Linial for lists
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Published:2022-05-17
Issue:6
Volume:35
Page:533-546
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ISSN:0178-2770
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Container-title:Distributed Computing
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language:en
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Short-container-title:Distrib. Comput.
Author:
Maus Yannic, Tonoyan TigranORCID
Abstract
AbstractLinial’s famous color reduction algorithm reduces a given m-coloring of a graph with maximum degree $$\varDelta $$
Δ
to an $$O(\varDelta ^2\log m)$$
O
(
Δ
2
log
m
)
-coloring, in a single round in the LOCAL model. We give a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m-coloring in a directed graph of maximum outdegree $$\beta $$
β
, if every node has a list of size $$\varOmega (\beta ^2 (\log \beta +\log \log m + \log \log |{\mathcal {C}}|))$$
Ω
(
β
2
(
log
β
+
log
log
m
+
log
log
|
C
|
)
)
from a color space $${\mathcal {C}}$$
C
then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial’s color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local$$({\text {deg}}+1)$$
(
deg
+
1
)
-list coloring algorithm from Barenboim et al. (PODC, pp 437–446, 2018) by slightly reducing the runtime to $$O(\sqrt{\varDelta \log \varDelta })+\log ^* n$$
O
(
Δ
log
Δ
)
+
log
∗
n
and significantly reducing the message size (from $$\varDelta ^{O(\log ^* \varDelta )}$$
Δ
O
(
log
∗
Δ
)
to roughly $$\varDelta $$
Δ
). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. (in: FOCS, pp 625–634, 2016).
Funder
Graz University of Technology
Publisher
Springer Science and Business Media LLC
Subject
Computational Theory and Mathematics,Computer Networks and Communications,Hardware and Architecture,Theoretical Computer Science
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