Computing of Blocks of Some Combinatorial Designs for Applications

In this paper, we propose a new class of local differential privacy (LDP) schemes based on combinatorial block designs for a discrete distribution estimation. This class not only recovers many known LDP schemes in a unified framework of combinatorial block design, but also suggests a novel way of finding new schemes achieving the optimal (or near-optimal) privacy-utility trade-off with lower communication costs. Indeed, we find many new LDP schemes that achieve both the optimal privacy-utility trade-off and the minimum communication cost among all the unbiased schemes for a certain set of input data size and LDP constraint. Furthermore, to partially solve the sparse existence issue of block design schemes, we consider a broader class of LDP schemes based on regular and pairwise-balanced designs, called RPBD schemes, which relax one of the symmetry requirements on block designs. By considering this broader class of RPBD schemes, we can find LDP schemes achieving near-optimal privacy-utility trade-off with reasonably low communication costs for a much larger set of input data size and LDP constraint.

Partially Balanced Incomplete Block (PBIB) designs are the primary subject of this investigation of independent dominance in block designs. In the combinatorial structure of PBIB-designs, where treatments and blocks are depicted as vertices and edges, respectively, the graph-theoretic parameter known as independent dominance is applied. Within the framework of PBIB-designs defined by different association schemes, the study delves into the theoretical foundations, computation, and consequences of independent domination. Finding autonomous dominating sets that effectively represent all treatments while minimizing redundancy is the goal of the study, which offers a paradigm for doing so by examining the links between blocks and treatments. Experiment design optimization using independent dominating parameters is demonstrated in case studies of 2-associate and 3-associate PBIB-designs. This study sheds light on the structural features of PBIB-designs by demonstrating the interaction between graph theory and combinatorial design theory. In situations where resources are limited, it highlights the value of independent dominance as a method for assessing and improving block designs’ efficiency. Our current understanding of block designs and how to optimize them for use in experimental planning, data analysis, and resource allocation is enhanced by these findings.

Objective: To design a key pre-distribution scheme for a heterogeneous group of sensor nodes with the help of combinatorial design for high resilience against node capture. Methods/Statistical analysis: Methodology used here is based on theory of balanced incomplete block design where a group of BIBD is used for each of the sensor groups and keys are distributed based on the orientation of these groups. Findings: Connectivity, Storage limitation and resilience are evaluated for the proposed scheme. Application/Improvements: Combinatorial design based scheme is proven to be most suitable method among various key pre-distribution schemes. Combinatorial design based scheme distributes keys to the sensor nodes according to Balanced Incomplete Block Design (BIBD). Use of combinatorial design improves the overall performance of the network.

A graph $G$ with vertex set $V$ is said to be $n$-existentially closed if, for every $S \subset V$ with $|S|=n$ and every $T \subseteq S$, there exists a vertex $x \in V-S$ such that $x$ is adjacent to each vertex of $T$ but is adjacent to no vertex of $S-T$. Given a combinatorial design ${\cal D}$ with block set ${\cal B}$, its block-intersection graph $G_{{\cal D}}$ is the graph having vertex set ${\cal B}$ such that two vertices $b_1$ and $b_2$ are adjacent if and only if $b_1$ and $b_2$ have non-empty intersection. In this paper we study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are $n$-existentially closed. We characterise the BIBDs with block size $k \geq 3$ and index $\lambda=1$ that have 2-e.c. block-intersection graphs and establish bounds on the parameters of BIBDs with index $\lambda=1$ that are $n$-e.c. where $n \geq 3$. For $\lambda \geq 2$ and $n \geq 2$, we prove that only simple $\lambda$-fold designs can have $n$-e.c. block-intersection graphs. In the case of $\lambda$-fold triple systems we show that $n \geq 3$ is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with $k=3$ and $\lambda=2$) have 2-e.c. block-intersection graphs.

In this paper, we aim at reducing the problems related to the use of a single channel by designing a multichannel protocol that allows interference mitigation, delay and collision reduction, as well as throughput improvement while still reducing idle listening through a receiver-based scheduling of communication slots. We design our protocol in a fully distributed way so that failure of nodes or links could be recovered by local message exchange. To achieve these features, we make use of combinatorial block design theory. Block Design is a system of subsets of a finite set which satisfies certain conditions related to the frequency of appearance of pairs of elements of the set in the subsets of the system. Our proposed solution increases the throughput, and completely eliminates the deafness problem without causing a large overhead. To validate our solution on large scale networks, we developed our proposal on ns3, which has improved features over traditional simulators. Simulation results show that our Latin Square based channel assignment significantly reduces collisions and end to end delay thereby making it suitable for most sensor network applications.

Conceptual modeling is drawing more attention among researchers from many organizations and different disciplines including high-performance low-power multicore systems. As the popularity and demand grow, more processing power is required by the mission critical systems. Several chip vendors have developed teraflop computing systems (like Intel 80-Core Research Chip). Even though IBM offers the Roadrunner supercomputer, NSF and other research organization are looking for more powerful petascale computer system. In a petascale computer system, thousands to millions of processing cores are expected to work together using petabytes of memory to produce the expected performance. A high performance system like Roadrunner requires a very sophisticated power management system. No wonder, such a gigantic system costs millions of dollars. Conceptual modeling offers the opportunity to conduct research on complex and expensive systems in an inexpensive way. In this work, we address some key conceptual modeling issues and discuss a methodology to develop conceptual models of petascale computer systems. Important features of this methodology include data acquisition to improve accuracy by capturing the important details about the system under consideration and (system-to-) model abstraction to make the modeling manageable without eliminating any significant characteristics of the target system. We believe our findings help develop sound conceptual models of complex high performance petascale computer systems.

In recent years, many approximate methods have been developed for analyzing queueing models of complex computer systems. These ad hoc methods usually focus on specific aspects of system operation, and appear to be different from one another, making it difficult to see the underlying principles of model development, to understand the relationship between different models of the same system, or to apply the existing methods to new situations. This book presents the first systematic study of approximation methods in queueing network modeling and of the way these methods are developed. Metamodeling identifies the underlying modeling process and provides tools and techniques for model development that will allow students and researchers to sort through the many different methods, understand them, and apply them to new problems. Using the metamodeling characterization, the book surveys and classifies a large number of approximation methods, catalogs a large number of useful model transformations, characterizes iterative solution procedures and gives theorems for proofs of convergence. This work has led to several other significant results, most notably: an approximation that works well for systems containing semaphores that serialize processes; and the discovery of multiple stable operating points for systems in which there are processes at several priority levels. Contents Queueing Network Models of Computer Systems • The Structure of the Modeling Process • Behavior Sequence Transformations and Models with Shadow Servers • State Space Transformations • Consistent Solution, Iteration, and Convergence This book inaugurates the MIT Press series in Computer Systems (Research Reports and Notes), edited by Herb Schwetman.

The c-secure code is an important tool in the anti-pirate of digital product. There are few papers on the c-secure code since it has been introduced in Crypo'95 and only several papers work for small c, i.e. c=2,3,4. The constructions of four kinds of c-secure codes have been proposed from the BIBD(balanced incomplete block design) and packing design in this paper. The codelengths of three of them are shorter than the well-known one given by Boneh and Shaw in Crypo'95 when the code number is same. It seems that this paper is the first one for any c from combinatorial designs and the new c-secure codes have shorter codelengths and simple traitor tracing algorithms.

Let Π be a finite projective plane of order n , and let G be a large abelian (or, more generally, quasiregular) collineation group of Π; to be specific, we assume ∣ G ∣ ( n 2 + n + 1)/2. Such planes have been classified into eight cases by Dembowski and Piper in 1967. We survey the present state of knowledge about the existence and structure of such planes. We also discuss some geometric applications, in particular to the construction of arcs and ovals. Technically, a recurrent theme will be the amazing strength of the approach using various types of difference sets and the machinery of integral group rings. Introduction A projective plane is a geometry consisting of points and lines such that any two distinct lines meet in exactly one point, any two distinct points are on exactly one common line, and there are four points no three of which are collinear. Well-known examples are provided by the desarguesian (or classical ) projective planes PG(2, q ), q a prime power, which can be defined over the finite field GF( q ) as follows: points and lines are the 1- and 2-dimensional subspaces of the vector space GF( q ) 3 , and a point is incident with a line if and only if it is a subset of the line. A standard reference for arbitrary projective planes is Hughes and Piper [104]; for the desarguesian case, see Hirschfeld [89].

In this paper we examine certain problems related to the use of diffusion approximations for the approximate modelling of computer systems. In particular we develop a model which allows us to handle waiting times and batch arrivals: these results are a new approach to the use of diffusion approximations. We also examine the effect of the distribution of holding times at boundaries: this question had remained open in earlier studies. We show that the stationary distributions associated with these diffusion models depend only on the average residence time of the process on the boundaries and not on the complete distribution function. This result justifies the use of exponential holding times as had been done in an earlier study.

An analytical approach is proposed for estimating characteristics of microprocessor computer systems with wide command words. Queueing networks are used as models of computer systems.

Many key pre-distribution (KPD) schemes based on combinatorial design were proposed for secure communication of wireless sensor networks (WSNs). Due to complexity of constructing the combinatorial design, it is infeasible to generate key rings using the corresponding combinatorial design in large scale deployment of WSNs. In this paper, we present a definition of new combinatorial design, termed “µ-partially balanced incomplete block design (µ-PBIBD)”, which is a refinement of partially balanced incomplete block design (PBIBD), and then describe a 2-D construction of µ-PBIBD which is mapped to KPD in WSNs. Our approach is of simple construction which provides a strong key connectivity and a poor network resilience. To improve the network resilience of KPD based on 2-D µ-PBIBD, we propose a KPD scheme based on 3-D Ex-µ-PBIBD which is a construction of µ-PBIBD from 2-D space to 3-D space. Ex-µ-PBIBD KPD scheme improves network scalability and resilience while has better key connectivity. Theoretical analysis and comparison with the related schemes show that key pre-distribution scheme based on Ex-µ-PBIBD provides high network resilience and better key scalability, while it achieves a trade-off between network resilience and network connectivity.

Pancyclic BIBD block-intersection graphs

Tiling sets and spectral sets over finite fields

Associahedra, Combinatorial Block Designs and Related Structures