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

Abstract

a b s t r a c tA local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r 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 r 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.

Survey of local algorithms

“…Algorithms from simple tilings. A simple approach for designing local algorithms in a geometric setting is to partition the two-dimensional plane into rectangles, and colour the rectangles with a constant number of colours [67,68,91,96,117,144,146,147]. Partitioning the two-dimensional plane into rectangles also partitions the network into clusters.…”

“…In Attiya et al's [10] terminology, the coloured rectangles provide a t-orientation of graph G for t = 3D. Now it is easy to design a local 3-approximation algorithm for vertex colouring [67,91,144,147]. We handle each connected component in each cluster independently in parallel.…”

“…A similar idea (with larger 3-coloured rectangles) can be used to design local 3-approximation algorithms for the following problems: edge colouring [67], vertex cover [67,68,144], and dominating set [67,68,144]. These are examples of local algorithms that unscrupulously exploit the assumption that local computation is free; nevertheless, Hunt et al [74] show how to solve the subproblem of finding a minimum-size dominating set or vertex cover within a rectangle in polynomial time.…”

Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

“…Other work on constant-time distributed algorithms for matching usually assumes either randomness [14,20,21,27,34] or geometric information [13,35]. We refer to the survey [31] for further information on local algorithms.…”

Dynamic energy aware aggregate Convergecast in a wireless sensor network

“…Collision is more to individual nodes but the effect is for the whole network. We looked at collision as a local problem in a tile and in a node's neighborhood (transmission range) [21][22][23].…”