206545024
Improved capacity bounds for wireless networks
We obtain improved upper and lower bounds on the best case and random case transport capacities of wireless networks under the Protocol Model of communication in Reference [1]. These results bracket the best case transport capacity to within a factor of for wireless networks on a disk. This is done by identifying larger exclusion regions for receivers. The general result on exclusion regions can also be applied to arbitrary wireless footprints, including those arising from directional antennas, thus obtaining superior bounds for such technologies too. Copyright © 2004 John Wiley & Sons, Ltd.
6349082
The Number of Neighbors Needed for Connectivity of Wireless Networks
Unlike wired networks, wireless networks do not come with links. Rather, links have to be fashioned out of the ether by nodes choosing neighbors to connect to. Moreover the location of the nodes may be random.The question that we resolve is: How many neighbors should each node be connected to in order that the overall network is connected in a multi-hop fashion? We show that in a network with n randomly placed nodes, each node should be connected to Θ(log n) nearest neighbors. If each node is connected to less than 0.074log n nearest neighbors then the network is asymptotically disconnected with probability one as n increases, while if each node is connected to more than 5.1774log n nearest neighbors then the network is asymptotically connected with probability approaching one as n increases. It appears that the critical constant may be close to one, but that remains an open problem.These results should be contrasted with some works in the 1970s and 1980s which suggested that the “magic number” of nearest neighbors should be six or eight.
63547373
The transport capacity of wireless networks over fading channels
We study the transport capacity of wireless networks with fading, in the high attenuation regime where either the medium is absorptive or the path loss exponent is greater than 3. For a network of n nodes we show that even were the states of all channels to be noncausally known a-priori, the transport capacity is no more than O(n). On the other hand, even when the channels are frequency selective and independent from time to time, G(n) is achievable through a multihop strategy
8476709
On the /spl theta/-coverage and connectivity of large random networks
Wireless planar networks have been used to model wireless networks in a tradition that dates back to 1961 to the work of E. N. Gilbert. Indeed, the study of connected components in wireless networks was the motivation for his pioneering work that spawned the modern field of continuum percolation theory. Given that node locations in wireless networks are not known, random planar modeling can be used to provide preliminary assessments of important quantities such as range, number of neighbors, power consumption, and connectivity, and issues such as spatial reuse and capacity. In this paper, the problem of connectivity based on nearest neighbors is addressed. The exact threshold function for /spl theta/-coverage is found for wireless networks modeled as n points uniformly distributed in a unit square, with every node connecting to its /spl phi//sub n/ nearest neighbors. A network is called /spl theta/-covered if every node, except those near the boundary, can find one of its /spl phi//sub n/ nearest neighbors in any sector of angle /spl theta/. For all /spl theta//spl isin/(0,2/spl pi/), if /spl phi//sub n/=(1+/spl delta/)log/sub 2/spl pi//2/spl pi/-/spl theta//n, it is shown that the probability of /spl theta/-coverage goes to one as n goes to infinity, for any /spl delta/>0; on the other hand, if /spl phi//sub n/=(1-/spl delta/)log/sub 2/spl pi//2/spl pi/-/spl theta//n, the probability of /spl theta/-coverage goes to zero. This sharp characterization of /spl theta/-coverage is used to show, via further geometric arguments, that the network will be connected with probability approaching one if /spl phi//sub n/=(1+/spl delta/)log/sub 2/n. Connections between these results and the performance analysis of wireless networks, especially for routing and topology control algorithms, are discussed.
999307
A counterexample in congestion control of wireless networks
One of the triumphs of wireline network research of the last decade has been the casting of the Internet congestion control problem within an optimization framework based on utility functions. Such an approach provides a sound understanding of the underlying stability and fairness issues, as well as a post-facto justification of TCP-like additive-increase multiplicative-decrease (AIMD) algorithms. This paper provides a counter-example showing that the same result cannot be extended to wireless networks, at least not in a straightforward manner.The fundamental difference is that wireless networks are of a broadcast nature. There is no strict notion of a "link," since transmissions from nearby nodes interfere with each other. Using a simple model of interference in wireless networks, a counter-example of a wireless network is presented in which the congestion control mechanism has an unstable equilibrium point at the desired fair solution. Further, ns-2 simulations of this counter-example manifest an oscillatory behavior. Surprisingly, this oscillatory behavior appears to be fairly typical in wireless networks, with most randomly chosen network examples manifesting it. This loss of stability suggests a possible need for the re-design of wireless TCP and wireless queue management to explicitly account for the wireless nature of the effects of interference. wireless interference can make this mechanism unstable. We present counter-example wireless graphs and flow patterns in which the congestion control mechanism fails to remain stable. ns-2 simulations indicate that this loss of stability manifests in practice as oscillatory behavior. Moreover, this oscillatory behavior is fairly typical in wireless networks. This loss of stability suggests a need for the re-design of wireless TCP and wireless queue management to explicitly account for the effects of interference.
2961249
The capacity of wireless networks
When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput /spl lambda/(n) obtainable by each node for a randomly chosen destination is /spl Theta/(W//spl radic/(nlogn)) bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission's range is optimally chosen, the bit-distance product that can be transported by the network per second is /spl Theta/(W/spl radic/An) bit-meters per second. Thus even under optimal circumstances, the throughput is only /spl Theta/(W//spl radic/n) bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signal-to-interference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance.
15224609
Codes for Optical CDMA
There has been a recent upsurge of interest in applying Code Division Multiple Access (CDMA) techniques to optical networks. Conventional spreading codes for OCDMA, known as optical orthogonal codes (OOC) spread the signal in the time domain only, which often results in the requirement of a large chip rate. By spreading in both time and wavelength using two-dimensional OOCs, the chip rate can be reduced considerably. This paper presents an overview of 1-D and 2-D optical orthogonal codes as well as some new results relating to bounds on code size and code construction.
7212084
Finding large sticks and potatoes in polygons
We study a class of optimization problems in polygons that seek to compute the "largest" subset of a prescribed type, e.g., a longest line segment ("stick") or a maximum-area triangle or convex body ("potato"). Exact polynomial-time algorithms are known for some of these problems, but their time bounds are high (e.g., O(n7) for the largest convex polygon in a simple n-gon). We devise efficient approximation algorithms for these problems. In particular, we give near-linear time algorithms for a (1 - ∈)-approximation of the biggest stick, an O(1)-approximation of the maximum-area convex body, and a (1 - ∈)-approximation of the maximum-area fat triangle or rectangle. In addition, we give efficient methods for computing large ellipses inside a polygon (whose vertices are a dense sampling of a closed smooth curve). Our algorithms include both deterministic and randomized methods, one of which has been implemented (for computing large area ellipses in a well sampled closed smooth curve).
42584661
Minimal Vertex Separators of Chordal Graphs
Chordal graphs form an important and widely studied subclass of perfect graphs. The set of minimal vertex separators constitute an unique class of separators of a chordal graph and capture the structure of the graph. In this paper, we explore the connection between perfect elimination orderings of a chordal graph and its minimal vertex separators. Specifically, we prove a characterization of these separators in terms of the monotone adjacency sets of the vertices of the graph, numbered by the maximum cardinality search (MCS) scheme. This leads to a simple linear-time algorithm to identify the minimal vertex separators of a chordal graph using the MCS scheme. We also introduce the notion of multiplicity of a minimal vertex separator which indicates the number of different pairs of vertices separated by it. We prove a useful property of the lexicographic breadth first scheme (LBFS) that enables us to determine the multiplicities of minimal vertex separators of a chordal graph.
45262024
Some new bounds on the spectral radius of graphs
The eigenvalues of a graph are the eigenvalues of its adjacency matrix. This paper presents some upper and lower bounds on the greatest eigenvalue and a lower bound on the smallest eigenvalue.
15090068
A simple provable algorithm for curve reconstruction
We present an algorithm that provably reconstructs a curve in the framework introduced by Amenta, Bern and Eppstein. The highlights of the algorithm are: (i) it is simple, (ii) it requires a sampling density better than previously known, (iii) it can be adapted for curve reconstruction in higher dimensions straightforwardly.