Pub Date : 2026-01-11DOI: 10.1016/j.tcs.2026.115754
Nicolas Bédaride , Valérie Berthé , Antoine Julien
This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of factors that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.
{"title":"On balance properties of hypercubic billiard words","authors":"Nicolas Bédaride , Valérie Berthé , Antoine Julien","doi":"10.1016/j.tcs.2026.115754","DOIUrl":"10.1016/j.tcs.2026.115754","url":null,"abstract":"<div><div>This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of factors that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115754"},"PeriodicalIF":1.0,"publicationDate":"2026-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980986","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 : 2026-01-11DOI: 10.1016/j.tcs.2026.115746
Zsuzsanna Lipták , Francesco Masillo , Gonzalo Navarro
We consider the problem of maintaining a collection of strings while efficiently supporting splits and concatenations on them, as well as comparing two substrings, and computing the longest common prefix between two suffixes. This problem can be solved in optimal time whp for the updates and worst-case time for the queries, where N is the total collection size [Gawrychowski et al., SODA 2018]. We present here a much simpler solution based on a forest of enhanced splay trees (FeST), where both the updates and the substring comparison take amortized time, n being the sum of the lengths of the strings involved in the operation. The length ℓ of the longest common prefix is computed in amortized time. Our query results are correct whp. Our simpler solution enables other more general updates in amortized time, such as reversing a substring and/or mapping its symbols. We can also make FeST use compact space, and extend it to regard substrings as circular or as their omega extension. A C++-implementation of our FeST data structure is available at https://github.com/fmasillo/FeST.
{"title":"A textbook solution for dynamic strings","authors":"Zsuzsanna Lipták , Francesco Masillo , Gonzalo Navarro","doi":"10.1016/j.tcs.2026.115746","DOIUrl":"10.1016/j.tcs.2026.115746","url":null,"abstract":"<div><div>We consider the problem of maintaining a collection of strings while efficiently supporting splits and concatenations on them, as well as comparing two substrings, and computing the longest common prefix between two suffixes. This problem can be solved in optimal time <span><math><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mi>N</mi><mo>)</mo></mrow></math></span> whp for the updates and <span><math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> worst-case time for the queries, where <em>N</em> is the total collection size [Gawrychowski et al., SODA 2018]. We present here a much simpler solution based on a forest of enhanced splay trees (FeST), where both the updates and the substring comparison take <span><math><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> amortized time, <em>n</em> being the sum of the lengths of the strings involved in the operation. The length ℓ of the longest common prefix is computed in <span><math><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mi>n</mi><mo>+</mo><msup><mi>log</mi><mn>2</mn></msup><mi>ℓ</mi><mo>)</mo></mrow></math></span> amortized time. Our query results are correct whp. Our simpler solution enables other more general updates in <span><math><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> amortized time, such as reversing a substring and/or mapping its symbols. We can also make FeST use compact space, and extend it to regard substrings as circular or as their omega extension. A <span>C++</span>-implementation of our FeST data structure is available at <span><span>https://github.com/fmasillo/FeST</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115746"},"PeriodicalIF":1.0,"publicationDate":"2026-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078801","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 : 2026-01-09DOI: 10.1016/j.tcs.2026.115747
Maxime Crochemore , Costas S. Iliopoulos , Jakub Radoszewski , Wojciech Rytter , Juliusz Straszyński , Tomasz Waleń , Wiktor Zuba
Internal pattern matching requires one to answer queries about factors of a given string. Many results are known on answering internal period queries, asking for the periods of a given factor. In this paper we investigate internal queries asking for covers (also known as quasiperiods) of a given factor. Let n denote the length of the string and m denote the length of the factor in question. We propose a data structure that answers such queries in time for the shortest cover and in time for a representation of all the covers, after time and space preprocessing.
This is a full version of a conference paper at SPIRE 2020 with query complexities improved by a log log n-factor and additional applications.
{"title":"Internal quasiperiod queries","authors":"Maxime Crochemore , Costas S. Iliopoulos , Jakub Radoszewski , Wojciech Rytter , Juliusz Straszyński , Tomasz Waleń , Wiktor Zuba","doi":"10.1016/j.tcs.2026.115747","DOIUrl":"10.1016/j.tcs.2026.115747","url":null,"abstract":"<div><div>Internal pattern matching requires one to answer queries about factors of a given string. Many results are known on answering internal period queries, asking for the periods of a given factor. In this paper we investigate internal queries asking for covers (also known as quasiperiods) of a given factor. Let <em>n</em> denote the length of the string and <em>m</em> denote the length of the factor in question. We propose a data structure that answers such queries in <span><math><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mi>m</mi><mo>)</mo></mrow></math></span> time for the shortest cover and in <span><math><mrow><mi>O</mi><mo>(</mo><mi>log</mi><mi>m</mi><mi>log</mi><mi>log</mi><mi>m</mi><mo>)</mo></mrow></math></span> time for a representation of all the covers, after <span><math><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mi>n</mi><mo>)</mo></mrow></math></span> time and space preprocessing.</div><div>This is a full version of a conference paper at SPIRE 2020 with query complexities improved by a log log <em>n</em>-factor and additional applications.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115747"},"PeriodicalIF":1.0,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980990","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 : 2026-01-07DOI: 10.1016/j.tcs.2026.115744
Zi-Yuan Liu , Masahiro Mambo , Raylin Tso , Yi-Fan Tseng
In this study, we revisit the cryptosystem termed public-key encryption with filtered equality test (PKEFET), originally devised by Huang et al. (J. Comput. Syst. Sci. 2017) and subsequently refined by Chen et al. (Des. Codes Cryptogr. 2021). This notion allows users to delegate the equality test functionality of ciphertexts to a tester, but only for ciphertexts associated with a selected message set. More specifically, consider two ciphertexts related to plaintexts m and m′, along with their respective tokens related to two plaintext sets m and m′. Assuming m ∈ m and m′ ∈ m′ hold simultaneously, then any tester has the ability to determine if the ciphertexts are associated with an identical plaintext (i.e., ). With this functionality, PKEFET is valuable in applications such as encrypted data search and spam filtering. However, there exist several challenges in the current PKEFET schemes, such as: (i) security is limited to indistinguishability against “non-adaptive” chosen-ciphertext attacks (CCA1) and (ii) the computational and storage complexities scale linearly with the maximum number of plaintexts that a token can authenticate. To address these challenges, by employing public-key encryption and key-policy attribute-based encryption (KPABE) supporting OR-gate policies as the foundational building blocks, we propose a generic construction of PKEFET. In particular, we show that the required KPABE can be obtained from identity-based encryption. We demonstrate that the proposed construction satisfies one-wayness and indistinguishability under strong filtered equality test against adaptive chosen-ciphertext attacks (CCA2). Moreover, the resulting scheme has significant computational and storage complexity advantages compared to existing PKEFET schemes.
{"title":"Public-key encryption with filtered equality test against adaptive chosen-ciphertext attacks","authors":"Zi-Yuan Liu , Masahiro Mambo , Raylin Tso , Yi-Fan Tseng","doi":"10.1016/j.tcs.2026.115744","DOIUrl":"10.1016/j.tcs.2026.115744","url":null,"abstract":"<div><div>In this study, we revisit the cryptosystem termed public-key encryption with filtered equality test (PKEFET), originally devised by Huang et al. (<em>J. Comput. Syst. Sci.</em> 2017) and subsequently refined by Chen et al. (<em>Des. Codes Cryptogr.</em> 2021). This notion allows users to delegate the equality test functionality of ciphertexts to a tester, but only for ciphertexts associated with a selected message set. More specifically, consider two ciphertexts related to plaintexts <em>m</em> and <em>m</em>′, along with their respective tokens related to two plaintext sets <strong><em>m</em></strong> and <strong><em>m</em></strong>′. Assuming <em>m</em> ∈ <strong><em>m</em></strong> and <em>m</em>′ ∈ <strong><em>m</em></strong>′ hold simultaneously, then any tester has the ability to determine if the ciphertexts are associated with an identical plaintext (i.e., <span><math><mrow><mi>m</mi><mo>=</mo><msup><mi>m</mi><mo>′</mo></msup></mrow></math></span>). With this functionality, PKEFET is valuable in applications such as encrypted data search and spam filtering. However, there exist several challenges in the current PKEFET schemes, such as: (i) security is limited to indistinguishability against “non-adaptive” chosen-ciphertext attacks (CCA1) and (ii) the computational and storage complexities scale linearly with the maximum number of plaintexts that a token can authenticate. To address these challenges, by employing public-key encryption and key-policy attribute-based encryption (KPABE) supporting OR-gate policies as the foundational building blocks, we propose a generic construction of PKEFET. In particular, we show that the required KPABE can be obtained from identity-based encryption. We demonstrate that the proposed construction satisfies one-wayness and indistinguishability under strong filtered equality test against adaptive chosen-ciphertext attacks (CCA2). Moreover, the resulting scheme has significant computational and storage complexity advantages compared to existing PKEFET schemes.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115744"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080049","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 : 2026-01-07DOI: 10.1016/j.tcs.2026.115743
Thomas W. Cusick , Younhwan Cheon
A Boolean function gn in n variables is rotation symmetric (RS) if it is invariant under powers of . An RS function is monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial, say where d is the degree of the function. An MRS function in n variables is called truncated rotation symmetric (TRS) if the function stops the expansion for the n-variable MRS function at the term where xn first occurs. Truncated functions are important because they are used in the computation of linear recursions which the weights of any RS functions are known to satisfy. Computing these recursions in general is very complex. This paper proves that for the quadratic TRS functions, an explicit formula for the generating function for the weights can be proved. This removes the need for the complex computation and makes the weight computation much simpler.
{"title":"Quadratic truncated rotation symmetric Boolean functions","authors":"Thomas W. Cusick , Younhwan Cheon","doi":"10.1016/j.tcs.2026.115743","DOIUrl":"10.1016/j.tcs.2026.115743","url":null,"abstract":"<div><div>A Boolean function <em>g<sub>n</sub></em> in <em>n</em> variables is rotation symmetric (RS) if it is invariant under powers of <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>,</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo></mrow></mrow></math></span>. An RS function is monomial rotation symmetric (MRS) if it is generated by applying powers of <em>ρ</em> to a single monomial, say <span><math><mrow><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mrow><mi>a</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msub><mo>…</mo><msub><mi>x</mi><mrow><mi>a</mi><mo>(</mo><mi>d</mi><mo>)</mo></mrow></msub><mo>,</mo></mrow></math></span> where <em>d</em> is the degree of the function. An MRS function in <em>n</em> variables is called truncated rotation symmetric (TRS) if the function stops the expansion for the <em>n</em>-variable MRS function at the term where <em>x<sub>n</sub></em> first occurs. Truncated functions are important because they are used in the computation of linear recursions which the weights of any RS functions are known to satisfy. Computing these recursions in general is very complex. This paper proves that for the quadratic TRS functions, an explicit formula for the generating function for the weights can be proved. This removes the need for the complex computation and makes the weight computation much simpler.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115743"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928841","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 : 2026-01-03DOI: 10.1016/j.tcs.2025.115720
Jonathan Wagner, Reshef Meir
We present a strategy-proof public goods budgeting mechanism where agents determine both the total volume of expanses and the specific allocation. It is constructed as a modification of VCG to a non-typical environment, namely where we do not assume quasi-linear utilities nor direct revelation. We further show that under plausible assumptions it satisfies strategyproofness in strictly dominant strategies, and consequently implements the social optimum as a Coalition-Proof Nash Equilibrium. A primary (albeit not an exclusive) motivation of our model is Participatory Budgeting, where members of a community collectively decide the spending policy of public tax dollars. While incentives alignment in our mechanism, as in classic VCG, is achieved via individual payments we charge from agents, in a PB context that seems unreasonable. Our second main result thus provides that, under further specifications relevant in that context, these payments will vanish in large populations. In the last section we expand the mechanism’s definition to a class of mechanisms in which the designer can prioritize certain outcomes she sees as desirable. In particular we give the example of favoring equitable/egalitarian allocations.
{"title":"Strategy-proof budgeting via a VCG-like mechanism","authors":"Jonathan Wagner, Reshef Meir","doi":"10.1016/j.tcs.2025.115720","DOIUrl":"10.1016/j.tcs.2025.115720","url":null,"abstract":"<div><div>We present a strategy-proof public goods budgeting mechanism where agents determine both the total volume of expanses and the specific allocation. It is constructed as a modification of VCG to a non-typical environment, namely where we do not assume quasi-linear utilities nor direct revelation. We further show that under plausible assumptions it satisfies strategyproofness in strictly dominant strategies, and consequently implements the social optimum as a Coalition-Proof Nash Equilibrium. A primary (albeit not an exclusive) motivation of our model is Participatory Budgeting, where members of a community collectively decide the spending policy of public tax dollars. While incentives alignment in our mechanism, as in classic VCG, is achieved via individual payments we charge from agents, in a PB context that seems unreasonable. Our second main result thus provides that, under further specifications relevant in that context, these payments will vanish in large populations. In the last section we expand the mechanism’s definition to a class of mechanisms in which the designer can prioritize certain outcomes she sees as desirable. In particular we give the example of favoring equitable/egalitarian allocations.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1067 ","pages":"Article 115720"},"PeriodicalIF":1.0,"publicationDate":"2026-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039653","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 : 2026-01-02DOI: 10.1016/j.tcs.2025.115725
Xiangping Chu, Qingguo Li
In this paper, we investigate the product of H-posets and their function spaces. Our main result establishes that the function space [X → Y], consisting of all Scott continuous functions from X to Y, is an H-poset when X is a continuous poset and Y is an H-poset with a smallest element. This finding generalizes earlier results reported by Kou, Liu, and Luo. Building on these insights, we derive a cartesian closed full subcategory of posets and Scott continuous functions. Additionally, we prove that the finite product of meet continuous H-posets remains a meet continuous H-poset.
{"title":"The category of H-posets","authors":"Xiangping Chu, Qingguo Li","doi":"10.1016/j.tcs.2025.115725","DOIUrl":"10.1016/j.tcs.2025.115725","url":null,"abstract":"<div><div>In this paper, we investigate the product of <em>H</em>-posets and their function spaces. Our main result establishes that the function space [<em>X</em> → <em>Y</em>], consisting of all Scott continuous functions from <em>X</em> to <em>Y</em>, is an <em>H</em>-poset when <em>X</em> is a continuous poset and <em>Y</em> is an <em>H</em>-poset with a smallest element. This finding generalizes earlier results reported by Kou, Liu, and Luo. Building on these insights, we derive a cartesian closed full subcategory of posets and Scott continuous functions. Additionally, we prove that the finite product of meet continuous <em>H</em>-posets remains a meet continuous <em>H</em>-poset.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115725"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145908870","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 : 2026-01-02DOI: 10.1016/j.tcs.2025.115730
Marcos E. González Laffitte , René González-Martínez , Amanda Montejano
We study local and global amoebas, which are graphs that have unique properties with respect to interpolation techniques in graphs. Our work includes a proof that almost every graph is not an amoeba, and the identification of a special type of edge-replacement called weird-edge-replacements. Additionally, we provide an infinite family of trees that are both weird local and global amoebas. Our contributions extend to the development and implementation of several algorithms for detecting local and global amoebas, which are made available in a public repository along with multiple examples.
{"title":"On the detection of local and global amoebas: Theoretical insights and practical algorithms","authors":"Marcos E. González Laffitte , René González-Martínez , Amanda Montejano","doi":"10.1016/j.tcs.2025.115730","DOIUrl":"10.1016/j.tcs.2025.115730","url":null,"abstract":"<div><div>We study local and global amoebas, which are graphs that have unique properties with respect to interpolation techniques in graphs. Our work includes a proof that almost every graph is not an amoeba, and the identification of a special type of edge-replacement called weird-edge-replacements. Additionally, we provide an infinite family of trees that are both weird local and global amoebas. Our contributions extend to the development and implementation of several algorithms for detecting local and global amoebas, which are made available in a public repository along with multiple examples.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115730"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928840","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 : 2026-01-02DOI: 10.1016/j.tcs.2025.115741
Hao Wang, Yan Wang, Baolei Cheng, Jianxi Fan
Spanning trees are completely independent spanning trees (CISTs) if and only if they are edge-disjoint and each node is internal in at most one tree. CISTs have a wide range of applications in routing protection, data transmission, etc., and can improve reliability, fault tolerance, and information security. Line graphs have received increasing attention in recent years for the construction of multiple CISTs, as they are more likely to satisfy the structural conditions required for their existence. This paper introduces an algorithm for constructing multiple CISTs in the line graph of the complete tripartite graph (denoted by ), using multiple two-dimensional matrices to guide the construction process. Furthermore, it presents a method for constructing multiple CISTs in the line graph of the complete multipartite graph (denoted by , when φ ≥ 4), utilizing edge-disjoint Hamiltonian cycles of the complete graph. In our simulation study, we employed multiple CISTs as transmission paths and compared their performance with shortest-path routing in terms of transmission latency and resilience to node failures.
{"title":"Completely independent spanning trees in the line graph of complete multipartite graphs","authors":"Hao Wang, Yan Wang, Baolei Cheng, Jianxi Fan","doi":"10.1016/j.tcs.2025.115741","DOIUrl":"10.1016/j.tcs.2025.115741","url":null,"abstract":"<div><div>Spanning trees <span><math><mrow><msub><mi>T</mi><mn>1</mn></msub><mo>,</mo><msub><mi>T</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>T</mi><mi>t</mi></msub></mrow></math></span> are completely independent spanning trees (CISTs) if and only if they are edge-disjoint and each node is internal in at most one tree. CISTs have a wide range of applications in routing protection, data transmission, etc., and can improve reliability, fault tolerance, and information security. Line graphs have received increasing attention in recent years for the construction of multiple CISTs, as they are more likely to satisfy the structural conditions required for their existence. This paper introduces an algorithm for constructing multiple CISTs in the line graph of the complete tripartite graph <span><math><msub><mi>K</mi><mrow><msub><mi>n</mi><mn>3</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub></math></span> (denoted by <span><math><mrow><mi>L</mi><mo>(</mo><msub><mi>K</mi><mrow><msub><mi>n</mi><mn>3</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub><mo>)</mo></mrow></math></span>), using multiple two-dimensional matrices to guide the construction process. Furthermore, it presents a method for constructing multiple CISTs in the line graph of the complete multipartite graph <span><math><msub><mi>K</mi><mrow><msub><mi>n</mi><mi>φ</mi></msub><mo>,</mo><msub><mi>n</mi><mrow><mi>φ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub></math></span> (denoted by <span><math><mrow><mi>L</mi><mo>(</mo><msub><mi>K</mi><mrow><msub><mi>n</mi><mi>φ</mi></msub><mo>,</mo><msub><mi>n</mi><mrow><mi>φ</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>n</mi><mn>1</mn></msub></mrow></msub><mo>)</mo></mrow></math></span>, when φ ≥ 4), utilizing edge-disjoint Hamiltonian cycles of the complete graph. In our simulation study, we employed multiple CISTs as transmission paths and compared their performance with shortest-path routing in terms of transmission latency and resilience to node failures.</div></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":"1066 ","pages":"Article 115741"},"PeriodicalIF":1.0,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928842","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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