Channel Assignment with Separation for Interference Avoidance in Wireless Networks

Given an integer\sigma 1 , a vector(\delta_1, \delta_2, \ldots, \delta_{\sigma-1})of nonnegative integers, and an undirected graphG=(V,E) , anL(\delta_1, \delta_2, \ldots,\delta_{\sigma-1}){\hbox{-}}\rm coloringofGis a functionffrom the vertex setVto a set of nonnegative integers such that| f(u) -f(v) | \ge \delta_i , ifd(u,v) = i, \ 1 \le i \le \sigma-1 , whered(u,v)is the distance (i.e., the minimum number of edges) between the verticesuandv . An optimalL(\delta_1, \delta_2, \ldots,\delta_{\sigma-1}){\hbox{-}}\rm coloringforGis one using the smallest range\lambdaof 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 distanceimust be at least\delta_iapart, while the same channel can be reused in vertices whose distance is at least\sigma . In particular, two versions of the coloring problem驴 L(2,1,1)andL(\delta_1, 1, \ldots,1) 驴are considered. Since these versions of the problem areNP{\hbox{-}}\rm hardfor general graphs, efficient algorithms for finding optimal colorings are provided for specific graphs modeling realistic wireless networks, including rings, bidimensional grids, and cellular grids.