Local 7-coloring for planar subgraphs of unit disk graphs

Abstract

The problem of computing locally a coloring of an arbitrary planar subgraph of a unit disk graph is studied. Each vertex knows its coordinates in the plane, can directly communicate 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 of hops away from it.

Survey of local algorithms

“…Wiese and Kranakis [144,147] present a local algorithm that constructs a connected, planar, vertex-coloured subgraph of a unit-disk graph. Czyzowicz et al [32] present a local algorithm for colouring the nodes in an arbitrary planar subgraph of a unit-disk graph. Czyzowicz et al [33] present a local algorithm for colouring the edges in certain subgraphs of unit-disk graphs.…”

Analysing local algorithms in location-aware quasi-unit-disk graphs

“…Indeed, quite a few local algorithms are known for location-aware graphs [6,[10][11][12]19,20,23,[25][26][27]30,48,51,52,[54][55][56][57][58][59]. In this work, we give a unified description and analysis of a class of local algorithms based on a simple ''tile-and-combine'' idea: decompose the plane into tiles, have each tile solve its subproblem optimally, and combine the solutions into a global output.…”