Isometric Embeddings in Trees and Their Use in Distance Problems

International audienceWe present powerful techniques for computing the diameter, all the eccentricities, and other related distance problems on some geometric graph classes, by exploiting their "tree-likeness" properties. We illustrate the usefulness of our approach as follows: (1) We propose a subquadratic-time algorithm for computing all eccentricities on partial cubes of bounded lattice dimension and isometric dimension O(n^{0.5−ε}). This is one of the first positive results achieved for the diameter problem on a subclass of partial cubes beyond median graphs. (2) Then, we obtain almost linear-time algorithms for computing all eccentricities in some classes of face-regular plane graphs, including benzenoid systems, with applications to chemistry. Previously, only a linear-time algorithm for computing the diameter and the center was known (and an O(n^{5/3})-time algorithm for computing all the eccentricities). (3) We also present an almost linear-time algorithm for computing the eccentricities in a polygon graph with an additive one-sided error of at most 2. (4) Finally, on any cube-free median graph, we can compute its absolute center in almost linear time. Independently from this work, Bergé and Habib have recently presented a linear-time algorithm for computing all eccentricities in this graph class (LAGOS'21), which also implies a linear-time algorithm for the absolute center problem. Our strategy here consists in exploiting the existence of some embeddings of these graphs in either a system or a product of trees, or in a single tree but where each vertex of the graph is embedded in a subset of nodes. While this may look like a natural idea, the way it can be done efficiently, which is our main technical contribution in the paper, is surprisingly intricate

The Graph Lottery Ticket Hypothesis: Finding Sparse, Informative Graph Structure

Graph learning methods help utilize implicit relationships among data items, thereby reducing training label requirements and improving task performance. However, determining the optimal graph structure for a particular learning task remains a challenging research problem. In this work, we introduce the Graph Lottery Ticket (GLT) Hypothesis - that there is an extremely sparse backbone for every graph, and that graph learning algorithms attain comparable performance when trained on that subgraph as on the full graph. We identify and systematically study 8 key metrics of interest that directly influence the performance of graph learning algorithms. Subsequently, we define the notion of a "winning ticket" for graph structure - an extremely sparse subset of edges that can deliver a robust approximation of the entire graph's performance. We propose a straightforward and efficient algorithm for finding these GLTs in arbitrary graphs. Empirically, we observe that performance of different graph learning algorithms can be matched or even exceeded on graphs with the average degree as low as 5

Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

In this paper we provide an $O(m (\log \log n)^{O(1)} \log(1/\epsilon))$-expected time algorithm for solving Laplacian systems on $n$-node $m$-edge graphs, improving improving upon the previous best expected runtime of $O(m \sqrt{\log n} (\log \log n)^{O(1)} \log(1/\epsilon))$ achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of $\ell_p$-stretch graph approximations with improved stretch and sparsity bounds. Additionally, as motivation for this work, we show that for every set of vectors in $\mathbb{R}^d$ (not just those induced by graphs) and all $k > 0$ there exist an ultra-sparsifiers with $d-1 + O(d/k)$ re-weighted vectors of relative condition number at most $k^2$. For small $k$, this improves upon the previous best known multiplicative factor of $k \cdot \tilde{O}(\log d)$, which is only known for the graph case.Comment: 52 pages, comments welcome

Stochastic Analysis of a Churn-Tolerant Structured Peer-to-Peer Scheme

We present and analyze a simple and general scheme to build a churn (fault)-tolerant structured Peer-to-Peer (P2P) network. Our scheme shows how to "convert" a static network into a dynamic distributed hash table(DHT)-based P2P network such that all the good properties of the static network are guaranteed with high probability (w.h.p). Applying our scheme to a cube-connected cycles network, for example, yields a $O(\log N)$ degree connected network, in which every search succeeds in $O(\log N)$ hops w.h.p., using $O(\log N)$ messages, where $N$ is the expected stable network size. Our scheme has an constant storage overhead (the number of nodes responsible for servicing a data item) and an $O(\log N)$ overhead (messages and time) per insertion and essentially no overhead for deletions. All these bounds are essentially optimal. While DHT schemes with similar guarantees are already known in the literature, this work is new in the following aspects: (1) It presents a rigorous mathematical analysis of the scheme under a general stochastic model of churn and shows the above guarantees; (2) The theoretical analysis is complemented by a simulation-based analysis that validates the asymptotic bounds even in moderately sized networks and also studies performance under changing stable network size; (3) The presented scheme seems especially suitable for maintaining dynamic structures under churn efficiently. In particular, we show that a spanning tree of low diameter can be efficiently maintained in constant time and logarithmic number of messages per insertion or deletion w.h.p. Keywords: P2P Network, DHT Scheme, Churn, Dynamic Spanning Tree, Stochastic Analysis