points to the current location. Whenever a robot moves, it may crash and disappear from the environment. Each vertex in the environment has limited physical space, so robots frequently obstruct each other. \u0000Our goal is to show that even under conditions of asynchronicity, frequent crashing, and limited physical space, the simple mobile robots complete their mission in linear time asymptotically almost surely, and time to completion degrades gracefully with the frequency of the crashes. Our model and analysis are based on the well-studied \"totally asymmetric simple exclusion process\" in statistical mechanics.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128179576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anaël Grandjean, Benjamin Hellouin de Menibus, Pascal Vanier
{"title":"Aperiodic points in $mathbb Z^2$-subshifts","authors":"Anaël Grandjean, Benjamin Hellouin de Menibus, Pascal Vanier","doi":"10.4230/LIPICS.ICALP.2018.496","DOIUrl":"https://doi.org/10.4230/LIPICS.ICALP.2018.496","url":null,"abstract":"We consider the structure of aperiodic points in $mathbb Z^2$-subshifts, and in particular the positions at which they fail to be periodic. We prove that if a $mathbb Z^2$-subshift contains points whose smallest period is arbitrarily large, then it contains an aperiodic point. This lets us characterise the computational difficulty of deciding if an $mathbb Z^2$-subshift of finite type contains an aperiodic point. Another consequence is that $mathbb Z^2$-subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some $mathbb Z$-subshift. Finally, we use this result to characterize sets of possible slopes of periodicity for $mathbb Z^3$-subshifts of finite type.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"368 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122059003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Periodicity of Generalized Fibonacci-like Sequences","authors":"Alexander V. Evako","doi":"10.12691/AMP-5-1-2","DOIUrl":"https://doi.org/10.12691/AMP-5-1-2","url":null,"abstract":"Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence Fn+2=A+BFn+1+CFn. It is shown that this sequence is periodic with period T>2, if C= -1, |B|<2.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121630507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Toward Quantum Combinatorial Games","authors":"P. Dorbec, M. Mhalla","doi":"10.4204/EPTCS.266.16","DOIUrl":"https://doi.org/10.4204/EPTCS.266.16","url":null,"abstract":"In this paper, we propose a Quantum variation of combinatorial games, generalizing the Quantum Tic-Tac-Toe proposed by Allan Goff. A combinatorial game is a two-player game with no chance and no hidden information, such as Go or Chess. In this paper, we consider the possibility of playing superpositions of moves in such games. We propose different rulesets depending on when superposed moves should be played, and prove that all these rulesets may lead similar games to different outcomes. We then consider Quantum variations of the game of Nim. We conclude with some discussion on the relative interest of the different rulesets.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115161822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Joint Entropy of $d$-Wise-Independent Variables","authors":"Dmitry Gavinsky, P. Pudlák","doi":"10.14712/1213-7243.2015.169","DOIUrl":"https://doi.org/10.14712/1213-7243.2015.169","url":null,"abstract":"How low can the joint entropy of $n$ $d$-wise independent (for $dge2$) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$, for $p<1$)? This question has been posed and partially answered in a recent work of Babai. \u0000In this paper we improve some of his bounds, prove new bounds in a wider range of parameters and show matching upper bounds in some special cases. In particular, we prove tight lower bounds for the min-entropy (as well as the entropy) of pairwise and three-wise independent balanced binary variables for infinitely many values of $n$.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"331 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132302162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On strongly spanning $k$-edge-colorable subgraphs","authors":"V. Mkrtchyan, Gagik N. Vardanyan","doi":"10.7494/OPMATH.2017.37.3.435","DOIUrl":"https://doi.org/10.7494/OPMATH.2017.37.3.435","url":null,"abstract":"A subgraph $H$ of a multigraph $G$ is called strongly spanning, if any vertex of $G$ is not isolated in $H$, while it is called maximum $k$-edge-colorable, if $H$ is proper $k$-edge-colorable and has the largest size. We introduce a graph-parameter $sp(G)$, that coincides with the smallest $k$ that a graph $G$ has a strongly spanning maximum $k$-edge-colorable subgraph. Our first result offers some alternative definitions of $sp(G)$. Next, we show that $Delta(G)$ is an upper bound for $sp(G)$, and then we characterize the class of graphs $G$ that satisfy $sp(G)=Delta(G)$. Finally, we prove some bounds for $sp(G)$ that involve well-known graph-theoretic parameters.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131959440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sasun Hambartsumyan, V. Mkrtchyan, Vahe L. Musoyan, H. Sargsyan
{"title":"The hardness of the independence and matching clutter of a graph","authors":"Sasun Hambartsumyan, V. Mkrtchyan, Vahe L. Musoyan, H. Sargsyan","doi":"10.7494/OPMATH.2016.36.3.375","DOIUrl":"https://doi.org/10.7494/OPMATH.2016.36.3.375","url":null,"abstract":"A {it clutter} (or {it antichain} or {it Sperner family}) $L$ is a pair $(V,E)$, where $V$ is a finite set and $E$ is a family of subsets of $V$ none of which is a subset of another. Usually, the elements of $V$ are called {it vertices} of $L$, and the elements of $E$ are called {it edges} of $L$. A subset $s_e$ of an edge $e$ of a clutter is called {it recognizing} for $e$, if $s_e$ is not a subset of another edge. The {it hardness} of an edge $e$ of a clutter is the ratio of the size of $etextrm{'s}$ smallest recognizing subset to the size of $e$. The hardness of a clutter is the maximum hardness of its edges. We study the hardness of clutters arising from independent sets and matchings of graphs.","PeriodicalId":111854,"journal":{"name":"arXiv: Discrete Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116427433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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