TY - GEN

T1 - Streaming Edge Coloring with Subquadratic Palette Size

AU - Chechik, Shiri

AU - Mukhtar, Doron

AU - Zhang, Tianyi

N1 - Publisher Copyright:
© Shiri Chechik, Doron Mukhtar, and Tianyi Zhang.

PY - 2024/7

Y1 - 2024/7

N2 - 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(∆2)-coloring w.h.p. (here ∆ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] showed that it is possible to deterministically compute an O(∆2/s)-coloring in O(ns) space. However, none of the improvements succeeded in reducing the number of colors to O(∆2−ϵ) while keeping the same space bound of Õ(n)1. In particular, no progress was made on the question of whether computing an O(∆)-coloring is possible with roughly O(n) space, which was stated in [Behnezhad et al., 2019] to be an interesting open problem. In this paper we bypass the quadratic bound by presenting a new randomized Õ(n)-space algorithm that uses Õ(∆1.5) colors.

AB - 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(∆2)-coloring w.h.p. (here ∆ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] showed that it is possible to deterministically compute an O(∆2/s)-coloring in O(ns) space. However, none of the improvements succeeded in reducing the number of colors to O(∆2−ϵ) while keeping the same space bound of Õ(n)1. In particular, no progress was made on the question of whether computing an O(∆)-coloring is possible with roughly O(n) space, which was stated in [Behnezhad et al., 2019] to be an interesting open problem. In this paper we bypass the quadratic bound by presenting a new randomized Õ(n)-space algorithm that uses Õ(∆1.5) colors.

KW - edge coloring

KW - graph algorithms

KW - streaming algorithms

UR - http://www.scopus.com/inward/record.url?scp=85198353297&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2024.40

DO - 10.4230/LIPIcs.ICALP.2024.40

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AN - SCOPUS:85198353297

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

A2 - Bringmann, Karl

A2 - Grohe, Martin

A2 - Puppis, Gabriele

A2 - Svensson, Ola

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

Y2 - 8 July 2024 through 12 July 2024

ER -