A @math -ruling set of a graph @math is a vertex-subset @math that is independent and satisfies the property that every vertex @math is at a distance of at most @math from some vertex in @math . A is a 1-ruling set. The problem of computing an MIS on a network is a fundamental problem in distributed algorithms and the fastest algorithm for this problem is the @math -round algorithm due to Luby (SICOMP 1986) and (J. Algorithms 1986) from more than 25 years ago. Since then the problem has resisted all efforts to yield to a sub-logarithmic algorithm. There has been recent progress on this problem, most importantly an @math -round algorithm on graphs with @math vertices and maximum degree @math , due to (Barenboim, Elkin, Pettie, and Schneider, April 2012, arxiv 1202.1983; to appear FOCS 2012). We approach the MIS problem from a different angle and ask if O(1)-ruling sets can be computed much more efficiently than an MIS? As an answer to this question, we show how to compute a 2-ruling set of an @math -vertex graph in @math rounds. We also show that the above result can be improved for special classes of graphs such as graphs with high girth, trees, and graphs of bounded arboricity. Our main technique involves randomized sparsification that rapidly reduces the graph degree while ensuring that every deleted vertex is close to some vertex that remains. This technique may have further applications in other contexts, e.g., in designing sub-logarithmic distributed approximation algorithms. Our results raise intriguing questions about how quickly an MIS (or 1-ruling sets) can be computed, given that 2-ruling sets can be computed in sub-logarithmic rounds.
The MIS problem on the class of growth-bounded graphs has attracted fair bit of attention @cite_11 @cite_5 @cite_14 . Growth-bounded graphs have the property that the @math -neighborhood of any vertex @math has at most @math independent vertices in it, for some constant @math . In other words, the rate of the growth of independent sets is polynomial in the radius of the ball'' around a vertex. Schneider and Wattenhofer @cite_14 showed that there is a deterministic MIS algorithm on growth-bounded graphs that runs in @math rounds. Growth-bounded graphs have been used to model wireless networks because the number of independent vertices in any spatial region is usually bounded by the area or volume of that region. In contrast to growth-bounded graphs, the graph subclasses we consider in this paper tend to have arbitrarily many independent vertices in any neighborhood.
@cite_5 @cite_14 @cite_11
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We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graph-theoretic structure, called Nash-Williams forests-decomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearly-tight lower bounds on the running time of any distributed algorithm for computing a forests-decomposition. We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size @math that halts in time @math with probability @math , each message containing @math bit: thus its bit complexity per channel is @math (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) for general graphs which is @math per channel (it halts in time @math and the size of each message is @math ). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices. We provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby's algorithm presented by Lynch and by Wattenhofer. This question remains open for the variant given by Peleg. Abstract We show that the maximum independent set problem on an n-vertex graph can be solved in 1.1996 n n O ( 1 ) time and polynomial space, which even is faster than Robson's 1.2109 n n O ( 1 ) -time exponential-space algorithm published in 1986. We also obtain improved algorithms for MIS in graphs with maximum degree 6 and 7, which run in time of 1.1893 n n O ( 1 ) and 1.1970 n n O ( 1 ) , respectively. Our algorithms are obtained by using fast algorithms for MIS in low-degree graphs in a hierarchical way and making a careful analysis on the structure of bounded-degree graphs. The efficient distributed construction of a maximal independent set (MIS) of a graph is of fundamental importance. We study the problem in the class of Growth-Bounded Graphs, which includes for example the well-known Unit Disk Graphs. In contrast to the fastest (time-optimal) existing approach [11], we assume that no geometric information (e.g., distances in the graph's embedding) is given. Instead, nodes employ randomization for their decisions. Our algorithm computes a MIS in O(log log n • log* n) rounds with very high probability for graphs with bounded growth, where n denotes the number of nodes in the graph. In view of Linial's Ω(log* n) lower bound for computing a MIS in ring networks [12], which was extended to randomized algorithms independently by Naor [18] and Linial [13], our solution is close to optimal. In a nutshell, our algorithm shows that for computing a MIS, randomization is a viable alternative to distance information.