Becker, F., Montealegre, P., Rapaport, I., & Todinca, I. (2020). The Impact Of Locality In The Broadcast Congested Clique Model. SIAM Discret. Math., 34(1), 682–700.
Abstract: The broadcast congested clique model (BCLIQUE) is a message-passing model of distributed computation where n nodes communicate with each other in synchronous rounds. First, in this paper we prove that there is a one-round, deterministic algorithm that reconstructs the input graph G if the graph is d-degenerate, and rejects otherwise, using bandwidth b = O(d . log n). Then, we introduce a new parameter to the model. We study the situation where the nodes, initially, instead of knowing their immediate neighbors, know their neighborhood up to a fixed radius r. In this new framework, denoted BCLIQuE[r], we study the problem of detecting, in G, an induced cycle of length at most k (CYCLE <= k) and the problem of detecting an induced cycle of length at least k +1 (CYCLE>k). We give upper and lower bounds. We show that if each node is allowed to see up to distance r = left perpendicular k/2 right perpendicular + 1, then a polylogarithmic bandwidth is sufficient for solving CYCLE>k with only two rounds. Nevertheless, if nodes were allowed to see up to distance r = left perpendicular k/3 right perpendicular, then any one-round algorithm that solves CYCLE>k needs the bandwidth b to be at least Omega(n/ log n). We also show the existence of a one-round, deterministic BCLIQUE algorithm that solves CYCLE <= k with bandwitdh b = O(n(1/left perpendicular k/2 right perpendicular). log n). On the negative side, we prove that, if epsilon <= 1/3 and 0 < r <= k/4, then any epsilon-error, R-round, b-bandwidth algorithm in the BCLIQUE[r] model that solves problem CYCLE(<= k )satisfies R . b = Omega(n(1/left perpendicular k/2 right perpendicular)).
|
Becker, F., Montealegre, P., Rapaport, I., & Todinca, I. (2021). The role of randomness in the broadcast congested clique model. Inf. Comput., 281, 104669.
Abstract: We study the role of randomness in the broadcast congested clique model. This is a message-passing model of distributed computation where the nodes of a network know their local neighborhoods and they broadcast, in synchronous rounds, messages that are visible to every other node.
This works aims to separate three different settings: deterministic protocols, randomized protocols with private coins, and randomized protocols with public coins. We obtain the following results:
If more than one round is allowed, public randomness is as powerful as private ran-domness.
One-round public-coin algorithms can be exponentially more powerful than determin-istic algorithms running in several rounds.
One-round public-coin algorithms can be exponentially more powerful than one-round private-coin algorithms.
One-round private-coin algorithms can be exponentially more powerful than one-round deterministic algorithms.
|
Montealegre, P., Perez-Salazar, S., Rapaport, I., & Todinca, I. (2020). Graph reconstruction in the congested clique. J. Comput. Syst. Sci., 113, 1–17.
Abstract: In this paper we study the reconstruction problem in the congested clique model. Given a class of graphs g, the problem is defined as follows: if G is not an element of g, then every node must reject; if G is an element of g, then every node must end up knowing all the edges of G. The cost of an algorithm is the total number of bits received by any node through one link. It is not difficult to see that the cost of any algorithm that solves this problem is Omega(log vertical bar g(n)vertical bar/n), where g(n) is the subclass of all n-node labeled graphs in g. We prove that the lower bound is tight and that it is possible to achieve it with only 2 rounds. (C) 2020 Elsevier Inc. All rights reserved.
|
