Unit Disk Graph Research Articles

Local 7-coloring for planar subgraphs of unit disk graphs

We study the problem of computing locally a coloring of an arbitrary planar subgraph of a unit disk graph. Each vertex knows its coordinates in the plane and can communicate directly with all its neighbors within unit distance. Using this setting, first a simple algorithm is given whereby each vertex can compute its color in a 9-coloring of the planar graph using only information on the subgraph located within at most 9 hops away from it in the original unit disk graph. A more complicated algorithm is then presented whereby each vertex can compute its color in a 7-coloring of the planar graph using only information on the subgraph located within a constant number (201, to be exact) of hops away from it.

On the Planarization of Wireless Sensor Networks

Network planarization has been an important technique in numerous sensornet protocols—such as Greedy Perimeter Stateless Routing (GPSR), topology discovery, data-centric storage, etc.—however the planarization process itself has been difficult. Known efficient planarization algorithms exist only for restrictive wireless network models: unit-disk graphs with accurately known location information. In this paper, we study efficient planarization of wireless sensor networks, and present a novel planarization method for a more general network model, where sensors can have non-uniform transmission ranges and no location information is needed. Our planarization algorithms also include a (2+ε)-approximation algorithm and an FPT algorithm for the bipartite planarization problem.

Constructing weakly connected dominating set for secure clustering in distributed sensor network

Secure clustering problem plays an important role in distributed sensor networks. Weakly Connected Dominating Set (WCDS) is used for solving this problem. Therefore, computing a minimum WCDS becomes an important topic of this research. In this paper, we compare the size of Maximal Independent Set (MIS) and minimum WCDS in unit disk graph. Our analysis shows that five is the least upper bound for this ratio. We also present a distributed algorithm to produce a weakly connected MIS within a factor 5 from the minimum WCDS.

A PTAS for the minimum weighted dominating set problem with smooth weights on unit disk graphs

In the minimum weighted dominating set problem (MWDS), we are given a unit disk graph with non-negative weight on each vertex. The MWDS seeks a subset of the vertices of the graph with minimum total weight such that each vertex of the graph is either in the subset or adjacent to some nodes in the subset. A weight function is called smooth, if the ratio of the weights of any two adjacent nodes is upper bounded by a constant. MWDS is known to be NP-hard. In this paper, we give the first polynomial time approximation scheme (PTAS) for MWDS with smooth weights on unit disk graphs, which achieves a (1+?)-approximation for MWDS, for any ?>0.

Polynomial time approximation scheme for t-latency bounded information propagation problem in wireless networks

Latency of information propagating in wireless network is gaining more and more attention recently. This paper studies the problem of t-Latency Bounded Information Propagation (t-LBIP) problem in wireless networks which are represented by unit-disk graphs. So far, no guaranteed approximation algorithm has been achieved for t-LBIP when t?2. In this paper, we propose a Polynomial Time Approximation Scheme for t-LBIP under the condition that the maximum degree is bounded by a constant.

On the Complexity of Scheduling in Wireless Networks

We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of -hop interference models, under which no two links within a -hop distance can successfully transmit at the same time. For a given , we can obtain a throughput-optimal scheduling policy by solving the well-known maximum weighted matching problem. We show that for , the resulting problems are NP-Hard that cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, for geometric unit-disk graphs that can be used to describe a wide range of wireless networks, the problems admit polynomial time approximation schemes within a factor arbitrarily close to 1. In these network settings, we also show that a simple greedy algorithm can provide a 49-approximation, and the maximal matching scheduling policy, which can be easily implemented in a distributed fashion, achieves a guaranteed fraction of the capacity region for "all ." The geometric constraints are crucial to obtain these throughput guarantees. These results are encouraging as they suggest that one can develop low-complexity distributed algorithms to achieve near-optimal throughput for a wide range of wireless networks.

Localized spanner construction for ad hoc networks with variable transmission range

This article presents an algorithm for constructing a spanner for ad hoc networks whose nodes have variable transmission range. Almost all previous spanner constructions for ad hoc networks assumed that all nodes in the network have the same transmission range. This allowed a succinct representation of the network as a unit disk graph, serving as the basis for the construction. In contrast, when nodes have variable transmission range, the ad hoc network must be modeled by a general disk graph. Whereas unit disk graphs are undirected, general disk graphs are directed. This complicates the construction of a spanner for the network, since currently there are no efficient constructions of low-stretch spanners for general directed graphs. Nevertheless, in this article it is shown that the class of disk graphs enjoys (efficiently constructible) spanners of quality similar to that of unit disk graph spanners. Moreover, it is shown that the new construction can be performed in a localized fashion. Our results use only simple packing arguments, hence all algorithms work for every metric space of constant doubling dimension.

Two algorithms for minimum 2-connected r-hop dominating set

For a graph G = ( V , E ) , a subset D ⊆ V is an r-hop dominating set if every vertex u ∈ V − D is at most r-hops away from D. It is a 2-connected r-hop dominating set if the subgraph of G induced by D is 2-connected. In this paper, we present two approximation algorithms to compute minimum 2-connected r-hop dominating set. The first one is a greedy algorithm using ear decomposition of 2-connected graphs. This algorithm is applicable to any 2-connected general graph. The second one is a three-phase algorithm which is only applicable to unit disk graphs. For both algorithms, performance ratios are given.

A Better Approximation Algorithm for Computing Connected Dominating Sets in Unit Ball Graphs

A Virtual Backbone (VB) of a wireless network is a subset of nodes such that only VB nodes are responsible for routing-related tasks. Since a smaller VB causes less overhead, size is the primary quality factor of VB. Frequently, Unit Disk Graphs (UDGs) are used to model 2D homogeneous wireless networks, and the problem of finding minimum VBs in the networks is abstracted as Minimum Connected Dominating Set (MCDS) problem in UDGs. In some applications, the altitude of nodes can be hugely different and UDG cannot abstract the networks accurately. Then, Unit Ball Graph (UBG) can replace UDG. In this paper, we study how to construct quality CDSs in UBGs in distributed environments. We first give an improved upper bound of the number of independent nodes in a UBG, and use this result to analyze the Performance Ratio (PR) of our new centralized algorithm C-CDS-UBG, which computes CDSs in UBGs. Next, we propose a distributed algorithm D-CDS-UBG originated from C-CDS-UBG and analyze its message and time complexities. Our theoretical analysis shows that the PR of D-CDS-UBG is 14.937, which is better than current best, 22. Our simulations also show that D-CDS-UBG outperforms the competitor, on average.

Separability and topology control of quasi unit disk graphs

A deep understanding of the structural properties of wireless networks is critical for evaluating the performance of network protocols and improving their designs. Many protocols for wireless networks—routing, topology control, information storage/retrieval and numerous other applications—have been based on the idealized unit-disk graph (UDG) network model. The significant deviation of the UDG model from many real wireless networks is substantially limiting the applicability of such protocols. A more general network model, the quasi unit-disk graph (quasi-UDG) model, captures much better the characteristics of wireless networks. However, the understanding of the properties of general quasi-UDGs has been very limited, which is impeding the designs of key network protocols and algorithms. In this paper, we present results on two important properties of quasi-UDGs: separability and the existence of power efficient spanners. Network separability is a fundamental property leading to efficient network algorithms and fast parallel computation. We prove that every quasi-UDG has a corresponding grid graph with small balanced separators that captures its connectivity properties. We also study the problem of constructing an energy-efficient backbone for a quasi-UDG. We present a distributed local algorithm that, given a quasi-UDG, constructs a nearly planar backbone with a constant stretch factor and a bounded degree. We demonstrate the excellent performance of these auxiliary graphs through simulations and show their applications in efficient routing.

On boundary recognition without location information in wireless sensor networks

Boundary recognition is an important and challenging issue in wireless sensor networks when no coordinates or distances are available. The distinction between inner and boundary nodes of the network can provide valuable knowledge to a broad spectrum of algorithms. This article tackles the challenge of providing a scalable and range-free solution for boundary recognition that does not require a high node density. We explain the challenges of accurately defining the boundary of a wireless sensor network with and without node positions and provide a new definition of network boundary in the discrete domain. Our solution for boundary recognition approximates the boundary of the sensor network by determining the majority of inner nodes using geometric constructions, which guarantee that for a given d , a node lies inside of the construction for a d -quasi unit disk graph model of the wireless sensor network. Moreover, such geometric constructions make it possible to compute a guaranteed distance from a node to the boundary. We present a fully distributed algorithm for boundary recognition based on these concepts and perform a detailed complexity analysis. We provide a thorough evaluation of our approach and show that it is applicable to dense as well as sparse deployments.

Minimum Latency Gossiping in Radio Networks

We studied the minimum latency gossiping (all-to-all broadcast) problem in multihop radio networks defined as follows: Each node in the network is preloaded with a message and the objective is to distribute each node's message to the entire network with minimum latency. We studied this problem in the unit-size message model and the unit disk graph model. The unit-size model means different messages cannot be combined as one message, and the unit disk graph model means a link exists between two nodes if and only if their euclidean distance is less than 1. The minimum latency gossiping problem is known to be NP-hard in these two models. In this work, we designed a gossiping scheme that significantly improved all current gossiping algorithms in terms of the approximation ratio. Our work has approximation ratio 27, a great improvement of the current state-of-the-art algorithm (which has ratio 1,947). We also discussed the single point of failure problem and its impact on our approximation ratio. We designed an amended gossiping algorithm with ratio 27 in case of a nonsource node failure. We also designed an amended gossiping algorithm with ratio 29 in case of source failure.

Approximations for node-weighted Steiner tree in unit disk graphs

Given a node-weighted connected graph and a subset of terminals, the problem node-weighted Steiner tree (NWST) seeks a lightest tree connecting a given set of terminals in a node-weighted graph. While NWST in general graphs are as hard as Set Cover, NWST restricted to unit-disk graphs (UDGs) admits constant-approximations. Recently, Zou et al. (Lecture notes in computer science, vol 5165, COCOA, 2008, pp 278–285) showed that any μ-approximation algorithm for the classical edge-weighted Steiner tree problem can be used to produce 2.5 μ-approximation algorithm for NWST restricted to UDGs. With the best known approximation bound 1.55 for the classical Steiner tree problem, they obtained an approximation bound 3.875 for NWST restricted to UDGs. In this paper, we present three approximation algorithms for NWST restricted to UDGs, the k-Restricted Relative Greedy Algorithm whose approximation bound converges to 1 + ln 5 ≈ 2.61 as k → ∞, the 3-Restricted Greedy Algorithm with approximation bound $${4\frac{1}{3}}$$ , and the k-Restricted Variable Metric Algorithm whose approximation bound converges to 3.9334 as k → ∞.

PTAS for minimum weighted connected vertex cover problem with c-local condition in unit disk graphs

Given a graph G =( V , E ) with node weight w : V R +, the minimum weighted connected vertex cover problem (MWCVC) is to seek a subset of vertices of the graph with minimum total weight, such that ...

MAXIMUM AREA INDEPENDENT SETS IN DISK INTERSECTION GRAPHS

Maximum Independent Set (MIS) and its relative Maximum Weight Independent Set (MWIS) are well-known problems in combinatorial optimization; they are NP-hard even in the geometric setting of unit disk graphs. In this paper, we study the Maximum Area Independent Set (MAIS) problem, a natural restricted version of MWIS in disk intersection graphs where the weight equals the disk area. We obtain: (i) Quantitative bounds on the maximum total area of an independent set relative to the union area; (ii) Practical constant-ratio approximation algorithms for finding an independent set with a large total area relative to the union area.

An Edge-Constrained Localized Delaunay Graph for Geographic Routing in Mobile Ad Hoc and Sensor Networks

In mobile ad hoc and sensor networks, since all nodes are mobile and there is no fixed infrastructure, the design of routing protocols becomes one of the most challenging issues. In recent years, geographic routing protocols have been widely used. Most of them, e.g., greedy-face-greedy routing protocols, need nodes to construct planar graphs as the underlying graphs. In this paper, we propose an edge constrained localized Delaunay graph, denoted by ECLDel, as the underlying graph for geographic routing. We prove that the ECLDel is a planar t-spanner of the unit-disk graph. Geographic routing on ECLDel is as efficient as on the previous work of PLDel in terms of path length (hop count). However, the construction of ECLDel graph is far more simple and it converges faster. This is due to the following two reasons: First, we significantly reduce the number of messages broadcast by each node from five rounds (each round may contain several messages) to only two messages; second, we define two new types of edges, the intersecting Gabriel (IG) edges and the unaware intersection (UI) edges, which are constrained in the ECLDel graph. These edges help significantly reduce the size of messages broadcast by each node. The decrease of both the number and the size of messages broadcast by each node reduces the communication cost, and saves the network bandwidth and node power, which are desirable for mobile ad hoc and sensor networks. Our simulation results show that the average number of messages and the average size of messages broadcast by each node are, respectively, 65 and 42 percent less in the construction of ECLDel than that in PLDel.

An optimal maximal independent set algorithm for bounded-independence graphs

We present a novel distributed algorithm for the maximal independent set problem (This is an extended journal version of Schneider and Wattenhofer in Twenty-seventh annual ACM SIGACT-SIGOPS symposium on principles of distributed computing, 2008). On bounded-independence graphs our deterministic algorithm finishes in O(log* n) time, n being the number of nodes. In light of Linial’s Ω(log* n) lower bound our algorithm is asymptotically optimal. Furthermore, it solves the connected dominating set problem for unit disk graphs in O(log* n) time, exponentially faster than the state-of-the-art algorithm. With a new extension our algorithm also computes a δ + 1 coloring and a maximal matching in O(log* n) time, where δ is the maximum degree of the graph.

ROMAN DOMINATION AND ITS VARIANTS IN UNIT DISK GRAPHS

Unit disk graphs are the intersection graphs of equal sized disks in the plane, they are widely used as a mathematical model for wireless ad-hoc networks and some problems in computational geometry. In this paper we first show that Roman dominating set and connected Roman dominating set problems in unit disk graphs are NP-complete, and then present two approximation algorithms for these problems.

Minimum Connected Dominating Set Using a Collaborative Cover Heuristic for Ad Hoc Sensor Networks

A minimum connected dominating set (MCDS) is used as virtual backbone for efficient routing and broadcasting in ad hoc sensor networks. The minimum CDS problem is NP-complete even in unit disk graphs. Many heuristics-based distributed approximation algorithms for MCDS problems are reported and the best known performance ratio has (4.8 + 1n 5). We propose a new heuristic called collaborative cover using two principles: 1) domatic number of a connected graph is at least two and 2) optimal substructure defined as subset of independent dominator preferably with a common connector. We obtain a partial Steiner tree during the construction of the independent set (dominators). A final postprocessing step identifies the Steiner nodes in the formation of Steiner tree for the independent set of G. We show that our collaborative cover heuristics are better than degree-based heuristics in identifying independent set and Steiner tree. While our distributed approximation CDS algorithm achieves the performance ratio of (4.8 + 1n 5) opt + 1.2, where opt is the size of any optimal CDS, we also show that the collaborative cover heuristic is able to give a marginally better bound when the distribution of sensor nodes is uniform permitting identification of the optimal substructures. We show that the message complexity of our algorithm is O(n¿ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), ¿ being the maximum degree of a node in graph and the time complexity is O(n).

Combining hierarchical and goal-directed speed-up techniques for dijkstra's algorithm

In recent years, highly effective hierarchical and goal-directed speed-up techniques for routing in large road networks have been developed. This article makes a systematic study of combinations of such techniques. These combinations turn out to give the best results in many scenarios, including graphs for unit disk graphs, grid networks, and time-expanded timetables. Besides these quantitative results, we obtain general insights for successful combinations.