Constructing Light Spanners Deterministically in Near-Linear Time

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [Shiri Chechik and Christian Wulff-Nilsen, 2018] improved the state-of-the-art for light spanners by constructing a (2k-1)(1+epsilon)-spanner with O(n^(1+1/k)) edges and O_epsilon(n^(1/k)) lightness. Soon after, Filtser and Solomon [Arnold Filtser and Shay Solomon, 2016] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn^(1+1/k)) (which is faster than [Shiri Chechik and Christian Wulff-Nilsen, 2018]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Omega_epsilon(kn^(1/k)), even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an O_epsilon(n^(2+1/k+epsilon\u27)) time spanner construction which achieves the state-of-the-art bounds. Our second result is an O_epsilon(m + n log n) time construction of a spanner with (2k-1)(1+epsilon) stretch, O(log k * n^(1+1/k) edges and O_epsilon(log k * n^(1/k)) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k=log n, for every constant epsilon>0, we provide an O(m+n^(1+epsilon)) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = omega(1). To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest

Spanner Approximations in Practice

A multiplicative $\alpha$-spanner $H$ is a subgraph of $G=(V,E)$ with the same vertices and fewer edges that preserves distances up to the factor $\alpha$, i.e., $d_H(u,v)\leq\alpha\cdot d_G(u,v)$ for all vertices $u$, $v$. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

In the decremental $(1+\epsilon)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s \in V$, and we are asked to process edge deletions efficiently and answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(s,v)$ for each $v \in V$, at any stage, such that $\mathbf{dist}_G(s,v) \leq \widetilde{\mathbf{dist}}_G(s,v) \leq (1+ \epsilon)\mathbf{dist}_G(s,v)$. In the decremental $(1+\epsilon)$-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates $\widetilde{\mathbf{dist}}_G(u,v)$ for every $u,v \in V$. In this article, we consider the problems for undirected, unweighted graphs. We present a new \emph{deterministic} algorithm for the decremental $(1+\epsilon)$-approximate SSSP problem that takes total update time $O(mn^{0.5 + o(1)})$. Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update time $\tilde{O}(n^2)$ and the best existing algorithm for sparse graphs with running time $\tilde{O}(n^{1.25}\sqrt{m})$ [SODA 2017] whenever $m = O(n^{1.5 - o(1)})$. In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic \emph{sparse} emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental $(1+\epsilon)$-approximate APSP problem with near-optimal total running time $\tilde{O}(mn /\epsilon)$ matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].Comment: Appeared in SODA'2

A Unified and Fine-Grained Approach for Light Spanners

Seminal works on light spanners from recent years provide near-optimal tradeoffs between the stretch and lightness of spanners in general graphs, minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a "truly optimal" tradeoff for Euclidean low-dimensional spaces. Some of these papers employ inherently different techniques than others. Moreover, the runtime of these constructions is rather high. In this work, we present a unified and fine-grained approach for light spanners. Besides the obvious theoretical importance of unification, we demonstrate the power of our approach in obtaining (1) stronger lightness bounds, and (2) faster construction times. Our results include: _ $K_r$-minor-free graphs: A truly optimal spanner construction and a fast construction. _ General graphs: A truly optimal spanner -- almost and a linear-time construction with near-optimal lightness. _ Low dimensional Euclidean spaces: We demonstrate that Steiner points help in reducing the lightness of Euclidean $1+\epsilon$-spanners almost quadratically for $d\geq 3$.Comment: We split this paper into two papers: arXiv:2106.15596 and arXiv:2111.1374

New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs

In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph $G=(V,E)$ subject to edge insertions and deletions and a source vertex $s\in V$, and the goal is to maintain the distance $d(s,t)$ for all $t\in V$. Fine-grained complexity has provided strong lower bounds for exact partially dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15]. Thus much focus has been directed towards finding efficient partially dynamic $(1+\epsilon)$-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14, FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20]. Despite this rich literature, for directed graphs there are no known deterministic algorithms for $(1+\epsilon)$-approximate dynamic SSSP that perform better than the classic ES-tree [JACM'81]. We present the first such algorithm. We present a \emph{deterministic} data structure for incremental SSSP in weighted digraphs with total update time $\tilde{O}(n^2 \log W)$ which is near-optimal for very dense graphs; here $W$ is the ratio of the largest weight in the graph to the smallest. Our algorithm also improves over the best known partially dynamic \emph{randomized} algorithm for directed SSSP by Henzinger et al. [STOC'14, ICALP'15] if $m=\omega(n^{1.1})$. We also provide improved conditional lower bounds. Henzinger et al. [STOC'15] showed that under the OMv Hypothesis, the partially dynamic exact $s$-$t$ Shortest Path problem in undirected graphs requires amortized update or query time $m^{1/2-o(1)}$, given polynomial preprocessing time. Under a hypothesis about finding Cliques, we improve the update and query lower bound for algorithms with polynomial preprocessing time to $m^{0.626-o(1)}$. Further, under the $k$-Cycle hypothesis, we show that any partially dynamic SSSP algorithm with $O(m^{2-\epsilon})$ preprocessing time requires amortized update or query time $m^{1-o(1)}$