Unit disk graph

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.

Abstract: Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

Abstract: In this paper we study a model for ad-hoc networks close enough to reality as to represent existing networks, being at the same time concise enough to promote strong theoretical results. The Quasi Unit Disk Graph model contains all edges shorter than a parameter d between 0 and 1 and no edges longer than 1.We show that .in comparison to the cost known on Unit Disk Graphs .the complexity results in this model contain the additional factor 1 /d2. We prove that in Quasi Unit Disk Graphs flooding is an asymptotically message-optimal routing technique, provide a geometric routing algorithm being more efficient above all in dense networks, and show that classic geometric routing is possible with the same performance guarantees as for Unit Disk Graphs if d = 1/v2.

Abstract: A connected dominating set in a graph is a subset of vertices such that every vertex is either in the subset or adjacent to a vertex in the subset and the subgraph induced by the subset is connected. A minimum-connected dominating set is such a vertex subset with minimum cardinality. An application in ad hoc wireless networks requires the study of the minimum-connected dominating set in unit-disk graphs. In this paper, we design a (1 + 1/s)-approximation for the minimum-connected dominating set in unit-disk graphs, running in time nO((s log s)2). © 2003 Wiley Periodicals, Inc.