Coloring in Graph Streams

Abstract: In this paper, we initiate the study of the vertex coloring problem of a graph in the semi streaming model. In this model, the input graph is defined by a stream of edges, arriving in adversarial order and any algorithm must process the edges in the order of arrival using space linear (up to polylogarithmic factors) in the number of vertices of the graph. In the offline settings, there is a simple greedy algorithm for $(\Delta+1)$-vertex coloring of a graph with maximum degree $\Delta$. We design a one pass randomized streaming algorithm for $(1+\varepsilon)\Delta$-vertex coloring problem for any constant $\varepsilon >0$ using $O(\varepsilon^{-1} n ~\mathrm{ poly} \log n)$ space where $n$ is the number of vertices in the graph. Much more color efficient algorithms are known for graphs with bounded arboricity in the offline settings. Specifically, there is a simple $2\alpha$-vertex coloring algorithm for a graph with arboricity $\alpha$. We present a $O(\varepsilon^{-1}\log n)$ pass randomized vertex coloring algorithm that requires at most $(2+\varepsilon)\alpha$ many colors for any constant $\varepsilon>0$ for a graph with arboricity $\alpha$ in the semi streaming model.