The best known solutions for kmessage broadcast in dynamic networks of size n require Ω(nk) rounds. In this paper, we see if these bounds can be improved by smoothed analysis. To do so, we study perhaps the most natural randomized algorithm for disseminating tokens in this setting: at every time step, choose a token to broadcast randomly from the set of tokens you know. We show that with even a small amount of smoothing (i.e., one random edge added per round), this natural strategy solves kmessage broadcast in Õ(n+k³) rounds, with high probability, beating the best known bounds for k = o(√n) and matching the Ω(n+k) lower bound for static networks for k = O(n^{1/3}) (ignoring logarithmic factors). In fact, the main result we show is even stronger and more general: given 𝓁smoothing (i.e., 𝓁 random edges added per round), this simple strategy terminates in O(kn^{2/3}log^{1/3}(n)𝓁^{1/3}) rounds. We then prove this analysis close to tight with an almostmatching lower bound. To better understand the impact of smoothing on information spreading, we next turn our attention to static networks, proving a tight bound of Õ(k√n) rounds to solve kmessage broadcast, which is better than what our strategy can achieve in the dynamic setting. This confirms the intuition that although smoothed analysis reduces the difficulties induced by changing graph structures, it does not eliminate them altogether. Finally, we apply tools developed to support our smoothed analysis to prove an optimal result for kmessage broadcast in socalled wellmixed networks in the absence of smoothing. By comparing this result to an existing lower bound for wellmixed networks, we establish a formal separation between oblivious and strongly adaptive adversaries with respect to wellmixed token spreading, partially resolving an open question on the impact of adversary strength on the kmessage broadcast problem.
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Distributed MST: A Smoothed Analysis
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., n¼) 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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 NSFPAR ID:
 10169205
 Date Published:
 Journal Name:
 ICDCN 2020: Proceedings of the 21st International Conference on Distributed Computing and Networking
 Page Range / eLocation ID:
 1 to 10
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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