Charles J. Colbourn

CRC Handbook of Combinatorial Designs

Balanced Incomplete Block Designs and t-Designs2-(v,k,l) Designs of Small OrderBIBDs with Small Block Sizet-Designs, t = 3Steiner SystemsSymmetric DesignsResolvable and Near Resolvable DesignsLatin Squares, MOLS, and Orthogonal ArraysLatin SquaresMutually Orthogonal Latin Squares (MOLS)Incomplete MOLSOrthogonal Arrays of Index More Than OneOrthogonal Arrays of Strength More Than TwoPairwise Balanced DesignsPBDs and GDDs: The BasicsPBDs: Recursive ConstructionsPBD-ClosurePairwise Balanced Designs as Linear SpacesPBDs and GDDs of Higher IndexPBDs, Frames, and ResolvabilityOther Combinatorial DesignsAssociation SchemesBalanced (Part) Ternary DesignsBalanced Tournament DesignsBhaskar Rao DesignsComplete Mappings and Sequencings of Finite GroupsConfigurationsCostas ArraysCoveringsCycle SystemsDifference FamiliesDifference MatricesDifference Sets: AbelianDifference Sets: NonabelianDifference Triangle SetsDirected DesignsD-Optimal MatricesEmbedding Partial QuasigroupsEquidistant Permutation ArraysFactorial DesignsFrequency SquaresGeneralized QuadranglesGraph Decompositions and DesignsGraphical DesignsHadamard Matrices and DesignsHall Triple SystemsHowell DesignsMaximal Sets of MOLSMendelsohn DesignsThe Oberwolfach ProblemOrdered Designs and Perpendicular ArraysOrthogonal DesignsOrthogonal Main Effect PlansPackingsPartial GeometriesPartially Balanced Incomplete Block DesignsQuasigroupsQuasi-Symmetric Designs(r,l)-DesignsRoom SquaresSelf-Orthogonal Latin Squares (SOLS)SOLS with a Symmetric Orthogonal Mate (SOLSSOM)Sequences with Zero AutocorrelationSkolem SequencesSpherical t-DesignsStartersTrades and Defining Sets(t,m,s)-NetsTuscan Squarest-Wise Balanced DesignsUniformly Resolvable DesignsVector Space DesignsWeighing Matrices and Conference MatricesWhist TournamentsYouden Designs, GeneralizedYouden SquaresApplicationsCodesComputer Science: Selected ApplicationsApplications of Designs to CryptographyDerandomizationOptimality and Efficiency: Comparing Block DesignsGroup TestingScheduling a TournamentWinning the LotteryRelated Mathematics and Computational MethodsFinite Groups and DesignsNumber Theory and Finite FieldsGraphs and MultigraphsFactorizations of GraphsStrongly Regular GraphsTwo-GraphsClassical GeometriesProjective Planes, NondesarguesianComputational Methods in Design TheoryIndex

Unit disk graphs

Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in computational geometry. We show that many standard graph theoretic problems remain NP-complete on unit disk graphs, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs. In contrast, we give a polynomial time algorithm for finding cliques when the geometric representation (circles in the plane) is provided.

Constructing test suites for interaction testing

Software system faults are often caused by unexpected interactions among components. Yet the size of a test suite required to test all possible combinations of interactions can be prohibitive in even a moderately sized project. Instead, we may use pairwise or t-way testing to provide a guarantee that all pairs or t-way combinations of components are tested together This concept draws on methods used in statistical testing for manufacturing and has been extended to software system testing. A covering array, CA(N; t, k, v), is an N/spl times/k array on v symbols such that every N x t sub-array contains all ordered subsets from v symbols of size t at least once. The properties of these objects, however do not necessarily satisfy real software testing needs. Instead we examine a less studied object, the mixed level covering array and propose a new object, the variable strength covering array, which provides a more robust environment for software interaction testing. Initial results are presented suggesting that heuristic search techniques are more effective than some of the known greedy methods for finding smaller sized test suites. We present a discussion of an integrated approach for finding covering arrays and discuss how application of these techniques can be used to construct variable strength arrays.

Handbook of Combinatorial Designs, Second Edition

PREFACE INTRODUCTION NEW! Opening the Door NEW! Design Theory: Antiquity to 1950 BLOCK DESIGNS 2-(v, k, ?) Designs of Small Order NEW! Triple Systems BIBDs with Small Block Size t-Designs with t = 3 Steiner Systems Symmetric Designs Resolvable and Near-Resolvable Designs LATIN SQUARES Latin Squares Quasigroups Mutually Orthogonal Latin Squares (MOLS) Incomplete MOLS Self-Orthogonal Latin Squares (SOLS) Orthogonal Arrays of Index More Than One Orthogonal Arrays of Strength More Than Two PAIRWISE BALANCED DESIGNS PBDs and GDDs: The Basics PBDs: Recursive Constructions PBD-Closure NEW! Group Divisible Designs PBDs, Frames, and Resolvability Pairwise Balanced Designs as Linear Spaces HADAMARD MATRICES AND RELATED DESIGNS Hadamard Matrices and Hadamard Designs Orthogonal Designs D-Optimal Matrices Bhaskar Rao Designs Generalized Hadamard Matrices Balanced Generalized Weighing Matrices and Conference Matrices Sequence Correlation Complementary, Base and Turyn Sequences NEW! Optical Orthogonal Codes OTHER COMBINATORIAL DESIGNS Association Schemes Balanced Ternary Designs Balanced Tournament Designs NEW! Bent Functions NEW! Block-Transitive Designs Complete Mappings and Sequencings of Finite Groups Configurations Correlation-Immune and Resilient Functions Costas Arrays NEW! Covering Arrays Coverings Cycle Decompositions Defining Sets NEW! Deletion-Correcting Codes Derandomization Difference Families Difference Matrices Difference Sets Difference Triangle Sets Directed Designs Factorial Designs Frequency Squares and Hypercubes Generalized Quadrangles Graph Decompositions NEW! Graph Embeddings and Designs Graphical Designs NEW! Grooming Hall Triple Systems Howell Designs NEW! Infinite Designs Linear Spaces: Geometric Aspects Lotto Designs NEW! Low Density Parity Check Codes NEW! Magic Squares Mendelsohn Designs NEW! Nested Designs Optimality and Efficiency: Comparing Block Designs Ordered Designs, Perpendicular Arrays and Permutation Sets Orthogonal Main Effect Plans Packings Partial Geometries Partially Balanced Incomplete Block Designs NEW! Perfect Hash Families NEW! Permutation Codes and Arrays NEW! Permutation Polynomials NEW! Pooling Designs NEW! Quasi-3 Designs Quasi-Symmetric Designs (r, ?)-designs Room Squares Scheduling a Tournament Secrecy and Authentication Codes Skolem and Langford Sequences Spherical Designs Starters Superimposed Codes and Combinatorial Group Testing NEW! Supersimple Designs Threshold and Ramp Schemes (t,m,s)-Nets Trades NEW! Turan Systems Tuscan Squares t-Wise Balanced Designs Whist Tournaments Youden Squares and Generalized Youden Designs RELATED MATHEMATICS Codes Finite Geometry NEW! Divisible Semiplanes Graphs and Multigraphs Factorizations of Graphs Computational Methods in Design Theory NEW! Linear Algebra and Designs Number Theory and Finite Fields Finite Groups and Designs NEW! Designs and Matroids Strongly Regular Graphs NEW! Directed Strongly Regular Graphs Two-Graphs BIBLIOGRAPHY INDEX

Prioritized interaction testing for pair-wise coverage with seeding and constraints☆

Interaction testing is widely used in screening for faults. In software testing, it provides a natural mechanism for testing systems to be deployed on a variety of hardware and software configurations. In many applications where interaction testing is needed, the entire test suite is not run as a result of time or budget constraints. In these situations, it is essential to prioritize the tests. Here, we adapt a ‘‘one-test-at-a-time’’ greedy method to take importance of pairs into account. The method can be used to generate a set of tests in order, so that when run to completion all pair-wise interactions are tested, but when terminated after any intermediate number of tests, those deemed most important are tested. In addition, practical concerns of seeding and avoids are addressed. Computational results are reported. � 2006 Elsevier B.V. All rights reserved.

The complexity of completing partial Latin squares

Abstract Completing partial Latin squares is shown to be NP-complete. Classical embedding techniques of Hall and Ryser underly a reduction from partitioning tripartite graphs into triangles. This in turn is shown to be NP-complete using a recent result of Holyer.

The density algorithm for pairwise interaction testing

There are many published algorithms for generating interaction test suites for software testing, exemplified by AETG, IPO, TCG, TConfig, simulated annealing and other heuristic search, and combinatorial design techniques. Among these, greedy one‐test‐at‐a‐time methods (such as AETG and TCG) have proven to be a reasonable compromise between the needs for small test suites, fast test‐suite generation, and flexibility to accommodate a variety of testing scenarios. However, such methods suffer from the lack of a worst‐case logarithmic guarantee on test suite size, while methods that provide such a guarantee at present are less efficient or flexible, or do not produce test suites that are competitive in size for practical testing scenarios. In this paper, a new algorithm establishes that efficient, greedy, one‐test‐at‐a‐time methods can indeed produce a logarithmic worst‐case guarantee on the test suite size. In addition, this can be done while still producing test suites that are of competitive size, and in a time that is comparable to the published methods. It is deterministic, guaranteeing reproducibility. It generates only one candidate test at a time, permits users to ‘seed’ the test suite with specified tests, and allows users to specify constraints of combinations that should be avoided. Further, statistical analysis examines the impact of five variables used to tune this density algorithm for execution time and test suite size: weighting of density for factors, scaling of density, tie‐breaking, use of multiple candidates, and multiple repetitions using randomization. Copyright

Constructions for Permutation Codes in Powerline Communications

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.

A new class of group divisible designs with block size three

Abstract The elementary necessary conditions for the existence of a group divisible design with block size three, t groups of size g , and one group of size u are shown to be sufficient for all choices of g , t , and u .

Optimal frequency-hopping sequences via cyclotomy

Using cyclotomic numbers, a simple construction is presented for frequency-hopping (FH) sequences having optimal autocorrelation with respect to the well-known Lempel-Greenberger bound. Some optimal families of FH sequences are constructed. The simplicity of this technique makes it attractive for practical use.