Theoretical computer science plays an important role in the understanding of social networks and their properties. We can model information rippling throughout social networks, or the opinions of social media users for example, using graph theory and Markov chains. In this thesis, we model social networks as graphs, and consider two such processes: 1. Nodes talk to other nodes and find middle ground, causing their opinions to come closer to consensus (the load balancing model) 2. All nodes take the maximum value of their neighbours in lockstep (the synchronous maximum model) We study the convergence behaviours of each process, such as the eventual state of the graph, the convergence time and the period. We provide proofs of the eventual states and periods for each of the above models, and theoretical bounds for the worst case convergence times. We verify these with experiments, and explore further questions such as the average case convergence time of various special classes of graphs, or the convergence times when the model is altered slightly.
{"title":"Convergence Properties of Dynamic Processes on Graphs","authors":"Timothy Horscroft","doi":"arxiv-2406.05147","DOIUrl":"https://doi.org/arxiv-2406.05147","url":null,"abstract":"Theoretical computer science plays an important role in the understanding of\u0000social networks and their properties. We can model information rippling\u0000throughout social networks, or the opinions of social media users for example,\u0000using graph theory and Markov chains. In this thesis, we model social networks\u0000as graphs, and consider two such processes: 1. Nodes talk to other nodes and find middle ground, causing their opinions\u0000to come closer to consensus (the load balancing model) 2. All nodes take the maximum value of their neighbours in lockstep (the\u0000synchronous maximum model) We study the convergence behaviours of each process, such as the eventual\u0000state of the graph, the convergence time and the period. We provide proofs of\u0000the eventual states and periods for each of the above models, and theoretical\u0000bounds for the worst case convergence times. We verify these with experiments,\u0000and explore further questions such as the average case convergence time of\u0000various special classes of graphs, or the convergence times when the model is\u0000altered slightly.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507058","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}
In the Bounded Multiple Traveling Salesman Problem (BMTSP), a tour for each salesman, that starts and ends at the depot and that respects the bounds on the number of cities that a feasible salesman tour should satisfy, is to be constructed. The objective is to minimize the total length of all tours. Already Euclidean traveling salesman problem is NP-hard. We propose a 3-Phase heuristic algorithm for the Euclidean BMTSP. We tested the algorithm for the 22 benchmark instances and 168 new problem instances that we created. We report 19 best known solutions for the 22 benchmark instances including the 12 largest ones. For the newly created instances, we compared the performance of our algorithm with that of an ILP-solver CPLEX, which was able to construct a feasible solution for 71% of the instances within the time limit of two hours imposed by us. For about 10% of the smallest new instances, CPLEX delivered slightly better solutions, where our algorithm took less than 180 seconds for the largest of these instances. For the remaining 61% of the instances solved by CPLEX, the solutions by our heuristic were, on average, about 21.5% better than those obtained by CPLEX.
{"title":"An Algorithm for the Euclidean Bounded Multiple Traveling Salesman Problem","authors":"Víctor Pacheco-Valencia, Nodari Vakhania","doi":"arxiv-2405.18615","DOIUrl":"https://doi.org/arxiv-2405.18615","url":null,"abstract":"In the Bounded Multiple Traveling Salesman Problem (BMTSP), a tour for each\u0000salesman, that starts and ends at the depot and that respects the bounds on the\u0000number of cities that a feasible salesman tour should satisfy, is to be\u0000constructed. The objective is to minimize the total length of all tours.\u0000Already Euclidean traveling salesman problem is NP-hard. We propose a 3-Phase\u0000heuristic algorithm for the Euclidean BMTSP. We tested the algorithm for the 22\u0000benchmark instances and 168 new problem instances that we created. We report 19\u0000best known solutions for the 22 benchmark instances including the 12 largest\u0000ones. For the newly created instances, we compared the performance of our\u0000algorithm with that of an ILP-solver CPLEX, which was able to construct a\u0000feasible solution for 71% of the instances within the time limit of two hours\u0000imposed by us. For about 10% of the smallest new instances, CPLEX delivered\u0000slightly better solutions, where our algorithm took less than 180 seconds for\u0000the largest of these instances. For the remaining 61% of the instances solved\u0000by CPLEX, the solutions by our heuristic were, on average, about 21.5% better\u0000than those obtained by CPLEX.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194474","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}
In this work, we study methodical decomposition of an undirected, unweighted complete graph ($K_n$ of order $n$, size $m$) into minimum number of edge-disjoint trees. We find that $x$, a positive integer, is minimum and $x=lceilfrac{n}{2}rceil$ as the edge set of $K_n$ is decomposed into edge-disjoint trees of size sequence $M = {m_1,m_2,...,m_x}$ where $m_ile(n-1)$ and $Sigma_{i=1}^{x} m_i$ = $frac{n(n-1)}{2}$. For decomposing the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed algorithm takes total $O(m)$ time.
{"title":"An Algorithm for the Decomposition of Complete Graph into Minimum Number of Edge-disjoint Trees","authors":"Antika Sinha, Sanjoy Kumar Saha, Partha Basuchowdhuri","doi":"arxiv-2405.18506","DOIUrl":"https://doi.org/arxiv-2405.18506","url":null,"abstract":"In this work, we study methodical decomposition of an undirected, unweighted\u0000complete graph ($K_n$ of order $n$, size $m$) into minimum number of\u0000edge-disjoint trees. We find that $x$, a positive integer, is minimum and\u0000$x=lceilfrac{n}{2}rceil$ as the edge set of $K_n$ is decomposed into\u0000edge-disjoint trees of size sequence $M = {m_1,m_2,...,m_x}$ where\u0000$m_ile(n-1)$ and $Sigma_{i=1}^{x} m_i$ = $frac{n(n-1)}{2}$. For decomposing\u0000the edge set of $K_n$ into minimum number of edge-disjoint trees, our proposed\u0000algorithm takes total $O(m)$ time.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194298","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}
Zachary Chase, Bogdan Chornomaz, Steve Hanneke, Shay Moran, Amir Yehudayoff
This work studies embedding of arbitrary VC classes in well-behaved VC classes, focusing particularly on extremal classes. Our main result expresses an impossibility: such embeddings necessarily require a significant increase in dimension. In particular, we prove that for every $d$ there is a class with VC dimension $d$ that cannot be embedded in any extremal class of VC dimension smaller than exponential in $d$. In addition to its independent interest, this result has an important implication in learning theory, as it reveals a fundamental limitation of one of the most extensively studied approaches to tackling the long-standing sample compression conjecture. Concretely, the approach proposed by Floyd and Warmuth entails embedding any given VC class into an extremal class of a comparable dimension, and then applying an optimal sample compression scheme for extremal classes. However, our results imply that this strategy would in some cases result in a sample compression scheme at least exponentially larger than what is predicted by the sample compression conjecture. The above implications follow from a general result we prove: any extremal class with VC dimension $d$ has dual VC dimension at most $2d+1$. This bound is exponentially smaller than the classical bound $2^{d+1}-1$ of Assouad, which applies to general concept classes (and is known to be unimprovable for some classes). We in fact prove a stronger result, establishing that $2d+1$ upper bounds the dual Radon number of extremal classes. This theorem represents an abstraction of the classical Radon theorem for convex sets, extending its applicability to a wider combinatorial framework, without relying on the specifics of Euclidean convexity. The proof utilizes the topological method and is primarily based on variants of the Topological Radon Theorem.
{"title":"Dual VC Dimension Obstructs Sample Compression by Embeddings","authors":"Zachary Chase, Bogdan Chornomaz, Steve Hanneke, Shay Moran, Amir Yehudayoff","doi":"arxiv-2405.17120","DOIUrl":"https://doi.org/arxiv-2405.17120","url":null,"abstract":"This work studies embedding of arbitrary VC classes in well-behaved VC\u0000classes, focusing particularly on extremal classes. Our main result expresses\u0000an impossibility: such embeddings necessarily require a significant increase in\u0000dimension. In particular, we prove that for every $d$ there is a class with VC\u0000dimension $d$ that cannot be embedded in any extremal class of VC dimension\u0000smaller than exponential in $d$. In addition to its independent interest, this result has an important\u0000implication in learning theory, as it reveals a fundamental limitation of one\u0000of the most extensively studied approaches to tackling the long-standing sample\u0000compression conjecture. Concretely, the approach proposed by Floyd and Warmuth\u0000entails embedding any given VC class into an extremal class of a comparable\u0000dimension, and then applying an optimal sample compression scheme for extremal\u0000classes. However, our results imply that this strategy would in some cases\u0000result in a sample compression scheme at least exponentially larger than what\u0000is predicted by the sample compression conjecture. The above implications follow from a general result we prove: any extremal\u0000class with VC dimension $d$ has dual VC dimension at most $2d+1$. This bound is\u0000exponentially smaller than the classical bound $2^{d+1}-1$ of Assouad, which\u0000applies to general concept classes (and is known to be unimprovable for some\u0000classes). We in fact prove a stronger result, establishing that $2d+1$ upper\u0000bounds the dual Radon number of extremal classes. This theorem represents an\u0000abstraction of the classical Radon theorem for convex sets, extending its\u0000applicability to a wider combinatorial framework, without relying on the\u0000specifics of Euclidean convexity. The proof utilizes the topological method and\u0000is primarily based on variants of the Topological Radon Theorem.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141173075","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}
We study a tree coloring model introduced by Guidon (2018), initially based on an analogy with a remote control system of a rail yard, seen as switches on a binary tree. For a given binary tree, we formalize the constraints on the coloring, in particular the distribution of the nodes among colors. Following Guidon, we are interested in balanced colorings i.e. colorings which minimize the maximum size of the subsets of the tree nodes distributed by color. With his method, we present balanced colorings for trees of height up to 7. But his method seems difficult to apply for trees of greater height. Also we present another method which gives solutions for arbitrarily large trees. We illustrate it with a balanced coloring for height 8. In the appendix, we give the exact formulas and the asymptotic behavior of the number of colorings as a function of the height of the tree.
{"title":"Remote control system of a binary tree of switches -- II. balancing for a perfect binary tree","authors":"Olivier Golinelli","doi":"arxiv-2405.16968","DOIUrl":"https://doi.org/arxiv-2405.16968","url":null,"abstract":"We study a tree coloring model introduced by Guidon (2018), initially based\u0000on an analogy with a remote control system of a rail yard, seen as switches on\u0000a binary tree. For a given binary tree, we formalize the constraints on the\u0000coloring, in particular the distribution of the nodes among colors. Following\u0000Guidon, we are interested in balanced colorings i.e. colorings which minimize\u0000the maximum size of the subsets of the tree nodes distributed by color. With\u0000his method, we present balanced colorings for trees of height up to 7. But his\u0000method seems difficult to apply for trees of greater height. Also we present\u0000another method which gives solutions for arbitrarily large trees. We illustrate\u0000it with a balanced coloring for height 8. In the appendix, we give the exact\u0000formulas and the asymptotic behavior of the number of colorings as a function\u0000of the height of the tree.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141173086","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}
We study a tree coloring model introduced by Guidon (2018), initially based on an analogy with a remote control system of a rail yard, seen as a switch tree. For a given rooted tree, we formalize the constraints on the coloring, in particular on the minimum number of colors, and on the distribution of the nodes among colors. We show that the sequence $(a_1,a_2,a_3,cdots)$, where $a_i$ denotes the number of nodes with color $i$, satisfies a set of inequalities which only involve the sequence $(n_0,n_1,n_2,cdots)$ where $n_i$ denotes the number of nodes with height $i$. By coloring the nodes according to their depth, we deduce that these inequalities also apply to the sequence $(d_0,d_1,d_2,cdots)$ where $d_i$ denotes the number of nodes with depth $i$.
{"title":"Remote control system of a binary tree of switches -- I. constraints and inequalities","authors":"Olivier Golinelli","doi":"arxiv-2405.16938","DOIUrl":"https://doi.org/arxiv-2405.16938","url":null,"abstract":"We study a tree coloring model introduced by Guidon (2018), initially based\u0000on an analogy with a remote control system of a rail yard, seen as a switch\u0000tree. For a given rooted tree, we formalize the constraints on the coloring, in\u0000particular on the minimum number of colors, and on the distribution of the\u0000nodes among colors. We show that the sequence $(a_1,a_2,a_3,cdots)$, where\u0000$a_i$ denotes the number of nodes with color $i$, satisfies a set of\u0000inequalities which only involve the sequence $(n_0,n_1,n_2,cdots)$ where $n_i$\u0000denotes the number of nodes with height $i$. By coloring the nodes according to\u0000their depth, we deduce that these inequalities also apply to the sequence\u0000$(d_0,d_1,d_2,cdots)$ where $d_i$ denotes the number of nodes with depth $i$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172903","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}
Michał Strada, Sebastian Ernst, Jacek Szybowski, Konrad Kułakowski
Most decision-making models, including the pairwise comparison method, assume the decision-makers honesty. However, it is easy to imagine a situation where a decision-maker tries to manipulate the ranking results. This paper presents three simple manipulation methods in the pairwise comparison method. We then try to detect these methods using appropriately constructed neural networks. Experimental results accompany the proposed solutions on the generated data, showing a considerable manipulation detection level.
{"title":"Detection of decision-making manipulation in the pairwise comparisons method","authors":"Michał Strada, Sebastian Ernst, Jacek Szybowski, Konrad Kułakowski","doi":"arxiv-2405.16693","DOIUrl":"https://doi.org/arxiv-2405.16693","url":null,"abstract":"Most decision-making models, including the pairwise comparison method, assume\u0000the decision-makers honesty. However, it is easy to imagine a situation where a\u0000decision-maker tries to manipulate the ranking results. This paper presents\u0000three simple manipulation methods in the pairwise comparison method. We then\u0000try to detect these methods using appropriately constructed neural networks.\u0000Experimental results accompany the proposed solutions on the generated data,\u0000showing a considerable manipulation detection level.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141173050","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}
Mohammad H. Shekarriz, Dhananjay Thiruvady, Asef Nazari, Rhyd Lewis
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{"title":"Soft happy colourings and community structure of networks","authors":"Mohammad H. Shekarriz, Dhananjay Thiruvady, Asef Nazari, Rhyd Lewis","doi":"arxiv-2405.15663","DOIUrl":"https://doi.org/arxiv-2405.15663","url":null,"abstract":"For $0<rholeq 1$, a $rho$-happy vertex $v$ in a coloured graph $G$ has at\u0000least $rhocdot mathrm{deg}(v)$ same-colour neighbours, and a $rho$-happy\u0000colouring (aka soft happy colouring) of $G$ is a vertex colouring that makes\u0000all the vertices $rho$-happy. A community is a subgraph whose vertices are\u0000more adjacent to themselves than the rest of the vertices. Graphs with\u0000community structures can be modelled by random graph models such as the\u0000stochastic block model (SBM). In this paper, we present several theorems\u0000showing that both of these notions are related, with numerous real-world\u0000applications. We show that, with high probability, communities of graphs in the\u0000stochastic block model induce $rho$-happy colouring on all vertices if certain\u0000conditions on the model parameters are satisfied. Moreover, a probabilistic\u0000threshold on $rho$ is derived so that communities of a graph in the SBM induce\u0000a $rho$-happy colouring. Furthermore, the asymptotic behaviour of $rho$-happy\u0000colouring induced by the graph's communities is discussed when $rho$ is less\u0000than a threshold. We develop heuristic polynomial-time algorithms for soft\u0000happy colouring that often correlate with the graphs' community structure.\u0000Finally, we present an experimental evaluation to compare the performance of\u0000the proposed algorithms thereby demonstrating the validity of the theoretical\u0000results.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141172981","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}
Non-monotone constrained submodular maximization plays a crucial role in various machine learning applications. However, existing algorithms often struggle with a trade-off between approximation guarantees and practical efficiency. The current state-of-the-art is a recent $0.401$-approximation algorithm, but its computational complexity makes it highly impractical. The best practical algorithms for the problem only guarantee $1/e$-approximation. In this work, we present a novel algorithm for submodular maximization subject to a cardinality constraint that combines a guarantee of $0.385$-approximation with a low and practical query complexity of $O(n+k^2)$. Furthermore, we evaluate the empirical performance of our algorithm in experiments based on various machine learning applications, including Movie Recommendation, Image Summarization, and more. These experiments demonstrate the efficacy of our approach.
{"title":"Practical $0.385$-Approximation for Submodular Maximization Subject to a Cardinality Constraint","authors":"Murad Tukan, Loay Mualem, Moran Feldman","doi":"arxiv-2405.13994","DOIUrl":"https://doi.org/arxiv-2405.13994","url":null,"abstract":"Non-monotone constrained submodular maximization plays a crucial role in\u0000various machine learning applications. However, existing algorithms often\u0000struggle with a trade-off between approximation guarantees and practical\u0000efficiency. The current state-of-the-art is a recent $0.401$-approximation\u0000algorithm, but its computational complexity makes it highly impractical. The\u0000best practical algorithms for the problem only guarantee $1/e$-approximation.\u0000In this work, we present a novel algorithm for submodular maximization subject\u0000to a cardinality constraint that combines a guarantee of $0.385$-approximation\u0000with a low and practical query complexity of $O(n+k^2)$. Furthermore, we\u0000evaluate the empirical performance of our algorithm in experiments based on\u0000various machine learning applications, including Movie Recommendation, Image\u0000Summarization, and more. These experiments demonstrate the efficacy of our\u0000approach.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141149173","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}
A correlation is a binary vector that encodes all possible positions of overlaps of two words, where an overlap for an ordered pair of words (u,v) occurs if a suffix of word u matches a prefix of word v. As multiple pairs can have the same correlation, it is relevant to count how many pairs of words share the same correlation depending on the alphabet size and word length n. We exhibit recurrences to compute the number of such pairs -- which is termed population size -- for any correlation; for this, we exploit a relationship between overlaps of two words and self-overlap of one word. This theorem allows us to compute the number of pairs with a longest overlap of a given length and to show that the expected length of the longest border of two words asymptotically diverges, which solves two open questions raised by Gabric in 2022. Finally, we also provide bounds for the asymptotic of the population ratio of any correlation. Given the importance of word overlaps in areas like word combinatorics, bioinformatics, and digital communication, our results may ease analyses of algorithms for string processing, code design, or genome assembly.
相关性是一个二进制向量，它编码了两个单词重叠的所有可能位置，其中，如果单词 u 的后缀与单词 v 的前缀相匹配，则有序单词对 (u,v) 会发生重叠。由于多个单词对可能具有相同的相关性，因此，根据字母表大小和单词长度 n 来计算有多少单词对具有相同的相关性是非常重要的。为此，我们利用了两个词的重叠和一个词的自重叠之间的关系。利用这一定理，我们可以计算出具有给定长度的最长重叠的词对数量，并证明两个词的最长边界的预期长度近似发散，从而解决了加布里克在 2022 年提出的两个悬而未决的问题。最后，我们还给出了任何相关性的人口比率的渐近边界。鉴于单词重叠在单词组合学、生物信息学和数字通信等领域的重要性，我们的结果可能有助于分析字符串处理、代码设计或基因组组装的算法。
{"title":"Counting overlapping pairs of strings","authors":"Eric Rivals, Pengfei Wang","doi":"arxiv-2405.09393","DOIUrl":"https://doi.org/arxiv-2405.09393","url":null,"abstract":"A correlation is a binary vector that encodes all possible positions of\u0000overlaps of two words, where an overlap for an ordered pair of words (u,v)\u0000occurs if a suffix of word u matches a prefix of word v. As multiple pairs can\u0000have the same correlation, it is relevant to count how many pairs of words\u0000share the same correlation depending on the alphabet size and word length n. We\u0000exhibit recurrences to compute the number of such pairs -- which is termed\u0000population size -- for any correlation; for this, we exploit a relationship\u0000between overlaps of two words and self-overlap of one word. This theorem allows\u0000us to compute the number of pairs with a longest overlap of a given length and\u0000to show that the expected length of the longest border of two words\u0000asymptotically diverges, which solves two open questions raised by Gabric in\u00002022. Finally, we also provide bounds for the asymptotic of the population\u0000ratio of any correlation. Given the importance of word overlaps in areas like\u0000word combinatorics, bioinformatics, and digital communication, our results may\u0000ease analyses of algorithms for string processing, code design, or genome\u0000assembly.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061725","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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