We study smoothed analysis of distributed graph algorithms, focusing on the fundamental minimum spanning tree (MST) problem. With the goal of studying the time complexity of distributed MST as a function of the "perturbation" of the input graph, we posit a smoothing model that is parameterized by a smoothing parameter 0 ≤ ϵ(n) ≤ 1 which controls the amount of random edges that can be added to an input graph G per round. Informally, ϵ(n) is the probability (typically a small function of n, e.g., n--¼) that a random edge can be added to a node per round. The added random edges, once they are added, can be used (only) for communication.
We show upper and lower bounds on the time complexity of distributed MST in the above smoothing model. We present a distributed algorithm that, with high probability, 1 computes an MST and runs in Õ(min{1/√ϵ(n)2O(√log n), D+ √n}) rounds2 where ϵ is the smoothing parameter, D is the network diameter and n is the network size.
To complement our upper bound, we also show a lower bound of Ω(min{1/√ϵ(n), D + √n}). We note that the upper and lower bounds essentially match except for a multiplicative 2O(√log n) polylog(n) factor.
Our work can be considered as a first step in understanding the smoothed complexity of distributed graph algorithms.
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Universally-optimal distributed algorithms for known topologies
Many distributed optimization algorithms achieve existentially-optimal running times, meaning that there exists some pathological worst-case topology on which no algorithm can do better. Still, most networks of interest allow for exponentially faster algorithms. This motivates two questions:
(i) What network topology parameters determine the complexity of distributed optimization?
(ii) Are there universally-optimal algorithms that are as fast as possible on every topology?
We resolve these 25-year-old open problems in the known-topology setting (i.e., supported CONGEST) for a wide class of global network optimization problems including MST, (1+є)-min cut, various approximate shortest paths problems, sub-graph connectivity, etc.
In particular, we provide several (equivalent) graph parameters and show they are tight universal lower bounds for the above problems, fully characterizing their inherent complexity. Our results also imply that algorithms based on the low-congestion shortcut framework match the above lower bound, making them universally optimal if shortcuts are efficiently approximable.
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- NSF-PAR ID:
- 10271616
- Date Published:
- Journal Name:
- Symposium on Theory of Computing (STOC)
- Page Range / eLocation ID:
- 1166 to 1179
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3406325.3451081
