We show how to construct an overlay network of constant degree and diameter $O(\log n)$ in time $O(\log n)$ 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. If the initial network's graph is weakly connected and has constant degree, then our algorithm constructs the desired topology with each node sending and receiving only $O(\log n)$ messages in each round in time $O(\log n)$, w.h.p., which beats the currently best $O(\log^{3/2} n)$ time algorithm of [G\"otte et al., SIROCCO'19]. 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, APPROX'14]. Additionally, we show how our algorithm can be used to efficiently solve graph problems in \emph{hybrid networks} [Augustine et al., SODA'20]. Motivated by the idea that nodes possess two different modes of communication, we assume that communication of the \emph{initial} edges is unrestricted, whereas only polylogarithmically many messages can be communicated 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 time $O(\log n)$, w.h.p. Furthermore, we show how to compute an MIS in time $O(\log d + \log \log n)$, w.h.p., where $d$ is the initial degree of $G$.

翻译：我们展示如何构建一个以恒定度和直径为no( log n) 的覆盖网络, 时间为$O( log n) 美元, 时间为$O( log n) 美元。 我们假设一个同步的通信网络, 节点可以将信息发送到节点中, 并可以通过发送节点标识建立新的连接。 如果初始网络的图形连接不力, 并且具有恒定度, 那么我们的算法会构建理想的表层, 每个节点发送和接收的只有$O( log n) 美元, 时间为$( log n) 美元, w. h. p. 美元, 这比当前最佳的 O( log log n) 通信网络同步化 。 我们通过快速的 O( log ) 14 14 美元, 时间算得比 点跳过 。