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arXiv 2002.05376cs.DCcs.DS
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分布式图实现

Distributed Graph Realizations

John Augustine, Keerti Choudhary, Avi Cohen, David Peleg, Sumathi Sivasubramaniam, Suman Sourav

摘要:

我们研究图实现问题从分布式的角度来看,我们研究它的节点容量限制集团(NCC)模型的分布式计算,最近推出的对等网络。我们专注于两个中心变量,度序列实现和最小阈值连接实现这两个结果在覆盖网络实现。覆盖网络实现可以是显式的或隐式的。显式实现要求实现图中任何边的两个端点都知道该边。另一方面,在隐式实现中,实现图的每条边的至少一个端点需要知道该边。我们提出的主要实现算法如下。1.度序列隐式实现的$\tilde{O}(\min\{\sqrt{m},Δ\})$时间算法。这里,$Δ= \max_v d(v)$是最大度,而$m = (1/2) \sum_v d(v)$是最终实现中的边的数量。度序列显式实现的$\tilde{O}(Δ)$时间算法。我们首先计算一个隐式实现,然后在$\tilde{O}(Δ)$附加轮中将其转换为显式实现。2.给出了一种求解门限连通性问题的$\tilde{O}(Δ)$时间算法和一种改进的$\tilde{O}(1)$算法,该算法在所有节点都知道彼此ID的情况下可得到显式解和隐式实现.这些算法是2-近似w.r.t.边缘的数量。我们补充我们的上界与下界,以表明上述算法是紧的$\log n$的因素。此外,我们提供了实现树的算法和近似度序列实现的$\tilde{O}(1)$轮算法。

英文摘要:

We study graph realization problems from a distributed perspective and we study it in the node capacitated clique (NCC) model of distributed computing, recently introduced for representing peer-to-peer networks. We focus on two central variants, degree-sequence realization and minimum threshold-connectivity realization both of which result in overlay network realizations. Overlay network realizations can be either explicit or implicit. Explicit realizations require both endpoints of any edge in the realized graph to be aware of the edge. In implicit realizations, on the other hand, at least one endpoint of each edge of the realized graph needs to be aware of the edge. The main realization algorithms we present are the following. 1. An $\tilde{O}(\min\{\sqrt{m},Δ\})$ time algorithm for implicit realization of a degree sequence. Here, $Δ= \max_v d(v)$ is the maximum degree and $m = (1/2) \sum_v d(v)$ is the number of edges in the final realization. An $\tilde{O}(Δ)$ time algorithm for an explicit realization of a degree sequence. We first compute an implicit realization and then transform it into an explicit one in $\tilde{O}(Δ)$ additional rounds. 2. An $\tilde{O}(Δ)$ time algorithm for the threshold connectivity problem that obtains an explicit solution and an improved $\tilde{O}(1)$ algorithm for implicit realization when all nodes know each other's IDs. These algorithms are 2-approximations w.r.t. the number of edges. We complement our upper bounds with lower bounds to show that the above algorithms are tight up to factors of $\log n$. Additionally, we provide algorithms for realizing trees and an $\tilde{O}(1)$ round algorithm for approximate degree sequence realization.

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