1493304
On random points in the unit disk
Let n be a positive integer and λ > 0 a real number. Let Vn be a set of n points in the unit disk selected uniformly and independently at random. Define G(λ, n) to be the graph with vertex set Vn, in which two vertices are adjacent if and only if their Euclidean distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ lnn/n and let X be the number of isolated points in G(λ, n). We prove that almost always X ∼ n1−c2 when 0 ≤ c < 1. It is known that if λ = √ (ln n + φ(n))/n where φ(n) → ∞, then G(λ, n) is connected. By extending a method of Penrose, we show that under the same condition on λ, there exists a constant K such that the diameter of G(λ, n) is bounded above by K · 2/λ. Furthermore, with a new geometric construction, we show that when λ = c √ lnn/n and c > 2.26164 · · · , the diameter of G(λ, n) is bounded by (4 + o(1))/λ; and we modify this construction to yield a function c(δ) > 0 such that the diameter is at most 2(1 + δ + o(1))/λ when c > c(δ). ∗This is a preprint of an article accepted for publication in Random Structures and Algorithms c ©2005. †Partially supported by NSF grant DMS-9977354. ‡Partially supported by NSF grant DMS-0245526, DMS-0308827 and a Sloan Fellowship. The author is also affiliated with Dalian University of Technology.
40163188
Triple Loop Networks with Minimal Transmission Delay
A multiloop network is a Cayley digraph of the additive group ℤm of integers modulo m generated by a set A of k ≥ 2 "steps". Because of their application to the computer network design, multiloop networks have been studied extensively. Given an integer d ≥ 2 and a real number r > 1, let m(d,3) be the maximum size of a triple loop network with transmission delay (diameter) less than or equal to d, and let m*(r,3) be the maximum size of a triple loop network with average transmission delay less than or equal to r. In this paper, the following lower bounds are proved: which improve earlier lower bounds by Jia and Aguilo, Fiol and Garcia.
41488878
On a Network Sensing Problem
Live traffic flow information can help improve the efficiency of a communication network. There are many ways available to monitor the traffic flow of a network. In this paper, we propose a very efficient monitoring strategy. This strategy not only reduces the number of nodes to be monitored but also determines the complete traffic information of the entire network using the information from the monitored nodes. The strategy is optimal for monitoring a network because it reduces the number of monitored nodes to a minimum. Fast algorithms are also presented in this paper to compute the traffic information for the entire network based on the information collected from the monitored nodes. The monitoring scheme discussed in this paper can be applied to the internet, telecommunication networks, wireless ad hoc networks, large scale multiprocessor computing systems, and other information systems where the transmission of information needs to be monitored.
20605726
Extremal Problems in the Construction of Distributed Loop Networks
Let $G(N, A)$ be the Cayley digraph associated with ${\bf Z}/(N)$ and $A$, where $N$ is a positive integer and $A$ is a subset of $\{1, 2, ..., N - 1\}$. Let $N(d, k)$ be the maximum $N$ such that the diameter of $G(N, A)$ is less than or equal to $d$ for some $A = \{a_{1}, a_{2},\ldots, a_{k}\}$ with $1 = a_{1}, < a_{2} < \cdots < a_{k}. An exact formula for $N(d, 2)$ is given, and $N(d, k)$ is estimated for $k \geq 3$. These results provide new bounds for minimal diameter in the construction of loop networks. A relation between this problem and the postage stamp problem in additive number theory is established to enhance the study of these problems.
2884321
Some remarks on minimal bases and maximal nonbases of integers
Abstract In this note, we generalize the concepts of minimal bases and maximal nonbases for integers, and prove some existence theorems for the generalized minimal bases and maximal nonbases, which generalize some results of Stohr, Deza and Erdos, and Nathanson.
15745921
Wireless networks and random geometric graphs
In this paper, mobile ad-hoc networks are studied. Chen and Jia (2001) proposed a routing algorithm by utilizing multiple communication channels and localizing the routing information (routing tables) for mobile stations within the wireless network. The routing algorithm relies on a spanning tree of the underlying network. It is proven that both the construction of the spanning tree and the routing algorithm are efficient. The routing algorithm is also reliable because of the frequent update of the spanning tree which the routing scheme is based on. The underlying network is a random geometric graph G(A,n) formed on a set of uniformly randomly located points in the unit disk, in which two vertices are adjacent if and only if their Euclidean distance is at most A.
6166247
Virtual backbone construction in multihop ad hoc wireless networks
Summary Recent research points out that the flooding mechanism for topology update or route request in existing ad hoc routing protocols greatly degrades the network capacity. If we restrict the broadcast of control packets within a small subset of hosts in the network, the protocol overhead can be substantially reduced. This motivates our research of constructing a virtual backbone by computing a connected dominating set (CDS) in unit-disk graphs. In this paper, we propose two distributed algorithms to approximate a minimum CDS. These algorithms take linear time. Their performance is verified by a complete theoretical analysis. Copyright # 2006 John Wiley & Sons, Ltd.
22388831
PROOF: A Parallel Object-Oriented Functional Computation Model
Abstract Despite the rapid development of parallel computer architectures in recent years, software development technology for parallel processing systems is still largely inadequate. An integrated object-oriented and functional computation model, called PROOF, is developed to facilitate the effective development of software for parallel processing systems. It will fully explore the parallelism in software systems and reduce the development effort by supporting software engineering principles, such as information hiding, modularity, modifiability, and reusability. In PROOF, we achieve the coherent integration of (1) parallelism and inheritance of objects and (2) referential transparency and history sensitivity of functions. The major features and the semantics of PROOF are presented. A parallel programming language and a software development approach based on PROOF are under development. Implementation issues are also discussed.
13944177
A pragmatic approach to formalizing object-oriented modeling and development
Despite recent developments in formal methods, these methods have yet to enter the mainstream of the software industry. We first examine the pragmatic obstacles to making formal methods practical and usable. Then we present a pragmatic approach to integrate a popular object oriented modeling notation UML and a popular formal notation Z. Our approach is distinctive in its emphasis on overcoming the practical obstacles. We also present a light weight supporting tool for our approach. A prototype of the supporting tool is completed. It allows us to demonstrate the feasibility and capability of our approach.
28697957
The maximum size of graphs satisfying a degree condition
Let G be a simple graph of order n without isolated vertices. If the integer h satisfies 
h ⩾ maxυ ϵ V(G)minu ψ N(V)(degGu) 
then G is called an Fn(h)-graph. In this note the maximum size of Fn(h)-graphs is determined. A result of Krol and Veldman on critically h-connected graphs follows as a corollary.
223086
Detecting null pointer violations in Java programs
The use of formal methods has been growing steadily and there have been a number of successful applications of formal methods in a range of application areas. It seems agreed that quality should be assured by applying testing, analysis and formal methods to rigorously defined precode artifacts. The detection of null pointer violation errors is definitely such a goal. This way of applying formal methods has a great potential to increase our confidence in the software. Our goal is to provide a practical mechanism to assist the application of formal methods in the early detection of null pointer violation errors in programs. Our solution is theorem proving based and is focused on the identification of the possible places in which a theorem prover could assist in the detection of null pointer violation errors and the formulation of the necessary proof obligations.