Streaming Edge Coloring with Subquadratic Palette Size
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In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an n-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors. Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in Õ(n) space a proper O(Δ^2)-coloring w.h.p. (here Δ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] shows that it is possible to deterministically compute an O(Δ^2/s)-coloring in O(ns) space. However, none of the improvements, succeeded in reducing the number of colors to O(Δ^2-ϵ) while keeping the same space bound of Õ(n). In particular, no progress was made on the question of whether computing an O(Δ)-coloring is possible with roughly O(n) space, which was stated in [Behnezhad et al., 2019] to be a major open problem. In this paper we bypass the quadratic bound by presenting a new randomized Õ(n)-space algorithm that uses Õ(Δ^1.5) colors.
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