Combinatorial Block Designs Research Articles

On Key and Multiple Space Key Pre-Distribution Schemes in Wireless Sensor Networks with a Combinatorial Block Design Structure

On Key and Multiple Space Key Pre-Distribution Schemes in Wireless Sensor Networks with a Combinatorial Block Design Structure

The structure of 3-pyramidal groups

A combinatorial block design D is called f-pyramidal if there exists a subgroup G of Aut(D) fixing f points and acting regularly on the other points. If this happens, we say that the design is f-pyramidal under G. In case D is a 3-pyramidal Kirkman triple system, it is known that such a group G has precisely 3 involutions, all conjugate to each other. In this paper, we obtain a classification of the groups with this property.

Scaling of Networks and their Key Systems Based on Combined Combinatorial Block Designs

A new method of scaling wireless sensor networks, as well as key distribution schemes and key spaces in them based on a combined combinatorial block design formed on the set of all pairs composed of blocks and corresponding to their elements dual blocks of a given combinatorial block design is presented. Estimates are given for the cardinalities of scaled circuits based on the combined projective plane, the combined modified residual block design, and the combined residual combinatorial transversal block design. It is shown that the size of the key memory of the nodes of such networks does not depend on the parameters of the combinatorial block design and is two key units, but in schemes based on the combined projective plane, the length of the shortest route between the vertices of the key matching graph increases to three. The consequences of node capture are evaluated and ways to limit them with a guaranteed security parameter are proposed. For schemes based on a combined cyclic projective plane, as applied to the w-safe WSN of a "smart home", communication schemes between nodes along routes of length from one to three and an example of a protocol are presented. An example of such a WSN containing 910 nodes that are child nodes from 31 to 91 routers is given.

Fixing Numbers of Point-Block Incidence Graphs

A vertex in a graph is referred to as fixed if it is mapped to itself under every automorphism of the vertices. The fixing number of a graph is the minimum number of vertices, when fixed, that fixes all of the vertices in the graph. Fixing numbers were first introduced by Laison and Gibbons, and independently by Erwin and Harary. Fixing numbers have also been referred to as determining numbers by Boutin. The main motivation is to remove all symmetries from a graph. A very simple application is in the creation of QR codes where the symbols must be fixed against any rotation. We determine the fixing number for several families of graphs, including those arising from combinatorial block designs. We also present several infinite families of graphs with an even stronger condition, where fixing any vertex in a graph fixes every vertex.

Balanced Nonadaptive Redundancy Scheduling

Distributed computing systems implement redundancy to reduce the job completion time and variability. Despite a large body of work about computing redundancy, the analytical performance evaluation of redundancy techniques in queuing systems is still an open problem. In this work, we take one step forward to analyze the performance of scheduling policies in systems with redundancy. In particular, we study the pattern of shared servers among replicas of different jobs. To this end, we employ combinatorics and graph theory and define and derive performance indicators using the statistics of the overlaps. We consider two classical nonadaptive scheduling policies: random and round-robin. We then propose a scheduling policy based on combinatorial block designs. Compared with conventional scheduling, the proposed scheduling improves the performance indicators. We study the expansion property of the graphs associated with round-robin and block design-based policies. It turns out the superior performance of the block design-based policy results from better expansion properties of its associated graph. As indicated by the performance indicators, the simulation results show that the block design-based policy outperforms random and round-robin scheduling in different scenarios. Specifically, it reduces the average waiting time in the queue to up to 25% compared to the random policy and up to 100% compared to the round-robin policy.

Number Cubes with Consecutive Line Sums

We settle the existence of certain “anti-magic” cubes using combinatorial block designs and graph decompositions to align a handful of small examples.

A mixed construction of variable-weight optical orthogonal code for optical code division multiple access networks

Optical orthogonal codes play an important role in optical code-division multiple access systems. In order to support heterogeneous quality of service, it is of interests to employ optical orthogonal codes in which different codewords may have different Hamming weights. In this paper a construction method of variable-weight optical orthogonal codes combining meta-heuristic search algorithm and combinatorial block designs is studied. This method is capable of producing variable-weight optical orthogonal codes with large and flexible parameters. We construct three infinite families of optimal variable-weight optical orthogonal codes using the proposed method.

Optimal arrangements of classical and quantum states with limited purity

We consider sets of trace-normalized non-negative operators in Hilbert-Schmidt balls that maximize their mutual Hilbert-Schmidt distance; these are optimal arrangements in the sets of purity-limited classical or quantum states on a finite-dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher-rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor-Steiner equiangular tight frame as the orbit of a projective representation of the Weyl-Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so-called binder, of the Gabor-Steiner equiangular tight frames are then characterized; under certain conditions these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor-Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.

MaxMinSum Steiner Systems for Access Balancing in Distributed Storage

Many code families such as low-density parity-check codes, fractional repetition codes, batch codes and private information retrieval codes with low storage overhead rely on the use of combinatorial block designs or derivatives thereof. In the context of distributed storage applications, one is often faced with system design issues that impose additional constraints on the coding schemes, and therefore on the underlying block designs. Here, we address one such problem, pertaining to server access frequency balancing, by introducing a new form of Steiner systems, termed MaxMinSum Steiner systems. MaxMinSum Steiner systems are characterized by the property that the minimum value of the sum of points (elements) within a block is maximized, or that the minimum sum of block indices containing some fixed point is maximized. We show that proper relabelings of points in the Bose and Skolem constructions for Steiner triple systems lead to optimal MaxMin values for the sums of interest; for the duals of the designs, we exhibit block labelings that are within a 3/4 multiplicative factor from the optimum.

Asymmetric Block Design-Based Neighbor Discovery Protocol in Sensor Networks

Neighbor discovery is one of the emerging research areas in a wireless sensor network. After sensors are distributed, neighbor discovery is the first process to set up a communication channel with neighboring sensors. This paper proposes a new block design–based asymmetric neighbor discovery protocol for sensor networks. We borrow the concept of combinatorial block designs for our block combination scheme for neighbor discovery. First, we introduce an asymmetric neighbor discovery problem and define a target research question. Second, we propose a new asymmetric block design–based neighbor discovery protocol and explain how it works. Third, we analyze the worst-case neighbor discovery latency numerically between our protocol and some well-known protocols in the literature, and compare and evaluate the performance between the proposed protocol and others. Our protocol reveals that the worst-case latency is much lower than that of Disco and U-Connect. Finally, we conclude that the minimum number of slots per a neighbor schedule shows the lowest discovery time in terms of discovery latency and energy consumption.

Energy-Efficient Distributed Topology Control Algorithm for Low-Power IoT Communication Networks

Topology control is one of the significant research topics in traditional wireless networks. The primary purpose of topology control ensures the connectivity of wireless nodes participated in the network. Low-power Internet of Things communication networks look like wireless network environments in which the main communication devices are wireless devices with limited energy like battery. In this paper, we propose a distributed topology control algorithm by merging the combinatorial block design from a design theory with the multiples of 2. The proposed technique especially focuses on asynchronous and asymmetric neighbor discovery. The concept of block design is used to generate the neighbor discovery schedule when a target duty cycle is given. In addition, the multiples of 2 are applied to overcome the challenge of the block design and support asymmetric operation. We analyze the worst case discovery latency and energy consumption numerically by calculating the total number of slots and wake-up slots based on the given duty cycle. It shows that our proposed method has the smallest total number of slots and wake-up slots among existing representative neighbor discovery protocols. The numerical analysis represents the proposed technique find neighbors quickly with minimum battery power compared with other protocols for distributed topology control. For future research direction, we could perform a simulation study or real experiment to investigate the best parameter for choosing the multiple of a certain number.

Switch-Centric Data Center Network Structures Based on Hypergraphs and Combinatorial Block Designs

Fat trees are considered suitable structures for data center interconnection networking. Such structures are rigid, and hard to scale up and scale out. A good data center network structure should have high scalability, efficient switch utilization, and high reliability. In this paper we present a class of data center network structures based on hypergraph theory and combinatorial block design theory. We show that our data center network structures are more flexible and scalable than fat trees. Using switches of the same size, our data center network structures can connect more nodes than fat trees, and it is possible to construct different structures with tradeoffs among inter-cluster communication capacity, reliability, the number of switches used, and the number of connected nodes.

Block Design-Based Asynchronous Neighbor Discovery Protocol for Wireless Sensor Networks

Neighbor discovery is a significant research topic in wireless sensor networks. After wireless sensor devices are deployed in specific areas, they attempt to determine neighbors within their communication range. This paper proposes a new Block design-based Asynchronous Neighbor Discovery protocol for sensor networks calledBAND. We borrow the concept of combinatorial block designs for neighbor discovery. First, we summarize a practical challenge and difficulty of using the original block designs. To address this challenge, we create a new block generation technique for neighbor discovery schedules and provide a mathematical proof of the proposed concept. A key aspect of the proposed protocol is that it combines two block designs in order to construct a new block for neighbor discovery. We analyze the worst-case neighbor discovery latency numerically between our protocol and some well-known protocols in the literature. Our protocol reveals that the worst-case latency is much lower than others. Finally, we evaluate the performance ofBANDand existing representative protocols through the simulation study. The results of our simulation study show that the average and maximum latency ofBANDis about 40% lower than that of existing protocols. Furthermore,BANDspends approximately 30% less energy than others during the neighbor discovery process.

Combinatorial design of congestion-free networks

This paper presents a new design methodology and tools to construct a packet switched network with bursty data sources. This network design combines two important properties for arbitrary traffic pattern: (1) the aggregate throughput is scalable and (2) there is no packet loss within the subnet. More specifically, given a bounded number of ports in every switching node, the design is based on the construction of multiple virtual rings under the following constraints: (1) the virtual rings are pairwise edge-disjoint and (2) there is at least one virtual ring between any pair of nodes. The target topology is obtained from the edge union of the multiple virtual rings. The two constraints ensure no loss due to congestion inside a network with arbitrary traffic pattern and that packets will reach (or converge) their destinations. The virtual rings are constructed by using combinatorial block designs together with an algorithm for realizing any size networks. It is shown that the bound on the maximum route length, under the two constraints, is O(/spl radic/N) for an N-node network, This sublinear bound facilitates the throughput scalability property.

On splitting sets in block designs and finding roots of polynomials

The general notion of t-splitting sets is introduced within the context of combinatorial block designs. A greatest lower bound on cardinality of such sets, and an upper bound on cardinality of the smallest such set in a given design are established. The abstraction of t-splitting sets is shown to provide a natural framework for the analysis of the problem of finding roots of polynomials over finite fields, and elementary concepts from design theory are applied to re-examine and extend some existing results in this area.

A new class of fault-tolerant static interconnection networks

The authors present a new class of interconnection networks using combinatorial block designs. These networks are highly structured and have strong fault-tolerant properties. They also have a free parameter that allows tradeoffs to be made between performance and cost in a fairly continuous way.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>