A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners

Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the corresponding distance in $G$ is upper bounded by $\sigma$. The minimum value of $\sigma$ for which $T$ is a $\sigma$-tree spanner of $G$ is also called the {\em stretch factor} of $T$. We address the fault-tolerant scenario in which each edge $e$ of a given tree spanner may temporarily fail and has to be replaced by a {\em best swap edge}, i.e. an edge that reconnects $T-e$ at a minimum stretch factor. More precisely, we design an $O(n^2)$ time and space algorithm that computes a best swap edge of every tree edge. Previously, an $O(n^2 \log^4 n)$ time and $O(n^2+m\log^2n)$ space algorithm was known for edge-weighted graphs [Bil\`o et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a $o(n^2)$ time and space algorithm would be considered a breakthrough.Comment: The paper has been accepted for publication at the 29th International Symposium on Algorithms and Computation (ISAAC 2018). 12 pages, 3 figure

Linear Time Distributed Swap Edge Algorithms

In this paper, we consider the all best swap edges problem in a distributed environment. We are given a 2-edge connected positively weighted network X, where all communication is routed through a rooted spanning tree T of X. If one tree edge e = {x, y} fails, the communication network will be disconnected. However, since X is 2-edge connected, communication can be restored by replacing e by non-tree edge e′, called a swap edge of e, whose ends lie in different components of T − e. Of all possible swap edges of e, we would like to choose the best, as defined by the application. The all best swap edges problem is to identify the best swap edge for every tree edge, so that in case of any edge failure, the best swap edge can be activated quickly. There are solutions to this problem for a number of cases in the literature. A major concern for all these solutions is to minimize the number of messages. However, especially in fault-transient environments, time is a crucial factor. In this paper we present a novel technique that addresses this problem from a time perspective; in fact, we present a distributed solution that works in linear time with respect to the height h of T for a number of differentcriteria, while retaining the optimal number of messages. To the best of our knowledge, all previous solutions solve the problem in O(h^2) time in the cases we consider

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

Community-aware network sparsification

Network sparsification aims to reduce the number of edges of a network while maintaining its structural properties; such properties include shortest paths, cuts, spectral measures, or network modularity. Sparsification has multiple applications, such as, speeding up graph-mining algorithms, graph visualization, as well as identifying the important network edges. In this paper we consider a novel formulation of the network-sparsification problem. In addition to the network, we also consider as input a set of communities. The goal is to sparsify the network so as to preserve the network structure with respect to the given communities. We introduce two variants of the community-aware sparsification problem, leading to sparsifiers that satisfy different connectedness community properties. From the technical point of view, we prove hardness results and devise effective approximation algorithms. Our experimental results on a large collection of datasets demonstrate the effectiveness of our algorithms.https://epubs.siam.org/doi/10.1137/1.9781611974973.48Accepted manuscrip

Fault Tolerant and Fully Dynamic DFS in Undirected Graphs: Simple Yet Efficient

We present an algorithm for a fault tolerant Depth First Search (DFS) Tree in an undirected graph. This algorithm is drastically simpler than the current state-of-the-art algorithms for this problem, uses optimal space and optimal preprocessing time, and still achieves better time complexity. This algorithm also leads to a better time complexity for maintaining a DFS tree in a fully dynamic environment

Improved Roundtrip Spanners, Emulators, and Directed Girth Approximation

Roundtrip spanners are the analog of spanners in directed graphs, where the roundtrip metric is used as a notion of distance. Recent works have shown existential results of roundtrip spanners nearly matching the undirected case, but the time complexity for constructing roundtrip spanners is still widely open. This paper focuses on developing fast algorithms for roundtrip spanners and related problems. For any $n$-vertex directed graph $G$ with $m$ edges (with non-negative edge weights), our results are as follows: - 3-roundtrip spanner faster than APSP: We give an $\tilde{O}(m\sqrt{n})$-time algorithm that constructs a roundtrip spanner of stretch $3$ and optimal size $O(n^{3/2})$. Previous constructions of roundtrip spanners of the same size either required $\Omega(nm)$ time [Roditty, Thorup, Zwick SODA'02; Cen, Duan, Gu ICALP'20], or had worse stretch $4$ [Chechik and Lifshitz SODA'21]. - Optimal roundtrip emulator in dense graphs: For integer $k\ge 3$, we give an $O(kn^2\log n)$-time algorithm that constructs a roundtrip \emph{emulator} of stretch $(2k-1)$ and size $O(kn^{1+1/k})$, which is optimal for constant $k$ under Erd\H{o}s' girth conjecture. Previous work of [Thorup and Zwick STOC'01] implied a roundtrip emulator of the same size and stretch, but it required $\Omega(nm)$ construction time. Our improved running time is near-optimal for dense graphs. - Faster girth approximation in sparse graphs: We give an $\tilde{O}(mn^{1/3})$-time algorithm that $4$-approximates the girth of a directed graph. This can be compared with the previous $2$-approximation algorithm in $\tilde{O}(n^2, m\sqrt{n})$ time by [Chechik and Lifshitz SODA'21]. In sparse graphs, our algorithm achieves better running time at the cost of a larger approximation ratio.Comment: To appear in SODA 202

Faster Parallel Algorithm for Approximate Shortest Path

We present the first $m\,\text{polylog}(n)$ work, $\text{polylog}(n)$ time algorithm in the PRAM model that computes $(1+\epsilon)$-approximate single-source shortest paths on weighted, undirected graphs. This improves upon the breakthrough result of Cohen~[JACM'00] that achieves $O(m^{1+\epsilon_0})$ work and $\text{polylog}(n)$ time. While most previous approaches, including Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent developments in \emph{continuous optimization}, studying the shortest path problem from the lens of the closely-related \emph{minimum transshipment} problem. To obtain our algorithm, we demonstrate a series of near-linear work, polylogarithmic-time reductions between the problems of approximate shortest path, approximate transshipment, and $\ell_1$-embeddings, and establish a recursive algorithm that cycles through the three problems and reduces the graph size on each cycle. As a consequence, we also obtain faster parallel algorithms for approximate transshipment and $\ell_1$-embeddings with polylogarithmic distortion. The minimum transshipment algorithm in particular improves upon the previous best $m^{1+o(1)}$ work sequential algorithm of Sherman~[SODA'17]. To improve readability, the paper is almost entirely self-contained, save for several staple theorems in algorithms and combinatorics.Comment: 53 pages, STOC 202

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum