Improved Parallel Algorithms for Spanners and Hopsets

We use exponential start time clustering to design faster and more work-efficient parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: * Linear work parallel algorithms that construct spanners with $O(k)$ stretch and size $O(n^{1+1/k})$ in unweighted graphs, and size $O(n^{1+1/k} \log k)$ in weighted graphs. * Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with $O(m\;\mathrm{polylog}\;n)$ work

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

Fault-tolerant additive weighted geometric spanners

Let S be a set of n points and let w be a function that assigns non-negative weights to points in S. The additive weighted distance d_w(p, q) between two points p,q belonging to S is defined as w(p) + d(p, q) + w(q) if p \ne q and it is zero if p = q. Here, d(p, q) denotes the (geodesic) Euclidean distance between p and q. A graph G(S, E) is called a t-spanner for the additive weighted set S of points if for any two points p and q in S the distance between p and q in graph G is at most t.d_w(p, q) for a real number t > 1. Here, d_w(p,q) is the additive weighted distance between p and q. For some integer k \geq 1, a t-spanner G for the set S is a (k, t)-vertex fault-tolerant additive weighted spanner, denoted with (k, t)-VFTAWS, if for any set S' \subset S with cardinality at most k, the graph G \ S' is a t-spanner for the points in S \ S'. For any given real number \epsilon > 0, we obtain the following results: - When the points in S belong to Euclidean space R^d, an algorithm to compute a (k,(2 + \epsilon))-VFTAWS with O(kn) edges for the metric space (S, d_w). Here, for any two points p, q \in S, d(p, q) is the Euclidean distance between p and q in R^d. - When the points in S belong to a simple polygon P, for the metric space (S, d_w), one algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges and another algorithm to compute a geodesic (k, (\sqrt{10} + \epsilon))-VFTAWS with O(kn(\lg{n})^2) edges. Here, for any two points p, q \in S, d(p, q) is the geodesic Euclidean distance along the shortest path between p and q in P. - When the points in $S$ lie on a terrain T, an algorithm to compute a geodesic (k, (2 + \epsilon))-VFTAWS with O(\frac{k n}{\epsilon^{2}}\lg{n}) edges.Comment: a few update