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Abstract-Given an integer ' > 1, a vector ð 1 ; 2 ; . . . ; 'À1 Þ of nonnegative integers, and an undirected graph G ¼ ðV ; EÞ, an Lð 1 ; 2 ; . . . ; 'À1 Þ-coloring of G is a function f from the vertex set V to a set of nonnegative integers such that jfðuÞ À fðvÞj ! i , if dðu; vÞ ¼ i; 1 i ' À 1, where dðu; vÞ is the distance (i.e., the minimum number of edges) between the vertices u and v. An optimal Lð 1 ; 2 ; . . . ; 'À1 Þ-coloring for G is one using the smallest range ! of integers over all such colorings. This problem has relevant application in channel assignment for interference avoidance in wireless networks, where channels (i.e., colors) assigned to interfering stations (i.e., vertices) at distance i must be at least i apart, while the same channel can be reused in vertices whose distance is at least '. In particular, two versions of the coloring problem-Lð2; 1; 1Þ and Lð 1 ; 1; . . . ; 1Þ-are considered. Since these versions of the problem are NP -hard for general graphs, efficient algorithms for finding optimal colorings are provided for specific graphs modeling realistic wireless networks, including rings, bidimensional grids, and cellular grids.

In this work we propose simple and efficient protocols for counting and leader election in mobile networks. For mobile networks with fixed base stations we provide a new and very efficient protocol for counting the number of mobile hosts. The main part of the work concentrates on ad-hoc networks (no fixed subnetwork).We provide a model for these networks and leader election (and a special form of counting) protocols for both named and anonymous mobile hosts. In this work we define two protocol classes, the Non-Compulsoryprotocols, which do not affect the motion of the hosts and the Compulsory, which determine the motion of some or all the hosts. By assuming that the mobile hosts move as if each one is doing a continuous random walk on their allowable space S of motions, and by assuming a universal time, we show that our leader election protocol terminates (with high probability and also on the average) in time asymptotically linear to the size of the space S, measured as its volume divided by the volume of the sphere defined by the range of transmission of each mobile host. We also provide a simple but very efficient Compulsory (forced random walks) Las Vegas protocol for leader election in ad-hoc networks, which also allows counting, with termination detection.Our analysis techniques for the meeting time of concurrent random walks extend the known facts and are tight. They may be used as an analysis tool in the design of many other distributed protocols. This is the first algorithmic and characterization work, to our knowledge, for ad-hoc networks.

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