Lenzen’s Distributed Routing Generalized: A Full Characterization of Constant-Time Routability
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3618260.3649627.pdf
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Author(s)
Ghaffari, Mohsen
Wang, Brandon
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Ghaffari, Mohsen and Wang, Brandon. 2024. "Lenzen’s Distributed Routing Generalized: A Full Characterization of Constant-Time Routability."
Version
Final published version
Abstract
A celebrated and widely used result of Lenzen and Wattenhofer
[STOC’11, PODC’13] shows a constant-round (deterministic) distributed routing algorithm for the complete-graph network: if each
node is the source or destination of at most Θ() packets, there is
a constant-round deterministic distributed algorithm that routes
all packets to their destinations in a constant number of rounds, on
the complete-graph network.
We study generalizations of this result to arbitrary network
graphs and show a necessary and su cient condition for the network so that it can route any such demand in constant rounds
distributedly. One can easily see that just for the existence of a
constant-round routing for all such demands, it is necessary that
any cut’s size, when normalized by the number of possible edges
in that cut, should be lower bounded by a positive constant. That
is, for any partition of nodes with exactly ∈ [1, /2] nodes on
one side, the cut should have at least Θ() edges. We call this a
graph with a positive minimum normalized cut, or a positive graph
for short.
We show that this necessary condition is also su cient. In particular, by tightening the Leighton-Rao multicommodity max- ow
min-cut theorem for positive graphs, we show the existence of a
constant-round routing in positive graphs (assuming the network
graph is known globally). Then, as the main technical contribution
of this paper, we also show that there is a (deterministic) distributed
algorithm that computes such a constant-round routing in constant
rounds in these graphs. This result allows us to vastly relax the conditions of the well-studied congested clique model of distributed
computing: Any distributed algorithm for the congested clique
model can be run in any positive graph network, without any asymptotic slow-down.
Our results are in fact more general and they give a distributed
routing bound for any network, as a function of its minimum normalized cut size (and without assuming it is a constant), within a
polynomial of the relevant lower bound.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Terms of Use
Creative Commons Attribution-NoDerivs License
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DOI of Published Version
https://doi.org/10.1145/3618260.3649627
