Pub Date : 2024-07-09DOI: 10.4230/LIPIcs.FUN.2024.26
Alessandro Panconesi, Pietro Maria Posta, Mirko Giacchini
In the video game"7 Billion Humans", the player is requested to direct a group of workers to various destinations by writing a program that is executed simultaneously on each worker. While the game is quite rich and, indeed, it is considered one of the best games for beginners to learn the basics of programming, we show that even extremely simple versions are already NP-Hard or PSPACE-Hard.
{"title":"Coordinating \"7 Billion Humans\" Is Hard","authors":"Alessandro Panconesi, Pietro Maria Posta, Mirko Giacchini","doi":"10.4230/LIPIcs.FUN.2024.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2024.26","url":null,"abstract":"In the video game\"7 Billion Humans\", the player is requested to direct a group of workers to various destinations by writing a program that is executed simultaneously on each worker. While the game is quite rich and, indeed, it is considered one of the best games for beginners to learn the basics of programming, we show that even extremely simple versions are already NP-Hard or PSPACE-Hard.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"46 2","pages":"26:1-26:16"},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141663780","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}
Pub Date : 2022-03-28DOI: 10.48550/arXiv.2203.14864
N. Aravind, Neeldhara Misra, Harshil Mittal
We introduce a generalization of"Solo Chess", a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 x 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e, make a sequence of captures after which only one piece is left. We generalize this game to unbounded boards with $n$ pieces, each of which have a given number of captures that they are permitted to make. We show that Generalized Solo Chess is NP-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be NP-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of Generalized Solo Chess can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board. Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game, which involves clearing a graph of tokens via captures along edges. This game subsumes Generalized Solo Chess played by knights. We show that the Graph Capture Game is NP-complete for undirected graphs and DAGs.
{"title":"Chess is hard even for a single player","authors":"N. Aravind, Neeldhara Misra, Harshil Mittal","doi":"10.48550/arXiv.2203.14864","DOIUrl":"https://doi.org/10.48550/arXiv.2203.14864","url":null,"abstract":"We introduce a generalization of\"Solo Chess\", a single-player variant of the game that can be played on chess.com. The standard version of the game is played on a regular 8 x 8 chessboard by a single player, with only white pieces, using the following rules: every move must capture a piece, no piece may capture more than 2 times, and if there is a King on the board, it must be the final piece. The goal is to clear the board, i.e, make a sequence of captures after which only one piece is left. We generalize this game to unbounded boards with $n$ pieces, each of which have a given number of captures that they are permitted to make. We show that Generalized Solo Chess is NP-complete, even when it is played by only rooks that have at most two captures remaining. It also turns out to be NP-complete even when every piece is a queen with exactly two captures remaining in the initial configuration. In contrast, we show that solvable instances of Generalized Solo Chess can be completely characterized when the game is: a) played by rooks on a one-dimensional board, and b) played by pawns with two captures left on a 2D board. Inspired by Generalized Solo Chess, we also introduce the Graph Capture Game, which involves clearing a graph of tokens via captures along edges. This game subsumes Generalized Solo Chess played by knights. We show that the Graph Capture Game is NP-complete for undirected graphs and DAGs.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115717221","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}
Pub Date : 2022-02-22DOI: 10.4230/LIPIcs.FUN.2022.13
Justin Dallant, J. Iacono
Consider a variant of Tetris played on a board of width w and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of O ( n ) rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction from the Multiphase problem [Pătraşcu, 2010] that on a board of width w = Θ( n ), if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time O ( n 1 / 2 − (cid:15) ) simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in O ( n 1 / 2 log 3 / 2 n ) time on boards of width n O (1) , matching the lower bound up to an n o (1) factor.
{"title":"How Fast Can We Play Tetris Greedily With Rectangular Pieces?","authors":"Justin Dallant, J. Iacono","doi":"10.4230/LIPIcs.FUN.2022.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2022.13","url":null,"abstract":"Consider a variant of Tetris played on a board of width w and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of O ( n ) rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction from the Multiphase problem [Pătraşcu, 2010] that on a board of width w = Θ( n ), if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time O ( n 1 / 2 − (cid:15) ) simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in O ( n 1 / 2 log 3 / 2 n ) time on boards of width n O (1) , matching the lower bound up to an n o (1) factor.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122545184","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}
Pub Date : 2020-11-01DOI: 10.4230/LIPIcs.FUN.2018.14
Michael J. Coulombe, J. Lynch
Abstract We show the undecidability of whether a team has a forced win in a number of well known video games including: Team Fortress 2, Super Smash Brothers: Brawl, and Mario Kart. To do so, we give a simplification of the Team Computation Game from Hearn and Demaine's “Games, Puzzles, and Computation” [7] , and use that to give an undecidable abstract game on graphs. This graph game framework better captures the geometry and common constraints in many games and is thus a powerful tool for showing their computational complexity.
{"title":"Cooperating in Video Games? Impossible! Undecidability of Team Multiplayer Games","authors":"Michael J. Coulombe, J. Lynch","doi":"10.4230/LIPIcs.FUN.2018.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2018.14","url":null,"abstract":"Abstract We show the undecidability of whether a team has a forced win in a number of well known video games including: Team Fortress 2, Super Smash Brothers: Brawl, and Mario Kart. To do so, we give a simplification of the Team Computation Game from Hearn and Demaine's “Games, Puzzles, and Computation” [7] , and use that to give an undecidable abstract game on graphs. This graph game framework better captures the geometry and common constraints in many games and is thus a powerful tool for showing their computational complexity.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123066426","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}
Pub Date : 2020-09-01DOI: 10.4230/LIPIcs.FUN.2021.6
Quentin Bramas, P. Lafourcade, Stéphane Devismes
In 2042, the exoplanet exploration program,1 launched in 2014 by NASA, finally discovers a new exoplanet so-called Poleless, due to the fact that it is not subject to any magnetism. A new generation of autonomous mobile robots, called M2C (for Melomaniac Myopic Chameleon), have been designed to find water on Poleless. To address this problem, we investigate optimal (w.r.t., visibility range and number of used colors) solutions to the infinite grid exploration problem (IGE) by a small team of M2C robots. Our first result shows that minimizing the visibility range and the number of used colors are two orthogonal issues: it is impossible to design a solution to the IGE problem that is optimal w.r.t. both parameters simultaneously. Consequently, we address optimality of these two criteria separately by proposing two algorithms; the former being optimal in terms of visibility range, the latter being optimal in terms of number of used colors. It is worth noticing that these two algorithms use a very small number of robots, respectively six and eight. 2012 ACM Subject Classification Theory of computation → Distributed algorithms
{"title":"Finding Water on Poleless Using Melomaniac Myopic Chameleon Robots","authors":"Quentin Bramas, P. Lafourcade, Stéphane Devismes","doi":"10.4230/LIPIcs.FUN.2021.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2021.6","url":null,"abstract":"In 2042, the exoplanet exploration program,1 launched in 2014 by NASA, finally discovers a new exoplanet so-called Poleless, due to the fact that it is not subject to any magnetism. A new generation of autonomous mobile robots, called M2C (for Melomaniac Myopic Chameleon), have been designed to find water on Poleless. To address this problem, we investigate optimal (w.r.t., visibility range and number of used colors) solutions to the infinite grid exploration problem (IGE) by a small team of M2C robots. Our first result shows that minimizing the visibility range and the number of used colors are two orthogonal issues: it is impossible to design a solution to the IGE problem that is optimal w.r.t. both parameters simultaneously. Consequently, we address optimality of these two criteria separately by proposing two algorithms; the former being optimal in terms of visibility range, the latter being optimal in terms of number of used colors. It is worth noticing that these two algorithms use a very small number of robots, respectively six and eight. 2012 ACM Subject Classification Theory of computation → Distributed algorithms","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134065038","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}
Pub Date : 2020-09-01DOI: 10.4230/LIPIcs.FUN.2021.20
Daiki Miyahara, Léo Robert, P. Lafourcade, So Takeshige, T. Mizuki, Kazumasa Shinagawa, A. Nagao, H. Sone
Takuzu and Juosan are logical Nikoli games in the spirit of Sudoku. In Takuzu, a grid must be filled with 0’s and 1’s under specific constraints. In Juosan, the grid must be filled with vertical and horizontal dashes with specific constraints. We give physical algorithms using cards to realize zero-knowledge proofs for those games. The goal is to allow a player to show that he/she has the solution without revealing it. Previous work on Takuzu showed a protocol with multiple instances needed. We propose two improvements: only one instance needed and a soundness proof. We also propose a similar proof for Juosan game. 2012 ACM Subject Classification Security and privacy → Information-theoretic techniques
{"title":"Card-Based ZKP Protocols for Takuzu and Juosan","authors":"Daiki Miyahara, Léo Robert, P. Lafourcade, So Takeshige, T. Mizuki, Kazumasa Shinagawa, A. Nagao, H. Sone","doi":"10.4230/LIPIcs.FUN.2021.20","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2021.20","url":null,"abstract":"Takuzu and Juosan are logical Nikoli games in the spirit of Sudoku. In Takuzu, a grid must be filled with 0’s and 1’s under specific constraints. In Juosan, the grid must be filled with vertical and horizontal dashes with specific constraints. We give physical algorithms using cards to realize zero-knowledge proofs for those games. The goal is to allow a player to show that he/she has the solution without revealing it. Previous work on Takuzu showed a protocol with multiple instances needed. We propose two improvements: only one instance needed and a soundness proof. We also propose a similar proof for Juosan game. 2012 ACM Subject Classification Security and privacy → Information-theoretic techniques","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125201156","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}
Pub Date : 2020-08-10DOI: 10.4230/LIPIcs.FUN.2021.12
Ross Dempsey, Charles D. Guinn
We study the average-case complexity of the classic Minesweeper game in which players deduce the locations of mines on a two-dimensional lattice. Playing Minesweeper is known to be co-NP-complete. We show empirically that Minesweeper exhibits a phase transition analogous to the well-studied SAT phase transition. Above the critical mine density it becomes almost impossible to play Minesweeper by logical inference. We use a reduction to Boolean unsatisfiability to characterize the hardness of Minesweeper instances, and show that the hardness peaks at the phase transition. Furthermore, we demonstrate algorithmic barriers at the phase transition for polynomial-time approaches to Minesweeper inference. Finally, we comment on expectations for the asymptotic behavior of the phase transition.
{"title":"A Phase Transition in Minesweeper","authors":"Ross Dempsey, Charles D. Guinn","doi":"10.4230/LIPIcs.FUN.2021.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2021.12","url":null,"abstract":"We study the average-case complexity of the classic Minesweeper game in which players deduce the locations of mines on a two-dimensional lattice. Playing Minesweeper is known to be co-NP-complete. We show empirically that Minesweeper exhibits a phase transition analogous to the well-studied SAT phase transition. Above the critical mine density it becomes almost impossible to play Minesweeper by logical inference. We use a reduction to Boolean unsatisfiability to characterize the hardness of Minesweeper instances, and show that the hardness peaks at the phase transition. Furthermore, we demonstrate algorithmic barriers at the phase transition for polynomial-time approaches to Minesweeper inference. Finally, we comment on expectations for the asymptotic behavior of the phase transition.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"205 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134516321","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}
Pub Date : 2020-03-22DOI: 10.4230/LIPIcs.FUN.2021.23
Jérémy Félix Barbay, Bernardo Subercaseaux
The game of Hangman is a classical asymmetric two player game in which one player, the setter, chooses a secret word from a language, that the other player, the guesser, tries to discover through single letter matching queries, answered by all occurrences of this letter if any. In the Evil Hangman variant, the setter can change the secret word during the game, as long as the new choice is consistent with the information already given to the guesser. We show that a greedy strategy for Evil Hangman can perform arbitrarily far from optimal, and most importantly, that playing optimally as an Evil Hangman setter is computationally difficult. The latter result holds even assuming perfect knowledge of the language, for several classes of languages, ranging from Finite to Turing Computable. The proofs are based on reductions to Dominating Set on 3-regular graphs and to the Membership problem, combinatorial problems already known to be computationally hard.
{"title":"The Computational Complexity of Evil Hangman","authors":"Jérémy Félix Barbay, Bernardo Subercaseaux","doi":"10.4230/LIPIcs.FUN.2021.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2021.23","url":null,"abstract":"The game of Hangman is a classical asymmetric two player game in which one player, the setter, chooses a secret word from a language, that the other player, the guesser, tries to discover through single letter matching queries, answered by all occurrences of this letter if any. In the Evil Hangman variant, the setter can change the secret word during the game, as long as the new choice is consistent with the information already given to the guesser. We show that a greedy strategy for Evil Hangman can perform arbitrarily far from optimal, and most importantly, that playing optimally as an Evil Hangman setter is computationally difficult. The latter result holds even assuming perfect knowledge of the language, for several classes of languages, ranging from Finite to Turing Computable. The proofs are based on reductions to Dominating Set on 3-regular graphs and to the Membership problem, combinatorial problems already known to be computationally hard.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117001339","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}
Pub Date : 2020-03-01DOI: 10.4230/LIPIcs.FUN.2021.7
Josh Brunner, Lily Chung, E. Demaine, D. Hendrickson, Adam Hesterberg, Adam Suhl, Avi Zeff
Consider $n^2-1$ unit-square blocks in an $n times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical $1 times 2$ and horizontal $2 times 1$ movable blocks and 4-color Subway Shuffle.
{"title":"1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete","authors":"Josh Brunner, Lily Chung, E. Demaine, D. Hendrickson, Adam Hesterberg, Adam Suhl, Avi Zeff","doi":"10.4230/LIPIcs.FUN.2021.7","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2021.7","url":null,"abstract":"Consider $n^2-1$ unit-square blocks in an $n times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical $1 times 2$ and horizontal $2 times 1$ movable blocks and 4-color Subway Shuffle.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124681830","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}
Pub Date : 2020-02-04DOI: 10.4230/LIPIcs.FUN.2021.22
Suthee Ruangwises, T. Itoh
Numberlink is a logic puzzle for which the player has to connect all pairs of cells with the same numbers by non-crossing paths in a rectangular grid. In this paper, we propose a physical protocol of zero-knowledge proof for Numberlink using a deck of cards, which allows a player to physically show that he/she knows a solution without revealing it. In particular, we develop a physical protocol to count the number of elements in a list that are equal to a given secret value without revealing that value, the positions of elements in the list that are equal to it, or the value of any other element in the list. Our protocol can also be applied to verify the existence of vertex-disjoint paths connecting all given pairs of endpoints in any graph.
{"title":"Physical Zero-Knowledge Proof for Numberlink","authors":"Suthee Ruangwises, T. Itoh","doi":"10.4230/LIPIcs.FUN.2021.22","DOIUrl":"https://doi.org/10.4230/LIPIcs.FUN.2021.22","url":null,"abstract":"Numberlink is a logic puzzle for which the player has to connect all pairs of cells with the same numbers by non-crossing paths in a rectangular grid. In this paper, we propose a physical protocol of zero-knowledge proof for Numberlink using a deck of cards, which allows a player to physically show that he/she knows a solution without revealing it. In particular, we develop a physical protocol to count the number of elements in a list that are equal to a given secret value without revealing that value, the positions of elements in the list that are equal to it, or the value of any other element in the list. Our protocol can also be applied to verify the existence of vertex-disjoint paths connecting all given pairs of endpoints in any graph.","PeriodicalId":293763,"journal":{"name":"Fun with Algorithms","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128905086","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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