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Balanced Black and White Coloring Problem on knights chessboards [article]

Luis Eduardo Urbán Rivero, Rafael López Bracho, Javier Ramírez Rodríguez
2018 arXiv   pre-print
In the same way, Berge in 1972 proposed the problem of placing b black queens and w white queens on a n × n chessboard such that no two queens of different color can attack to each other, the complexity  ...  In this work we deal with the knight piece under the balance property, since this special case is the most difficult for brute force algorithms.  ...  One way to answer the Problem 1.3 with knights is to fix the value of b and maximize the size of white knights set.  ... 
arXiv:1803.07693v2 fatcat:qqdvqfwrgvck7c66m7gsw64yoe

Balanced Black and White Coloring Problem on Knight's Chessboards

Luis E. Urban-Rivero, Javier Ramirez-Rodriguez, Rafael Lopez-Bracho
2018 2018 15th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE)  
In the same way, Berge in 1972 [2] proposed the problem of placing b black queens and w white queens on a n × n chessboard such that no two queens of different color can attack to each other, the complexity  ...  Graph anticoloring problem is partial coloring problem where the main feature is the opposite rule of the graph coloring problem, i.e., if two vertices are adjacent, their assigned colors must be the same  ...  The balanced BWC problem with knights and queens remained unsolved until this work.  ... 
doi:10.1109/iceee.2018.8533915 dblp:conf/iceee/RiveroRL18 fatcat:xlgt3unpljb2tnmsrgc676jfsa

Matching problems

G. Katona, D. Sza´sz
1971 Journal of combinatorial theory. Series B (Print)  
Two types of figures are considered: 1. parallelepiped, 2. figure consisting of two relatively fixed, not necessarily connected, unit cubes, e.g. the field on the chess-board where the knight stands and  ...  a field attacked by the knight.  ...  An n-dimensional parallelepiped R can be covered with knight figures of type a x 0 x ... x 0 if and only if one of its sizes is divisible by 2a. THEOREM 14.  ... 
doi:10.1016/0095-8956(71)90067-0 fatcat:txdkxxurpzbptkzeqfsinghqly

Minimal Block Knight's Tour and Edge with LSB Pixel Replacement Based Encrypted Image Steganography

B. S. Shashikiran, K. Shaila, K. R. Venugopal
2021 SN Computer Science  
In the proposed algorithm, the image encryption and steganography are performed using Knight's move in the game of chess called Knight's Tour Algorithm.  ...  Minimum block or square required for a knight's tour to reach all the squares is 5 × 5 block. The 5 × 5 blocks' pattern generated is used for image encryption.  ...  This article is part of the topical collection "Cyber Security and Privacy in Communication Networks" guest edited by Rajiv Misra, R. K.  ... 
doi:10.1007/s42979-021-00542-7 pmid:33748775 pmcid:PMC7955696 fatcat:vhlkzsgtxvdt3nmkq4mzty7uxy

Polyominoes on the infinite checkerboard

M.S Klamkin, A Liu
1980 Journal of combinatorial theory. Series A  
ACKNOWLEDGMENT The authors are grateful to the referee for several helpful suggestions.  ...  Now every copy of R lies entirely on the board, and must cover at least t monominoes.  ...  FIGURE 9 To prove that b(5, n; 2,3) > n -1, consider a 5 x PZ checkerboard with b(5, n; 2, 3) monominoes placed on it to block off 2 x 3 rectangles.  ... 
doi:10.1016/0097-3165(80)90055-2 fatcat:k3y2zpoauzb2jpyja2lnapi3pq

Taming the Knight's Tour: Minimizing Turns and Crossings [article]

Juan Jose Besa, Timothy Johnson, Nil Mamano, Martha C. Osegueda
2019 arXiv   pre-print
We show lower bounds of (6-ε)n, for any ε>0, and 4n-O(1) on the respective problems of minimizing these metrics. Hence, we achieve approximation ratios of 19/12+o(1) and 13/4+o(1).  ...  We give a novel algorithm that produces tours with 9.5n+O(1) turns and 13n+O(1) crossings on a n× n board.  ...  Then, the board is covered by these smaller tours like a mosaic. The small tours are connected into a single one by swapping a few knight moves.  ... 
arXiv:1904.02824v1 fatcat:y5dqaeitknevloxxwcgqrh5ydy

Tourneys and the Fast Generation and Obfuscation of Closed Knight's Tours [article]

Ian Parberry
2020 arXiv   pre-print
New algorithms for generating closed knight's tours are obtained by generating a vertex-disjoint cycle cover of the knight's graph and joining the resulting cycles.  ...  Tourney Generation Following Tutte [9] , the problem of finding a cycle cover of the knight's graph K n can be reduced in O(n 2 ) time to the problem of finding a maximum cardinality matching in an undirected  ...  Right: The knight's graph K 6 on a 6 × 6 chessboard. operations, since vert(i) is in row y i = i/n and column x i = i mod n.  ... 
arXiv:2001.06044v1 fatcat:u3hajipw7ndqnhbrrvpxozzt34

Three Is a Crowd: SAT, SMT and CLP on a Chessboard [chapter]

Sebastian Krings, Michael Leuschel, Philipp Körner, Stefan Hallerstede, Miran Hasanagić
2017 Lecture Notes in Computer Science  
We study various high-level and low-level encodings and solutions, covering SAT, SMT and CLP-based solutions of the puzzle.  ...  In this article we evaluate the idea of using very high-level declarative models themselves to express constraint satisfaction problems.  ...  Following, we can prevent knights from attacking each other: ∀i ∈ 1 . . . n, j ∈ 1 . . . n, (x, y) ∈ reachable2 i,j,n knights i,j = ⇒ knights x,y = ⊥ In contrast to movement, encoding the number of figures  ... 
doi:10.1007/978-3-319-73305-0_5 fatcat:ufqnvnfykjaivefmkbeboagpea

Page 5973 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
, CO) On the 7 x n knight cover problem.  ...  Summary: “A set of knights covers a board if a knight attacks every unoccupied square. What is the minimum number of knights in a cover of an n xn board?  ... 

Taming the Knight's Tour: Minimizing Turns and Crossings

Juan Jose Besa, Timothy Johnson, Nil Mamano, Martha C. Osegueda, Ryuhei Uehara, Martin Farach-Colton, Giuseppe Prencipe
2020 Fun with Algorithms  
We give a novel algorithm that produces tours with 9.5n+O(1) turns and 13n+O(1) crossings on a n× n board, and we show lower bounds of (6-ε)n and 4n-O(1) on the respective problems of minimizing these  ...  In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.  ...  Then, the board is covered by these smaller tours like a mosaic. The small tours are connected into a single one by swapping a few knight moves.  ... 
doi:10.4230/lipics.fun.2021.4 dblp:conf/fun/VialJMO21 fatcat:m6p3dnvmofh37dmfpwoqzou5gm

Relation-algebraic modeling and solution of chessboard independence and domination problems

Rudolf Berghammer
2012 The Journal of Logic and Algebraic Programming  
We describe a simple computing technique for solving independence and domination problems on rectangular chessboards.  ...  It rests upon relational modeling and uses the BDD-based specific purpose computer algebra system RelView for the evaluation of the relation-algebraic expressions that specify the problems' solutions and  ...  Acknowledgement I want to thank the unknown referees for carefully reading the paper and especially one of them for the valuable remarks and suggestions.  ... 
doi:10.1016/j.jlap.2012.05.001 fatcat:nkp4yznmyfhg3eij7xqdmjkeeu

Lattice Paths Between Diagonal Boundaries

Heinrich Niederhausen
1998 Electronic Journal of Combinatorics  
Hook moves differ from knight moves (covered by the first theorem) by being blocked by any piece of the barrier they encounter along their three part move.  ...  Depending on the application, the boundary conditions can take different forms.  ...  Replace the contents of those cells (framed in the board below) by zero, and obtain an equivalent problem for knight moves in the band x − l + 1 < y < x + u − 1. d[m, n] = d[m − 1, n − 2] + d[m − 2,  ... 
doi:10.37236/1368 fatcat:wo3x6mvcy5fi3ktgmt3slokgs4

Weakly Monotonic Propagators [chapter]

Christian Schulte, Guido Tack
2009 Lecture Notes in Computer Science  
This paper introduces the first realistic model of constraint propagation by assuming a propagator to be weakly monotonic (complying with the constraint it implements).  ...  The important insight is that weak monotonicity makes propagation in combination with search well behaved.  ...  The alldifferent part of circuit uses domain-consistency. knights-n finds a Knights tour on an n × n board and tsp-* finds an optimal traveling salesman tour.  ... 
doi:10.1007/978-3-642-04244-7_56 fatcat:v2t7z5nmsva2laicqkvksqo4fu

Page 271 of National Union Catalog Vol. 15, Issue [page]

1963 National Union Catalog  
Knight" pamphlet laid in. 1.Knight, Douglas Maitland, 1921- I, Title, ViU NUC67-13861 Duke University, Durham, N.C. Observance of the one hundredth anniversary of the birth of James Buchanan Duke.  ...  At head of title: His Royal Highness the Duke of Edinburgh's Study Conference on the Human Problems of Industrial Communities Within the Commonwealth and pool London, New York, Oxford ye. 2 has title:  ... 

Page 92 of Manufacturing Engineering Vol. 11, Issue 8 [page]

1942 Manufacturing Engineering  
part side ot the draw ne ind sn. tions Large Investments | 37 x n¢ eo + , t € IC D ry rT RK C r n A ¢ ¢ Di + € + | y Ma ¢ | a 5 nwn q tnat 4 NA n \ Ve 2 wr n > af KDE } ) € Ww Constantly contronting inspecto  ...  on the spot.  ... 
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