Faster Distributed Shortest Path Approximations via Shortcuts

A long series of recent results and breakthroughs have led to faster and better distributed approximation algorithms for single source shortest paths (SSSP) and related problems in the CONGEST model. The runtime of all these algorithms, however, is Omega~(sqrt{n}), regardless of the network topology, even on nice networks with a (poly)logarithmic network diameter D. While this is known to be necessary for some pathological networks, most topologies of interest are arguably not of this type. We give the first distributed approximation algorithms for shortest paths problems that adjust to the topology they are run on, thus achieving significantly faster running times on many topologies of interest. The running time of our algorithms depends on and is close to Q, where Q is the quality of the best shortcut that exists for the given topology. While Q = Theta~(sqrt{n} + D) for pathological worst-case topologies, many topologies of interest have Q = Theta~(D), which results in near instance optimal running times for our algorithm, given the trivial Omega(D) lower bound. The problems we consider are as follows: - an approximate shortest path tree and SSSP distances, - a polylogarithmic size distance label for every node such that from the labels of any two nodes alone one can determine their distance (approximately), and - an (approximately) optimal flow for the transshipment problem. Our algorithms have a tunable tradeoff between running time and approximation ratio. Our fastest algorithms have an arbitrarily good polynomial approximation guarantee and an essentially optimal O~(Q) running time. On the other end of the spectrum, we achieve polylogarithmic approximations in O~(Q * n^epsilon) rounds for any epsilon > 0. It seems likely that eventually, our non-trivial approximation algorithms for the SSSP tree and transshipment problem can be bootstrapped to give fast Q * 2^O(sqrt{log n log log n}) round (1+epsilon)-approximation algorithms using a recent result by Becker et al

A Faster Distributed Single-Source Shortest Paths Algorithm

We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $\tilde \Omega (\sqrt{n} + D)$ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $n$ is the number of nodes in the network and $D$ is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs $\tilde O(n^{2/3} D^{1/3} + n^{5/6})$ rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing $\tilde O (\sqrt{n D})$ rounds and the second performing $\tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D)$ rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a $(1 + \epsilon)$-approximation $\tilde O ((\sqrt{n} D^{1/4} + D) / \epsilon)$-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018

Dynamic Low-Stretch Trees via Dynamic Low-Diameter Decompositions

Spanning trees of low average stretch on the non-tree edges, as introduced by Alon et al. [SICOMP 1995], are a natural graph-theoretic object. In recent years, they have found significant applications in solvers for symmetric diagonally dominant (SDD) linear systems. In this work, we provide the first dynamic algorithm for maintaining such trees under edge insertions and deletions to the input graph. Our algorithm has update time $n^{1/2 + o(1)}$ and the average stretch of the maintained tree is $n^{o(1)}$, which matches the stretch in the seminal result of Alon et al. Similar to Alon et al., our dynamic low-stretch tree algorithm employs a dynamic hierarchy of low-diameter decompositions (LDDs). As a major building block we use a dynamic LDD that we obtain by adapting the random-shift clustering of Miller et al. [SPAA 2013] to the dynamic setting. The major technical challenge in our approach is to control the propagation of updates within our hierarchy of LDDs: each update to one level of the hierarchy could potentially induce several insertions and deletions to the next level of the hierarchy. We achieve this goal by a sophisticated amortization approach. We believe that the dynamic random-shift clustering might be useful for independent applications. One of these applications is the dynamic spanner problem. By combining the random-shift clustering with the recent spanner construction of Elkin and Neiman [SODA 2017]. We obtain a fully dynamic algorithm for maintaining a spanner of stretch $2k - 1$ and size $O (n^{1 + 1/k} \log{n})$ with amortized update time $O (k \log^2 n)$ for any integer $2 \leq k \leq \log n$. Compared to the state-of-the art in this regime [Baswana et al. TALG '12], we improve upon the size of the spanner and the update time by a factor of $k$.Comment: To be presented at the 51st Annual ACM Symposium on the Theory of Computing (STOC 2019); abstract shortened to respect the arXiv limit of 1920 character