We consider networks of processes which interact with beeps. Various beeping models are used. The basic one, defined by Cornejo and Kuhn [CK10], assumes that a process can choose either to beep or to listen; if it listens it can distinguish between silence or the presence of at least one beep. The aim of this paper is the study of the resolution of paradigms such as collision detection, computation of the degree of a vertex, colouring, or 2-hop-colouring in the framework of beeping models. For each of these problems we present Las Vegas or Monte Carlo algorithms and we analyse their complexities expressed in terms of the number of slots. We present also efficient randomised emulations of more powerful beeping models on the basic one. We illustrate emulation procedures with an efficient degree computation algorithm in the basic beeping model; this algorithm was given initially in a more powerful model.
In @cite_6 , Cornejo and Kuhn study the interval colouring problem: an interval colouring assigns to each vertex an interval (contiguous fraction) of resources such that neighbouring vertices do not share resources (it is a variant of vertex colouring). They assume that each node knows its degree and an upper bound of the maximum degree @math of the graph. They present in the beeping model @math a probabilistic algorithm which never stops and stabilises with a correct @math -interval coloring in @math periods w.h.p., where: @math is the size of the graph, and a period is @math time slots with @math , thus it stabilises in @math slots.
@cite_6
2951682038 2179569280 2062111496 2105546154
We present the communication model, which assumes nodes have minimal knowledge about their environment and severely limited communication capabilities. Specifically, nodes have no information regarding the local or global structure of the network, don't have access to synchronized clocks and are woken up by an adversary. Moreover, instead on communicating through messages they rely solely on carrier sensing to exchange information. We study the problem of , a variant of vertex coloring specially suited for the studied beeping model. Given a set of resources, the goal of interval coloring is to assign every node a large contiguous fraction of the resources, such that neighboring nodes share no resources. To highlight the importance of the discreteness of the model, we contrast it against a continuous variant described in [17]. We present an O(1 @math (T ) @math T @math O( n) @math (T ) @math O( n) @math ( n)$ on the time required to solve interval coloring for this model against randomized algorithms. This lower bound implies that our algorithm is asymptotically optimal for constant degree graphs. In this paper, we study a colouring problem motivated by a practical frequency assignment problem and, up to our best knowledge, new. In wireless networks, a node interferes with other nodes, the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph (G,w) where the weight function w on the edges of G represents the noise (interference) between the two end-vertices. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted t-improper k-colouring of (G,w) is a k-colouring of the nodes of G (assignment of k frequencies) such that the interference at each node does not exceed the threshold t. We consider here the Weighted Improper Colouring problem which consists in determining the weighted t-improper chromatic number defined as the minimum integer k such that (G,w) admits a weighted t-improper k-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where, given a number k of colours, we want to determine the minimum real t such that (G,w) admits a weighted t-improper k-colouring. We first present general upper bounds for both problems; in particular we show a generalisation of Lovasz@?s Theorem for the weighted t-improper chromatic number. Motivated by the original application, we then study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1 2 for the nodes at distance two. We derive the weighted t-improper chromatic number for all values of t. We present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time O(logn) with probability 1-o(n^-^1). Each message containing 1 bit, its bit complexity per channel is O(logn). From this algorithm, we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree @D and size n that uses at most @D+1 colours and halts in time O(logn) with probability 1-o(n^-^1). We also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time O(logn) with probability 1-o(n^-^1). We study some parameters such as the number, the size and the radius of trees of the spanning forest. We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available.