Roundtrip Spanners with (2k-1) Stretch

A roundtrip spanner of a directed graph $G$ is a subgraph of $G$ preserving roundtrip distances approximately for all pairs of vertices. Despite extensive research, there is still a small stretch gap between roundtrip spanners in directed graphs and undirected graphs. For a directed graph with real edge weights in $[1,W]$, we first propose a new deterministic algorithm that constructs a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log (nW))$ edges for every integer $k> 1$, then remove the dependence of size on $W$ to give a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log n)$ edges. While keeping the edge size small, our result improves the previous $2k+\epsilon$ stretch roundtrip spanners in directed graphs [Roditty, Thorup, Zwick'02; Zhu, Lam'18], and almost matches the undirected $(2k-1)$-spanner with $O(n^{1+1/k})$ edges [Alth\"ofer et al. '93] when $k$ is a constant, which is optimal under Erd\"os conjecture.Comment: 12 page

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

A Fast Algorithm for Source-Wise Round-Trip Spanners

In this paper, we study the problem of efficiently constructing source-wise round-trip spanners in weighted directed graphs. For a source vertex set $S\subseteq V$ in a digraph $G(V,E)$, an $S$-source-wise round-trip spanner of $G$ of stretch $k$ is a subgraph $H$ of $G$ such that for every $u\in S,v\in V$, the round-trip distance between $u$ and $v$ in $H$ is at most $k$ times of the original distance in $G$. We show that, for a digraph $G(V,E)$ with $n$ vertices, $m$ edges and nonnegative edge weights, an $s$-sized source vertex set $S\subseteq V$ and a positive integer $k$, there exists an algorithm, in time $O(ms^{1/k}\log^5n)$, with high probability constructing an $S$-source-wise round-trip spanner of stretch $O(k\log n)$ and size $O(ns^{1/k}\log^2n)$. Compared with the state of the art for constructing source-wise round-trip spanners, our algorithm significantly improves their construction time $\Omega(\min\{ms,n^\omega\})$ (where $\omega \in [2,2.373)$ and 2.373 is the matrix multiplication exponent) to nearly linear $O(ms^{1/k}\log^5n)$, while still keeping a spanner stretch $O(k\log n)$ and size $O(ns^{1/k}\log^2n)$, asymptotically similar to their stretch $2k+\epsilon$ and size $O((k^2/\epsilon)ns^{1/k}\log(nw))$, respectively

Relaxed spanners for directed disk graphs

Let $(V,\delta)$ be a finite metric space, where $V$ is a set of $n$ points and $\delta$ is a distance function defined for these points. Assume that $(V,\delta)$ has a constant doubling dimension $d$ and assume that each point $p\in V$ has a disk of radius $r(p)$ around it. The disk graph that corresponds to $V$ and $r(\cdot)$ is a \emph{directed} graph $I(V,E,r)$, whose vertices are the points of $V$ and whose edge set includes a directed edge from $p$ to $q$ if $\delta(p,q)\leq r(p)$. In \cite{PeRo08} we presented an algorithm for constructing a (1+\eps)-spanner of size O(n/\eps^d \log M), where $M$ is the maximal radius $r(p)$. The current paper presents two results. The first shows that the spanner of \cite{PeRo08} is essentially optimal, i.e., for metrics of constant doubling dimension it is not possible to guarantee a spanner whose size is independent of $M$. The second result shows that by slightly relaxing the requirements and allowing a small perturbation of the radius assignment, considerably better spanners can be constructed. In particular, we show that if it is allowed to use edges of the disk graph I(V,E,r_{1+\eps}), where r_{1+\eps}(p) = (1+\eps)\cdot r(p) for every $p\in V$, then it is possible to get a (1+\eps)-spanner of size O(n/\eps^d) for $I(V,E,r)$. Our algorithm is simple and can be implemented efficiently

Spanning Properties of Theta-Theta Graphs

We study the spanning properties of Theta-Theta graphs. Similar in spirit with the Yao-Yao graphs, Theta-Theta graphs partition the space around each vertex into a set of k cones, for some fixed integer k > 1, and select at most one edge per cone. The difference is in the way edges are selected. Yao-Yao graphs select an edge of minimum length, whereas Theta-Theta graphs select an edge of minimum orthogonal projection onto the cone bisector. It has been established that the Yao-Yao graphs with parameter k = 6k' have spanning ratio 11.67, for k' >= 6. In this paper we establish a first spanning ratio of $7.82$ for Theta-Theta graphs, for the same values of $k$. We also extend the class of Theta-Theta spanners with parameter 6k', and establish a spanning ratio of $16.76$ for k' >= 5. We surmise that these stronger results are mainly due to a tighter analysis in this paper, rather than Theta-Theta being superior to Yao-Yao as a spanner. We also show that the spanning ratio of Theta-Theta graphs decreases to 4.64 as k' increases to 8. These are the first results on the spanning properties of Theta-Theta graphs.Comment: 20 pages, 6 figures, 3 table

Fully Dynamic Spanners with Worst-Case Update Time

An alpha-spanner of a graph G is a subgraph H such that H preserves all distances of G within a factor of alpha. In this paper, we give fully dynamic algorithms for maintaining a spanner H of a graph G undergoing edge insertions and deletions with worst-case guarantees on the running time after each update. In particular, our algorithms maintain: - a 3-spanner with ~O(n^{1+1/2}) edges with worst-case update time ~O(n^{3/4}), or - a 5-spanner with ~O(n^{1+1/3}) edges with worst-case update time ~O (n^{5/9}). These size/stretch tradeoffs are best possible (up to logarithmic factors). They can be extended to the weighted setting at very minor cost. Our algorithms are randomized and correct with high probability against an oblivious adversary. We also further extend our techniques to construct a 5-spanner with suboptimal size/stretch tradeoff, but improved worst-case update time. To the best of our knowledge, these are the first dynamic spanner algorithms with sublinear worst-case update time guarantees. Since it is known how to maintain a spanner using small amortized}but large worst-case update time [Baswana et al. SODA\u2708], obtaining algorithms with strong worst-case bounds, as presented in this paper, seems to be the next natural step for this problem

Parameterized Complexity of Directed Spanner Problems

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $m-k$ arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $m-k$ arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $\mathcal {O}(k^4t^5)$ and can be solved in randomized $(4t)^k\cdot n^{\mathcal {O}(1)}$ time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $k^{2k}\cdot n^{\mathcal {O}(1)}$ time on directed acyclic graphs, (iii) Directed Additive Spanner is ${{\,\mathrm{\mathsf{W}}\,}}[1]$-hard when parameterized by k for every fixed $t\ge 1$ even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is ${{\,\mathrm{\mathsf{FPT}}\,}}$ when parameterized by t and k.publishedVersio