On the Complexity of Distributed Greedy Coloring

Distributed Greedy Coloring is an interesting and intuitive variation of the standard Coloring problem. It still consists in coloring in a distributed setting each node of a given graph in such a way that two adjacent nodes do not get the same color, but it adds a further constraint. Given an order among the colors, a coloring is said to be greedy if there does not exist a node for which its associated color can be replaced by a color of lower position in this order without violating the coloring property. We provide lower and upper bounds for this problem in Linial’s model and we relate them to other well-known problems, namely Coloring, Maximal Independent Set (MIS), and Largest First Coloring. Whereas the best known upper bound for Coloring, MIS, and Greedy Coloring are the same, we prove a lower bound which is strong in the sense that it now makes a difference between Greedy Coloring and MIS. We discuss the vertex coloring problem in a distributed network. Such a network consists of a set V of processors and a set E of bidirectional communication links between pairs of processors. It can be modeled by an undirected graph G = (V,E). We denote n = |V |, m = |E| and for each vertex v define its neighborhood Nv = {u : {u, v} ∈ E} and vertex degree degG v = |Nv|. The set of neighbours of high degree is denoted by N≥ v = {u ∈ Nv : deg u ≥ deg v}. To color the vertices of G means to give each vertex a positive integer color value in such a way that no two adjacent vertices get the same color. If at most k colors are used, the result is called a k-coloring. In many practical considerations, such as code assignment in wireless networks [1], it is desirable to minimise the number of used colors. The smallest possible positive integer k for which there exists a k-coloring of G is called the chromatic number χ(G). This value is bounded from above by Δ + 1, where Δ denotes the maximal vertex degree of the graph; consequently, a graph always admits a (Δ+ 1)-coloring. The research was partially funded by the European projects COST Action 293, “Graphs and Algorithms in Communication Networks” (GRAAL) and, COST Action 295, “Dynamic Communication Networks” (DYNAMO). A. Pelc (Ed.): DISC 2007, LNCS 4731, pp. 482–484, 2007. c © Springer-Verlag Berlin Heidelberg 2007 On the Complexity of Distributed Greedy Coloring 483 The problems discussed in this paper can be formulated using a local definition: the goal is to achieve a state of the system in which the local state variables associated with each node fulfill certain constraints with respect to the local state variables of its neighbours. Definition 1. The problems are defined by the following constraints on the local variable c at vertex v: (Δ+ 1)-Coloring (COL): c(v) ∈ {1, . . . ,Δ+ 1} \ c(Nv) Greedy Coloring (G-COL): c(v) = min {1, . . . ,Δ+ 1} \ c(Nv) Largest-First Coloring (LF-COL): c(v) = min {1, . . . ,Δ+ 1} \ c(N≥ v ) Maximal Independent Set (MIS): c(v) = 0 ⇔ c(Nv) = {0} We provide lower and upper bounds on the deterministic distributed (Linial’s model) time complexity of Greedy Coloring (G-COL) with respect to Coloring (COL), Maximal Independent Set (MIS) and Largest First Coloring (LF-COL). A summary of the results is contained in Table 1, where (*) indicates the new results obtained in this paper. In particular, we derive new upper bounds for GCOL and LF-COL and a new lower bound for G-COL. Whereas the upper bounds for the COL, MIS, and G-COL are the same, we prove a strong lower bound in the sense that our lower bound now makes a difference between G-COL and MIS. Table 1. The time complexity of Greedy Coloring with respect to other well-known problems in the distributed setting. The table can be read also vertically as (Δ + 1)Coloring ≤ Maximal Independent Set ≤ Greedy Coloring ≤ LF-Coloring. Problem Lower Bound Upper Bound (Δ+ 1)-Coloring (COL) Ω(log∗ n) [2] 2 √ log n) [3,4] Maximal Independent Set (MIS) Ω (√ log n log log n ) [5] 2 √ log n) [3,4] O(Δ+ COL)