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

A local algorithm with local horizon rr is a distributed algorithm that runs in rr synchronous communication rounds; here rr is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within rr hops from the node.We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set.We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.

► Local distributed algorithms: constant number of communication rounds.
► Geometric graphs with location-aware nodes: each node knows its coordinates.
► Approximation algorithms for many combinatorial problems in UDGs and quasi-UDGs.
► Simple algorithms based on the tile-and-combine strategy.
► Tight analysis of the local horizon: exactly how many rounds are sufficient.