We consider the distributed and parallel construction of low-diameter decompositions with strong diameter. We present algorithms for arbitrary undirected, weighted graphs and also for undirected, weighted graphs that can be separated through k ∈ Õ(1) shortest paths. This class of graphs includes planar graphs, graphs of bounded treewidth, and graphs that exclude a fixed minor K_r. Our algorithms work in the PRAM, CONGEST, and the novel HYBRID communication model and are competitive in all relevant parameters. Given 𝒟 > 0, our low-diameter decomposition algorithm divides the graph into connected clusters of strong diameter 𝒟. For an arbitrary graph, an edge e ∈ E of length 𝓁_e is cut between two clusters with probability O(𝓁_e⋅log(n)/𝒟). If the graph can be separated by k ∈ Õ(1) paths, the probability improves to O(𝓁_e⋅log(log n)/𝒟). In either case, the decompositions can be computed in Õ(1) depth and Õ(m) work in the PRAM and Õ(1) time in the HYBRID model. In CONGEST, the runtimes are Õ(HD + √n) and Õ(HD) respectively. All these results hold w.h.p. Broadly speaking, we present distributed and parallel implementations of sequential divide-and-conquer algorithms where we replace exact shortest paths with approximate shortest paths. In contrast to exact paths, these can be efficiently computed in the distributed and parallel setting [STOC '22]. Further, and perhaps more importantly, we show that instead of explicitly computing vertex-separators to enable efficient parallelization of these algorithms, it suffices to sample a few random paths of bounded length and the nodes close to them. Thereby, we do not require complex embeddings whose implementation is unknown in the distributed and parallel setting.

We consider the problem of computing routing schemes in the HYBRID model of distributed computing where nodes have access to two fundamentally different communication modes. In this problem nodes have to compute small labels and routing tables that allow for efficient routing of messages in the local network, which typically offers the majority of the throughput. Recent work has shown that using the HYBRID model admits a significant speed-up compared to what would be possible if either communication mode were used in isolation. Nonetheless, if general graphs are used as the input graph the computation of routing schemes still takes polynomial rounds in the HYBRID model. We bypass this lower bound by restricting the local graph to unit-disc-graphs and solve the problem deterministically with running time O(|H|2+log⁡n), label size O(log⁡n), and size of routing tables O(|H|2⋅log⁡n) where |H| is the number of “radio holes” in the network. Our work builds on recent work by Coy et al., who obtain this result in the much simpler setting where the input graph has no radio holes. We develop new techniques to achieve this, including a decomposition of the local graph into path-convex regions, where each region contains a shortest path for any pair of nodes in it.

We consider the problem of computing a compact routing scheme for a weighted undirected planar graph G := (V, E, w) in several models. For a given parameter ϵ > 0, we compute a routing scheme with stretch 1 + ϵ and labels and routing tables of size Õ(ϵ−1). In CONGEST, the construction takes Õ(ϵ−3 · HD) time, where HD denotes the network's hop-diameter. Further, it takes Õ(ϵ−3) time in a PRAM with O(n) processors and the novel HYBRID model. Thus, our algorithms are almost optimal in all relevant parameters. To achieve these results, we extend the divide-and-conquer framework of Li and Parter [STOC '19] and combine it with state-of-the-art distributed distance approximation algorithms [STOC '22].

Abstract This article shows how to construct an overlay network of constant degree and diameter $$O(\log n)$$ O ( log n ) in $$O(\log n)$$ 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 $$O(\log n)$$ O ( log n ) messages in each round in $$O(\log n)$$ O ( log n ) time w.h.p., which beats the currently best $$O(\log ^{3/2} n)$$ 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 $$O(\log n)$$ O ( log n ) rounds (which would even require each node to communicate $$\Omega (n)$$ Ω ( 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 $$O(\log n)$$ O ( log n ) time w.h.p. Furthermore, we show how to compute an MIS in $$O(\log d + \log \log n)$$ O ( log d + log log n ) time w.h.p., where d is the initial degree of G .

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 establish new connections 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(log3/2 n) time algorithm of [Götte 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 Ω(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].