This article shows how to construct an overlay network of constant degree and diameter \(O(\log n)\) in \(O(\log n)\) time starting from an arbitrary weakly connected graph. We assume a synchronous communication network in which nodes can send messages to nodes they know the identifier of, and new connections can be established by sending node identifiers. Suppose the initial network’s graph is weakly connected and has constant degree. In that case, our algorithm constructs the desired topology with each node sending and receiving only \(O(\log n)\) messages in each round in \(O(\log n)\) time w.h.p., which beats the currently best \(O(\log ^{3/2} n)\) time algorithm of Götte et al. (International colloquium on structural information and communication complexity (SIROCCO), Springer, 2019). Since the problem cannot be solved faster than by using pointer jumping for \(O(\log n)\) rounds (which would even require each node to communicate \(\Omega (n)\) bits), our algorithm is asymptotically optimal. We achieve this speedup by using short random walks to repeatedly establish random connections between the nodes that quickly reduce the conductance of the graph using an observation of Kwok and Lau (Approximation, randomization, and combinatorial optimization. Algorithms and techniques (APPROX/RANDOM 2014), Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014). Additionally, we show how our algorithm can be used to efficiently solve graph problems in hybrid networks (Augustine et al. in Proceedings of the fourteenth annual ACM-SIAM symposium on discrete algorithms, SIAM, 2020). Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the initial edges is unrestricted, whereas only polylogarithmically many messages can be sent over edges that have been established throughout an algorithm’s execution. For an (undirected) graph G with arbitrary degree, we show how to compute connected components, a spanning tree, and biconnected components in \(O(\log n)\) time w.h.p. Furthermore, we show how to compute an MIS in \(O(\log d + \log \log n)\) time w.h.p., where d is the initial degree of G.

本文展示了如何从任意弱连通图开始在\(O(\log n)\)时间内构建一个度数和直径为 \(O (\log n)\)的覆盖网络。我们假设一个同步通信网络，其中节点可以向它们知道其标识符的节点发送消息，并且可以通过发送节点标识符来建立新的连接。假设初始网络的图是弱连通的并且具有常数度。在那种情况下，我们的算法构建了所需的拓扑结构，每个节点在\ (O(\log n)\ ) 时间 whp 内的每一轮中仅发送和接收 \(O(\log n) \) 消息，这超过了当前最好的\(O(\log ^{3/2} n)\)Götte 等人的时间算法。（关于结构信息和通信复杂性的国际学术讨论会 (SIROCCO)，施普林格，2019 年）。由于问题不能比使用指针跳转\(O(\log n)\)轮更快地解决（这甚至需要每个节点通信\(\Omega (n)\)位），我们的算法是渐近最优的。我们通过使用短随机游走在节点之间重复建立随机连接来实现这种加速，这些节点使用 Kwok 和 Lau 的观察（近似、随机化和组合优化。算法和技术（APPROX/RANDOM 2014）快速降低图形的电导） , Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014)。此外，我们展示了我们的算法如何用于有效解决混合网络中的图问题（Augustine 等人在第十四届年度 ACM-SIAM 离散算法研讨会论文集，SIAM，2020 年）。受节点拥有两种不同通信模式这一想法的启发，我们假设初始通信edges 是不受限制的，而只有多对数数量的消息可以通过在整个算法执行过程中建立的边发送。对于具有任意度数的（无向）图G ，我们展示了如何在\(O(\log n)\)时间内计算连通分量、生成树和双连通分量whp 此外，我们展示了如何在\ (O(\log d + \log \log n)\)时间 whp，其中d是G的初始度。