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Column replacement techniques for creating covering arrays rely on the construction of perfect and distributing hash families with few rows, having as many columns as possible for a specified number of symbols. To construct distributing hash families in which the number of rows is less than the strength, we examine a method due to Blackburn and extend it in three ways. First, the method is generalized from homogeneous hash families (in which every row has the same number of symbols) to heterogeneous ones. Second, the extension treats distributing hash families, in which only separation into a prescribed number of parts is required, rather than perfect hash families, in which columns must be completely separated. Third, the requirements on one of the main ingredients are relaxed to permit the use of a large class of distributing hash families, which we call fractal. Constructions for fractal perfect and distributing hash families are given, and applications to the construction of perfect hash families of large strength are developed. [ABSTRACT FROM AUTHOR]

We construct orthogonal arrays OA λ (k , n) (of strength two) having a row that is repeated m times, where m is as large as possible. In particular, we consider OAs where the ratio m ∕ λ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any k ≥ n + 1 , albeit with large λ. We also study basic OAs; these are optimal OAs in which gcd (m , λ) = 1. We construct a basic OA with n = 2 and k = 4 t + 1 , provided that a Hadamard matrix of order 8 t + 4 exists. This completely solves the problem of constructing basic OAs with n = 2 , modulo the Hadamard matrix conjecture. [ABSTRACT FROM AUTHOR]

• IPO strategy generates Covering Perfect Hash Families efficiently. • Fast covering array generation for large scale combinatorial testing. • Many improvements to upper bounds on CAN. • CPHFs with subspace restrictions lead to many different small CAs. Combinatorial testing makes it possible to test large systems effectively while maintaining certain coverage guarantees. At the same time, the construction of optimized covering arrays (CAs) with a large number of columns is a challenging task. Heuristic and Metaheuristic approaches often become inefficient when applied to large instances, as the computation of the quality for new moves or solutions during the search becomes too slow. Recently, the generation of covering perfect hash families (CPHFs) has led to vast improvements to the state of the art for many different instances of covering arrays. CPHFs can be considered a compact form of a specific family of covering arrays. Their compact representation makes it possible to apply heuristic methods for instances with a much larger number of columns. In this work, we adapt the ideas of the well-known In-Parameter-Order (IPO) strategy for covering array generation to efficiently construct CPHFs, and therefore implicitly covering arrays. We design a way to realize the concept of vertical extension steps in the context of CPHFs and discuss how a horizontal extension can be implemented in an efficient manner. Further, we develop a horizontal extension strategy for CPHFs with subspace restrictions that identifies candidate columns greedily based on conditional expectation. Then using a local optimization strategy, a candidate may be adjoined to the solution or may replace one of the existing columns. An extensive set of computational results yields many significant improvements on the sizes of the smallest known covering arrays. [ABSTRACT FROM AUTHOR]

Abstract: Necessary and sufficient conditions are established for an integer vector to be the -vector of some pure simplicial complex of rank three, and also for an integer vector to be the -vector of some pure simplicial multicomplex of rank three. For specified numbers of sets of cardinality one and cardinality two, an upper bound on the number of sets of cardinality three is established using shifting arguments. Then techniques from combinatorial design theory are used to establish a lower bound. Then it is shown that every number of sets of cardinality three between the lower and the upper bound can be realized. This characterization is restated to determine the precise spectrum of possible numbers of sets of cardinality two for specified numbers of sets of cardinality one and three. For simplicial complexes, these spectra are not always intervals, and the gaps are determined precisely. For simplicial multicomplexes, an alternative proof is given that these spectra are always intervals. [Copyright &y& Elsevier]

Abstract: The deterministic construction of measurement matrices for compressive sensing is a challenging problem, for which a number of combinatorial techniques have been developed. One of them employs a widely used column replacement technique based on hash families. It is effective at producing larger measurement matrices from smaller ones, but it can only preserve the strength (level of sparsity supported), not increase it. Column replacement is extended here to produce measurement matrices with larger strength from ingredient arrays with smaller strength. To do this, a new type of hash family, called a strengthening hash family, is introduced. Using these hash families, column replacement is shown to increase strength under two standard notions of recoverability. Then techniques to construct strengthening hash families, both probabilistically and deterministically, are developed. Using a variant of the Stein–Lovász–Johnson theorem, a deterministic, polynomial time algorithm for constructing a strengthening hash family of fixed strength is derived. [Copyright &y& Elsevier]

Abstract: Grooming uniform all-to-all traffic in optical ring networks with grooming ratio requires the determination of graph decompositions of the complete graph into subgraphs each having at most edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum drop cost is determined for grooming ratio 9. Previously this bound was shown to be met when with two exceptions and eleven additional possible exceptions for , and also when with one exception and one possible exception for . In this paper it is shown that the bound is met for all with four exceptions for and one possible exception for . Using this result, it is further shown that when and is sufficiently large, the bound is also met. [Copyright &y& Elsevier]

Abstract: Grooming uniform all-to-all traffic in optical ring networks with grooming ratio requires the determination of graph decompositions of the complete graph into subgraphs each having at most edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum possible drop cost is determined for grooming ratio 8, and this cost is shown to be realized with six exceptions, and 37 possible exceptions, the largest being 105. [Copyright &y& Elsevier]

Abstract: Covering arrays have been extensively studied, in part because of their applications in testing interacting software components. In this setting, fast and flexible methods are needed to construct covering arrays of close-to-minimum size. However testing scenarios often impose additional structure on the tests that can be selected. We extend a greedy method to construct test suites for complex systems that have a hierarchical structure in which components combine to form subsystems, which in turn form larger subsystems, until the entire system is formed. The algorithm for merging covering arrays that we propose is then shown to have further potential application in the compression of multiple sequence alignments of genomic data. [Copyright &y& Elsevier]

Abstract: In this paper, we exploit space as a new dimension in collision resolution for a carrier sense multiple access (CSMA) protocol. Most contention-based medium access control protocols resolve collisions by backing off in time. We introduce power backoff (PB), the use of transmission power control to resolve collisions by backing off in space, and incorporate it into a CSMA protocol as CSMA/PB. Through analysis and simulation, we show that collision resolution using power backoff can be remarkably successful. Simulation results show that CSMA/PB outperforms IEEE 802.11 in both static and mobile ad hoc network scenarios. CSMA/PB improves end-to-end throughput and uses less energy. The resulting gains in throughput per unit energy can be substantial. [Copyright &y& Elsevier]

Abstract: Reliability is a major concern in the design of large disk arrays. In this paper, we examine the effect of encountering more failures than that for which the RAID array was initially designed. Erasure codes are incorporated to enable system recovery from a specified number of disk erasures, and strive beyond that threshold to recover the system as frequently, and as thoroughly, as is possible. Erasure codes for tolerating two disk failures are examined. For these double erasure codes, we establish a correspondence between system operation and acyclicity of its graph model. For the most compact double erasure code, the full 2-code, this underlies an efficient algorithm for the computation of system operation probability (all disks operating or recoverable). When the system has failed, some disks are nonetheless recoverable. We extend the graph model to determine the probability that d disks have failed, a of which are recoverable by solving one linear equation, b of which are further recoverable by solving systems of linear equations, and of which cannot be recovered. These statistics are efficiently calculated for the full 2-code by developing a three variable ordinary generating function whose coefficients give the specified values. Finally, examples are given to illustrate the probability that an individual disk can be recovered, even when the system is in a failed state. [Copyright &y& Elsevier]

Abstract: A constant composition code over a k-ary alphabet has the property that the numbers of occurrences of the k symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. In this paper, direct and recursive methods are developed for the construction of constant composition codes. [Copyright &y& Elsevier]

In wavelength division multiplexing for unidirectional rings, traffic grooming is used to pack low rate signals into higher rate streams to share a wavelength. The grooming chosen determines the number of add-drop multiplexers used for the optical-to-electronic conversion. The determination of groomings to use the fewest multiplexers is equivalent to a graph design problem, which has been solved when up to four signals can be packed into a stream. We completely settle the case here when five signals pack into one stream, using old and new results on (K4-e)-designs. [Copyright &y& Elsevier]

In a systematic erasure code for the correction of two simultaneous erasures, each information symbol must have two associated parity symbols. When implemented in a redundant array of independent disks (RAID), performance requirements on the update penalty necessitate that each information symbol be associated with no more parity symbols than the two required. This leads to a simple graph model of the erasure codes, with parity symbols as vertices and information symbols as edges. Based on simulations of RAID performance, an ordering of the edges in which every sequence of three consecutive edges in the order induces as few vertices as possible is found to optimize access performance of the disk array. The ladder orderings to optimize performance are shown to exist for the complete graph Kn, except possibly when n∈{15,18,22}. [Copyright &y& Elsevier]