Showing papers on "Unit disk graph published in 2004"
TL;DR: This paper presents their own distributed algorithm that outperforms the existing algorithms for minimum CDS and establishes the Ω(nlog n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, thus message-optimal.
Abstract: Connected dominating set (CDS) has been proposed as virtual backbone or spine of wireless ad hoc networks. Three distributed approximation algorithms have been proposed in the literature for minimum CDS. In this paper, we first reinvestigate their performances. None of these algorithms have constant approximation factors. Thus these algorithms cannot guarantee to generate a CDS of small size. Their message complexities can be as high as O(n2), and their time complexities may also be as large as O(n2) and O(n3). We then present our own distributed algorithm that outperforms the existing algorithms. This algorithm has an approximation factor of at most 8, O(n) time complexity and O(n log n) message complexity. By establishing the Ω(n log n) lower bound on the message complexity of any distributed algorithm for nontrivial CDS, our algorithm is thus message-optimal.
909 citations
Journal Article•
TL;DR: A polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs is presented and is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph.
Abstract: A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1 + e)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.
62 citations
16 Dec 2004
TL;DR: This work combines the properties of WCDS with other ideas to obtain the following interesting applications: An online distributed algorithm for collision-free, low latency, low redundancy and high throughput broadcasting, and Distributed capacity preserving backbones for unicast routing and scheduling.
Abstract: We present fast distributed algorithms for coloring and (connected) dominating set construction in wireless ad hoc networks. We present our algorithms in the context of Unit Disk Graphs which are known to realistically model wireless networks. Our distributed algorithms take into account the loss of messages due to contention from simultaneous interfering transmissions in the wireless medium.
We present randomized distributed algorithms for (conflict-free) Distance-2 coloring, dominating set construction, and connected dominating set construction in Unit Disk Graphs. The coloring algorithm has a time complexity of O(Δ log2n) and is guaranteed to use at most O(1) times the number of colors required by the optimal algorithm. We present two distributed algorithms for constructing the (connected) dominating set; the former runs in time O(Δ log 2n) and the latter runs in time O(log 2n). The two algorithms differ in the amount of local topology information available to the network nodes.
Our algorithms are geared at constructing Well Connected Dominating Sets (WCDS) which have certain powerful and useful structural properties such as low size, low stretch and low degree. In this work, we also explore the rich connections between WCDS and routing in ad hoc networks. Specifically, we combine the properties of WCDS with other ideas to obtain the following interesting applications:
An online distributed algorithm for collision-free, low latency, low redundancy and high throughput broadcasting.
Distributed capacity preserving backbones for unicast routing and scheduling.
59 citations
25 Jul 2004
TL;DR: This paper presents a load-balanced routing algorithm with the stretch factor constraint in an online setting, in which the routing requests come one by one, and shows the application and extension of the trade-off to the wireless network routing and VLSI layout design.
Abstract: A graph has growth rate k if the number of nodes in any subgraph with diameter r is bounded by O(rk). The communication graphs of wireless networks and peer-to-peer networks often have small growth rate. In this paper we study the tradeoff between two quality measures for routing in growth restricted graphs. The two measures we consider are the stretch factor, which measures the lengths of the routing paths, and the load balancing ratio, which measures how evenly the traffic is distributed. We show that if the routing algorithm is required to use paths with stretch factor c, then its load balancing ratio is bounded by O((n/c)1-1/k), where k is the graph's growth rate. We illustrate our results by focusing on the unit disk graph for modeling wireless networks in which two nodes have direct communication if their distance is under certain threshold. We show that if the maximum density of the nodes is bounded by ρ, there exists routing scheme such that the stretch factor of routing paths is at most c, and the maximum load on the nodes is at most O(min(√ρn/c, n/c)) times the optimum. In addition, the bound on the load balancing ratio is tight in the worst case. As a special case, when the density is bounded by a constant, the shortest path routing has a load balancing ratio of O(√n). The result extends to k-dimensional unit ball graphs and graphs with growth rate k. We also discuss algorithmic issues for load balanced short path routing and for load balanced routing in spanner graphs.
37 citations
14 Sep 2004
TL;DR: A novel randomized algorithm for computing a dominating set based clustering in wireless ad-hoc and sensor networks that captures the characteristics of the set-up phase of such multi-hop radio networks: asynchronous wake-up, the hidden terminal problem, and scarce knowledge about the topology of the network graph.
Abstract: We propose a novel randomized algorithm for computing a dominating set based clustering in wireless ad-hoc and sensor networks. The algorithm works under a model which captures the characteristics of the set-up phase of such multi-hop radio networks: asynchronous wake-up, the hidden terminal problem, and scarce knowledge about the topology of the network graph. When modelling the network as a unit disk graph, the algorithm computes a dominating set in polylogarithmic time and achieves a constant approximation ratio.
36 citations
TL;DR: This work establishes the complexity of the minimum clique partition problem and the maximum independent set problem for penny graphs, both NP-complete, and presents two approximation algorithms for finding clique partitions.
32 citations
TL;DR: This work proposes and analyzes several online and offiine labeling algorithms for the class of disk graphs and gives a traditional coloring of G, the intersection graph of a set of disks in the plane.
28 citations
13 Dec 2004
TL;DR: In this paper, a distributed approximation algorithm that constructs a minimum connected dominating set (MCDS) for wireless ad hoc networks based on a maximal independent set (MIS) is presented.
Abstract: In this paper, we present a new distributed approximation algorithm that constructs a minimum connected dominating set (MCDS) for wireless ad hoc networks based on a maximal independent set (MIS), Our algorithm, which is fully localized, has a constant approximation ratio, and O(n) time and O(n) message complexity. In this algorithm each node only requires the knowledge of its one-hop neighbors and there are only one shortest path connecting two dominators that are at most three hops away. Compared with other MCDS approximation algorithms, our algorithm shows better efficiency and performance than them.
22 citations
1 Jan 2004
TL;DR: The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1+)-approximate independent set or a certificate showing that the input graph is no unit disk graph.
Abstract: A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers).
The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1+)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.
21 citations
1 Jan 2004
TL;DR: In this article, a polynomial-time approximation scheme for the minimum dominating set problem on unit disk graphs is presented, which does not require a geometric representation of the vertices to be given as part of the input.
Abstract: We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input.
12 citations
29 Sep 2004
TL;DR: Upper bounds are found (involving λ but independent of p) for the diameter of Gp(λ,n), building on a method originally due to M. Penrose.
Abstract: Let n be a positive integer, λ>0 a real number, and 1≤ p≤ ∞. We study the unit disk random geometric graphGp(λ,n), defined to be the random graph on n vertices, independently distributed uniformly in the standard unit disk in ${\mathbb R}^2$, with two vertices adjacent if and only if their lp-distance is at most λ. Let $\lambda=c\sqrt{\ln n/n}$, and let ap be the ratio of the (Lebesgue) areas of the lp- and l2-unit disks. Almost always, Gp(λ,n) has no isolated vertices and is also connected if c>ap−−1/2, and has $n^{1-a_pc^2}(1+o(1))$ isolated vertices if c
1 Jan 2004
TL;DR: Analytical results about the performance of various Connected Dominating Set algorithms: MultiPoint Relaying (MPR) flooding, MPR-CDS, Generalized Wu Li CDS (GWL-C DS), in particular on the 1D unit disk graph model are provided.
Abstract: We provide analytical results about the performance of various Connected Dominating Set (CDS) algorithms: MultiPoint Relaying (MPR) flooding, MPR-CDS, Generalized Wu Li CDS (GWL-CDS). In particular we focuse on the 1D unit disk graph model.
19 May 2004
TL;DR: This paper presents a localized Shortest Path Tree (SPT) based algorithm for constructing a sub-network with the minimum-energy property for a given wireless ad hoc network.
Abstract: In this paper, we present a localized Shortest Path Tree (SPT) based algorithm for constructing a sub-network with the minimum-energy property for a given wireless ad hoc network. Each mobile node determines its own transmission power based only on its local information. The proposed algorithm constructs local shortest path trees from the unit disk graph. The performance improvements of our algorithm are demonstrated through simulations.
21 Jun 2004
TL;DR: This work presents a few connections between graph theory and wireless networking, such as topology control, clustering, and geo-routing, which are supplemented with an open problem.
Abstract: Wireless and mobile networks are an excellent playground for graph theoreticians Many research challenges turn out to be variants of classic graph theory problems In particular the rapidly growing areas of ad-hoc and sensor networks demand new solutions for timeless graph theory problems, because i) wireless devices have lower bandwidth and ii) wireless devices are mobile and therefore the topology of the network changes rather frequently As a consequence, algorithms for wireless and mobile networks should have i) as little communication as possible and should ii) run as fast as possible Both goals can only be achieved by developing algorithms requiring a small number of communication rounds only (so-called local algorithms) In this work we present a few connections between graph theory and wireless networking, such as topology control, clustering, and geo-routing Each section is supplemented with an open problem
Patent•
30 Sep 2004TL;DR: In this paper, the authors proposed an approximation method for channel allocation in a wireless, local area network (WLAN) in a reasonable time period using approximation methods, provided the interference pattern associated with APs within a given WLAN conform to a unit disk graph interference pattern.
Abstract: Channels are allocated to access points (APs) within a wireless, local area network (WLAN) in a reasonable time period using approximation methods. One approximation method guarantees channel allocations will be no less than 1/6 of an optimal channel allocation scheme provided the interference pattern associated with APs within a given WLAN conform to a unit disk graph interference pattern.
21 Jun 2004
TL;DR: In this article, a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs is presented, which is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph.
Abstract: A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers).
The approximation algorithm presented is robust in the sense that it accepts any graph as input and either returns a (1+e)-approximate independent set or a certificate showing that the input graph is no unit disk graph. The algorithm can easily be extended to other families of intersection graphs of geometric objects.
1 Jan 2004
TL;DR: This chapter proposes a new efficient heuristic algorithm for the minimum connected dominating set problem that reduces the size of the CDS by excluding some vertices using a greedy criterion and discusses a distributed version of this algorithm.
Abstract: Given a graph G = (V, E), a dominating set D is a subset of V such that any vertex not in D is adjacent to at least one vertex in D. Efficient algorithms for computing the minimum connected dominating set (MCDS) are essential for solving many practical problems, such as finding a minimum size backbone in ad hoc networks. Wireless ad hoc networks appear in a wide variety of applications, including mobile commerce, search and discovery, and military battlefield. In this chapter we propose a new efficient heuristic algorithm for the minimum connected dominating set problem. The algorithm starts with a feasible solution containing all vertices of the graph. Then it reduces the size of the CDS by excluding some vertices using a greedy criterion. We also discuss a distributed version of this algorithm. The results of numerical testing show that, despite its simplicity, the proposed algorithm is competitive with other existing approaches.
26 Sep 2004
TL;DR: This paper model the network as a multi-hop quasi unit disk graph and allows nodes to wake up asynchronously at any time, and shows that even for this restricted model, a good clustering can be computed efficiently.
Abstract: A newly deployed multi-hop radio network is unstructured and lacks a reliable and efficient communication scheme. In this paper, we take a step towards analyzing the problems existing during the initialization phase of ad hoc and sensor networks. Particularly, we model the network as a multi-hop quasi unit disk graph and allow nodes to wake up asynchronously at any time. Further, nodes do not feature a reliable collision detection mechanism, and they have only limited knowledge about the network topology. We show that even for this restricted model, a good clustering can be computed efficiently. Our algorithm efficiently computes an asymptotically optimal clustering. Based on this algorithm, we describe a protocol for quickly establishing synchronized sleep and listen schedule between nodes within a cluster. Additionally, we provide simulation results in a variety of settings.
15 Dec 2004
TL;DR: A localized algorithm that builds a graph called planar localized Delaunay triangulation, PLDel, known to be a good spanner of the unit disk graph, UDG, in a single communication step, maintaining a communication cost of O(n log n), which is within a constant of the optimum.
Abstract: A localized Delaunay triangulation owns the following interesting properties in a wireless ad hoc setting: it can be built with localized information, the communication cost imposed by control information is limited and it supports geographical routing algorithms that offer guaranteed convergence. This paper presents a localized algorithm that builds a graph called planar localized Delaunay triangulation, PLDel, known to be a good spanner of the unit disk graph, UDG. Unlike previous work, our algorithm builds PLDel in a single communication step, maintaining a communication cost of O(n log n), which is within a constant of the optimum. This represents a significant practical improvement over previous algorithms with similar theoretical bounds. Furthermore, the small cost of our algorithm makes feasible to use PLDel in real systems, instead of the Gabriel or the Relative Neighborhood graphs, which are not good spanners of UDG.
26 Apr 2004
TL;DR: The XTC ad-hoc network topology control algorithm introduced shows three main advantages over previously proposed algorithms: it is extremely simple and strictly local, it does not assume the network graph to be a unit disk graph, and XTC proves correct also on general weighted network graphs.
Abstract: Summary form only given. The XTC ad-hoc network topology control algorithm introduced shows three main advantages over previously proposed algorithms. First, it is extremely simple and strictly local. Second, it does not assume the network graph to be a unit disk graph; XTC proves correct also on general weighted network graphs. Third, the algorithm does not require availability of node position information. Instead, XTC operates with a general notion of order over the neighbors' link qualities. In the special case of the network graph being a unit disk graph, the resulting topology proves to have bounded degree, to be a planar graph, and - on average-case graphs - to be a good spanner.
1 Oct 2004
TL;DR: A non-approximability result is proved for the problem of embedding a given unit disk graph that if non-neighboring nodes are not allowed to be closer to each other than distance 1, then two neighbors can be as far apart as √3/2 - ε, where ε goes to 0 as n goes to infinity, unless P=NP.
Abstract: Finding a good embedding of a unit disk graph given by its connectivity information is a problem of practical importance in a variety of fields. In wireless ad hoc and sensor networks, such an embedding can be used to obtain virtual coordinates. In this paper, we prove a non-approximability result for the problem of embedding a given unit disk graph. Particularly, we show that if non-neighboring nodes are not allowed to be closer to each other than distance 1, then two neighbors can be as far apart as √3/2 - e, where e goes to 0 as n goes to infinity, unless P=NP. We further show that finding a realization of a d-quasi unit disk graph with d ≥ 1/√2 is NP-hard.