Weakly-connected dominating sets and sparse spanners in wireless ad hoc networks

A set S is dominating if each node in the graph G = (V, E) is either in S or adjacent to at least one of the nodes in S. The subgraph weakly induced by S is the graph G' = (V, E') such that each edge in E' has at least one end point in S. The set S is a weakly-connected dominating set (WCDS) of G if S is dominating and G' is connected G' is a sparse spanner if it has linear edges. In this paper, we present two distributed algorithms for finding a WCDS in O(n) time. The first algorithm has an approximation ratio of 5, and requires O(n log n) messages. The second algorithm has a larger approximation ratio, but it requires only O(n) messages. The graph G' generated by the second algorithm forms a sparse spanner with a topological dilation of 3, and a geometric dilation of 6.

[1]  Brad Karp,et al.  GPSR: greedy perimeter stateless routing for wireless networks , 2000, MobiCom '00.

[2]  Stefano Basagni,et al.  A performance comparison of protocols for clustering and backbone formation in large scale ad hoc networks , 2004, 2004 IEEE International Conference on Mobile Ad-hoc and Sensor Systems (IEEE Cat. No.04EX975).

[3]  David P. Dobkin,et al.  On sparse spanners of weighted graphs , 1993, Discret. Comput. Geom..

[4]  Jie Wu,et al.  A Dominating-Set-Based Routing Scheme in Ad Hoc Wireless Networks , 2001, Telecommun. Syst..

[5]  R. Rajaraman,et al.  An efficient distributed algorithm for constructing small dominating sets , 2002 .

[6]  Rajmohan Rajaraman,et al.  Topology control and routing in ad hoc networks: a survey , 2002, SIGA.

[7]  Peng-Jun Wan,et al.  Message-optimal connected dominating sets in mobile ad hoc networks , 2002, MobiHoc '02.

[8]  Luca Trevisan,et al.  Non-approximability results for optimization problems on bounded degree instances , 2001, STOC '01.

[9]  Harry B. Hunt,et al.  Simple heuristics for unit disk graphs , 1995, Networks.

[10]  N Linial,et al.  Low diameter graph decompositions , 1993, Comb..

[11]  RNG and internal node based broadcasting algorithms for wireless one-to-one networks , 2001, MOCO.

[12]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[13]  Johannes H. Hattingh,et al.  On weakly connected domination in graphs , 1997, Discret. Math..

[14]  Panganamala Ramana Kumar,et al.  The Number of Neighbors Needed for Connectivity of Wireless Networks , 2004, Wirel. Networks.

[15]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1993, STOC.

[16]  Peng-Jun Wan,et al.  Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[17]  Israel Cidon,et al.  Propagation and Leader Election in a Multihop Broadcast Environment , 1998, DISC.

[18]  Samir Khuller,et al.  Approximation Algorithms for Connected Dominating Sets , 1996, Algorithmica.

[19]  Vijay V. Vazirani,et al.  Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[20]  Charles J. Colbourn,et al.  Unit disk graphs , 1991, Discret. Math..

[21]  Peng-Jun Wan,et al.  Distributed heuristics for connected dominating sets in wireless ad hoc networks , 2002, Journal of Communications and Networks.

[22]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[23]  Ivan Stojmenovic,et al.  Routing with Guaranteed Delivery in Ad Hoc Wireless Networks , 2001, Wirel. Networks.

[24]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[25]  Peng-Jun Wan,et al.  New distributed algorithm for connected dominating set in wireless ad hoc networks , 2002, Proceedings of the 35th Annual Hawaii International Conference on System Sciences.

[26]  Baruch Awerbuch,et al.  Sparse partitions , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[27]  Vaduvur Bharghavan,et al.  Routing in ad-hoc networks using minimum connected dominating sets , 1997, Proceedings of ICC'97 - International Conference on Communications.

[28]  Peng-Jun Wan,et al.  Virtual backbones in wireless ad hoc networks , 2002 .

[29]  Arthur L. Liestman,et al.  Approximating minimum size weakly-connected dominating sets for clustering mobile ad hoc networks , 2002, MobiHoc '02.