Errol L. Lloyd

Scheduling algorithms for multihop radio networks

Algorithms for transmission scheduling in multihop broadcast radio networks are presented. Both link scheduling and broadcast scheduling are considered. In each instance, scheduling algorithms are given that improve upon existing algorithms both theoretically and experimentally. It is shown that tree networks can be scheduled optimally and that arbitrary networks can be scheduled so that the schedule is bounded by a length that is proportional to a function of the network thickness times the optimum. Previous algorithms could guarantee only that the schedules were bounded by a length no worse than the maximum node degree times optimum. Since the thickness is typically several orders of magnitude less than the maximum node degree, the algorithms presented represent a considerable theoretical improvement. Experimentally, a realistic model of a radio network is given and the performance of the new algorithms is studied. These results show that, for both types of scheduling, the new algorithms (experimentally) perform consistently better than earlier methods. >

Relay Node Placement in Wireless Sensor Networks

A wireless sensor network consists of many low-cost, low-power sensor nodes, which can perform sensing, simple computation, and transmission of sensed information. Long distance transmission by sensor nodes is not energy efficient since energy consumption is a superlinear function of the transmission distance. One approach to prolonging network lifetime while preserving network connectivity is to deploy a small number of costly, but more powerful, relay nodes whose main task is communication with other sensor or relay nodes. In this paper, we assume that sensor nodes have communication range r>0, while relay nodes have communication range Rgesr, and we study two versions of relay node placement problems. In the first version, we want to deploy the minimum number of relay nodes so that, between each pair of sensor nodes, there is a connecting path consisting of relay and/or sensor nodes. In the second version, we want to deploy the minimum number of relay nodes so that, between each pair of sensor nodes, there is a connecting path consisting solely of relay nodes. We present a polynomial time 7-approximation algorithm for the first problem and a polynomial time (5+epsi)-approximation algorithm for the second problem, where epsi>0 can be any given constant

Algorithmic aspects of topology control problems for ad hoc networks

Topology control problems are concerned with the assignment of power values to the nodes of an ad hoc network so that the power assignment leads to a graph topology satisfying some specified properties. This paper considers such problems under several optimization objectives, including minimizing the maximum power and minimizing the total power. A general approach leading to a polynomial algorithm is presented for minimizing maximum power for a class of graph properties called textbf monotone properties. The difficulty of generalizing the approach to properties that are not monotone is discussed. Problems involving the minimization of total power are known to be bf NP -complete even for simple graph properties. A general approach that leads to an approximation algorithm for minimizing the total power for some monotone properties is presented. Using this approach, a new approximation algorithm for the problem of minimizing the total power for obtaining a 2-node-connected graph is obtained. It is shown that this algorithm provides a constant performance guarantee. Experimental results from an implementation of the approximation algorithm are also presented.

Fault-Tolerant Relay Node Placement in Heterogeneous Wireless Sensor Networks

Existing work on placing additional relay nodes in wireless sensor networks to improve network connectivity typically assumes homogeneous wireless sensor nodes with an identical transmission radius. In contrast, this paper addresses the problem of deploying relay nodes to provide fault tolerance with higher network connectivity in heterogeneous wireless sensor networks, where sensor nodes possess different transmission radii. Depending on the level of desired fault tolerance, such problems can be categorized as: 1) full fault-tolerant relay node placement, which aims to deploy a minimum number of relay nodes to establish k(k ? 1) vertexdisjoint paths between every pair of sensor and/or relay nodes and 2) partial fault-tolerant relay node placement, which aims to deploy a minimum number of relay nodes to establish k(k ? 1) vertex-disjoint paths only between every pair of sensor nodes. Due to the different transmission radii of sensor nodes, these problems are further complicated by the existence of two different kinds of communication paths in heterogeneous wireless sensor networks, namely, two-way paths, along which wireless communications exist in both directions; and one-way paths, along which wireless communications exist in only one direction. Assuming that sensor nodes have different transmission radii, while relay nodes use the same transmission radius, this paper comprehensively analyzes the range of problems introduced by the different levels of fault tolerance (full or partial) coupled with the different types of path (one-way or two-way). Since each of these problems is NP-hard, we develop O(?k2)-approximation algorithms for both one-way and two-way partial fault-tolerant relay node placement, as well as O(?k3)-approximation algorithms for both one-way and two-way full fault-tolerant relay node placement (? is the best performance ratio of existing approximation algorithms for finding a minimum k-vertex connected spanning graph). To facilitate the applications in higher dimensions, we also extend these algorithms and derive their performance ratios in d-dimensional heterogeneous wireless sensor networks (d ? 3). Finally, heuristic implementations of these algorithms are evaluated via QualNet simulations.

Scheduling algorithms for multi-hop radio networks

New algorithms for transmission scheduling in multihop broadcast radio networks are presented. Both link scheduling and broadcast scheduling are considered. In each instance scheduling algorithms are given that improve upon existing algorithms both theoretically and experimentally. Theoretically, it is shown that tree networks can be scheduled optimally, and that arbitrary networks can be scheduled so that the schedule is bounded by a length that is proportional to a function of the network thickness times the optimum. Previous algorithms could guarantee only that the schedules were bounded by a length no worse than the maximum node degree, the algorithms presented here represent a considerable theoretical improvement. Experimentally, a realistic model of a radio network is given and the performance of the new algorithms is studied. These results show that, for both types of scheduling, the new algorithms (experimentally) perform consistently better than earlier methods.

On the k -coloring of intervals

Abstract The problem of coloring a set of n intervals (from the real line) with a set of k colors is studied. In such a coloring, two intersecting intervals must receive distinct colors. Our main result is an O ( k + n ) algorithm for k - coloring a maximum cardinality subset of the intervals, assuming that the endpoints of the intervals are presorted. Previous methods are linear only in n , and assume that k is a fixed constant. In addition to the main result, we provide an O ( kS ( n )) algorithm for k - coloring a set of weighted intervals of maximum total weight. Here, S ( n ) is the running time of any algorithm for finding shortest paths in graphs with O ( n ) edges. The best previous algorithm for this problem required time O ( nS ( n )). Since in most applications, k is substantially smaller than n , the saving is significant.

CLTC: a cluster-based topology control for ad hoc networks

The topology of an ad hoc network has a significant impact on its performance in that a dense topology may induce high interference and low capacity, while a sparse topology is vulnerable to link failure and network partitioning. Topology control aims to maintain a topology that optimizes network performance while minimizing energy consumption. Existing topology control algorithms utilize either a purely centralized or a purely distributed approach. A centralized approach, although able to achieve strong connectivity (k-connectivity for k /spl ges/ 2), suffers from scalability problems. In contrast, a distributed approach, although scalable, lacks strong connectivity guarantees. We propose a hybrid topology control framework, cluster-based topology control (CLTC) that achieves both scalability and strong connectivity. By varying the algorithms utilized in each of the three phases of the framework, a variety of optimization objectives and topological properties can be achieved. In this paper, we present the CLTC framework; describe topology control algorithms based on CLTC and prove that k-connectivity is achieved using those algorithms; analyze the message complexity of an implementation of CLTC, namely, CLTC-A, and present simulation studies that evaluate the effectiveness of CLTC-A for a range of networks.

On triangulations of a set of points in the plane

A set, V, of points in the plane is triangulated by a subset T, of the straight-line segments whose endpoints are in V, if T is a maximal subset such that the line segments in T intersect only at their endpoints. The weight of any triangulation is the sum of the Euclidean lengths of the line segments in the triangulation. We examine two problems involving triangulations. We discuss the problem of finding a minimum weight triangulation among all triangulations of a set of points and give counterexamples to two published solutions to this problem. Secondly, we show that the problem of determining the existence of a triangulation, in a given subset of the line segments whose endpoints are in V, is NP-Complete.

Deploying Directional Sensor Networks with Guaranteed Connectivity and Coverage

In contrast to existing work on the connected coverage problem in wireless sensor networks which assumes omnidirectional sensors with disk-like sensing range, this paper investigates a suite of novel problems related to connected coverage in directional sensor networks where sensors only sense directionally and have a sector-like sensing range. We first consider the problems of deploying a minimum number of directional sensors to form a connected network to cover either a set of point- locations (connected point-coverage deployment (CPD)) or the entire target sensing area (connected region-coverage deployment (CRD)). CPD is NP-hard as its subproblem of geometric sector cover (GSC) is NP-hard. We present two approximation algorithms for GSC as subroutines, and develop a general solution framework for CPD with approximation ratio sigma + O(1), where sigma is the approximation ratio of the selected GSC subroutine. We also describe two efficient deployment patterns with guaranteed covering density for CRD, and analyze their performance bounds with respect to arbitrary non-crossing deployment patterns. Extensive simulation results validate the correctness and merits of the presented algorithms and analysis.

Feedback vertex sets and cyclically reducible graphs

The problem of finding a minimum cardinality feedback vertex set of a directed graph is considered. Of the classic NP-complete problems, this is one of the least understood. Although Karp showed the general problem to be NP-complete, a linear algorithm for its solution on reducible flow graphs was given by Shamir. The class of reducible flow graphs is the only nontrivial class of graphs for which a polynomial-time algorithm to solve this problem is known. The main result of this paper is to present a new class of graphs—the cyclically reducible graphs—for which minimum feedback vertex sets can be found in polynomial time. This class is not restricted to flow graphs, and most small graphs (10 or fewer nodes) fall into this class. The identification of this class is particularly important since there do not exist approximation algorithms for this problem having a provably good worst case performance. Along with the class and a simple polynomial-time algorithm for finding minimum feedback vertex sets of graphs in the class, several related results are presented. It is shown that there is no “forbidden subgraph” characterization of the class and that there is no particular inclusion relationship between this class and the reducible flow graphs. In addition, it is shown that a class of (general) graphs, which are related to the reducible flow graphs, are contained in the cyclically reducible class.