Local Edge Colouring of Yao-Like Subgraphs of Unit Disk Graphs

The focus of the present paper is on providing a local deterministic algorithm for colouring the edges of Yao-like subgraphs of Unit Disc Graphs. These are geometric graphs such that for some positive integers l, k the following property holds at each node v: if we partition the unit circle centered at v into 2k equally sized wedges then each wedge can contain at most l points different from v. We assume that the nodes are location aware, i.e. they know their Cartesian coordinates in the plane. The algorithm presented is local in the sense that each node can receive information emanating only from nodes which are at most a constant (depending on k and l, but not on the size of the graph) number of hops away from it, and hence the algorithm terminates in a constant number of steps. The number of colours used is 2kl + 1 and this is optimal for local algorithms (since the maximal degree is 2kl and a colouring with 2kl colours can only be constructed by a global algorithm), thus showing that in this class of graphs the price for locality is only one additional colour.

[1]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[2]  Andrew Chi-Chih Yao,et al.  On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems , 1977, SIAM J. Comput..

[3]  Jorge Urrutia,et al.  Local Construction of Planar Spanners in Unit Disk Graphs with Irregular Transmission Ranges , 2006, LATIN.

[4]  Xiang-Yang Li,et al.  Localized Delaunay Triangulation with Application in Ad Hoc Wireless Networks , 2003, IEEE Trans. Parallel Distributed Syst..

[5]  Aravind Srinivasan,et al.  End-to-end packet-scheduling in wireless ad-hoc networks , 2004, SODA '04.

[6]  Madhav V. Marathe,et al.  Strong edge coloring for channel assignment in wireless radio networks , 2006, Fourth Annual IEEE International Conference on Pervasive Computing and Communications Workshops (PERCOMW'06).

[7]  Milind Dawande,et al.  Link scheduling in sensor networks: distributed edge coloring revisited , 2005, Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies..

[8]  Tomomi Matsui,et al.  Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs , 1998, JCDCG.

[9]  P. Erdös.  On circuits and subgraphs of chromatic graphs , 1962 .

[10]  Xiang-Yang Li,et al.  Localized Construction of Bounded Degree and Planar Spanner for Wireless Ad Hoc Networks , 2003, DIALM-POMC '03.

[11]  S. Ramanathan,et al.  A unified framework and algorithm for channel assignment in wireless networks , 1999, Wirel. Networks.

[12]  Moni Naor,et al.  What Can be Computed Locally? , 1995, SIAM J. Comput..

[13]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[14]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

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

[16]  Jorge Urrutia,et al.  Half-Space Proximal: A New Local Test for Extracting a Bounded Dilation Spanner of a Unit Disk Graph , 2005, OPODIS.

[17]  Murali S. Kodialam,et al.  Characterizing achievable rates in multi-hop wireless mesh networks with orthogonal channels , 2005, IEEE/ACM Transactions on Networking.