An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions

Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given k ? 2, can be used to compute a spanner of stretch 2k-1 and expected size O(n^{1+1/k}) in k rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG\u2719]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the total number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given ? ? (0,1], we compute a low diameter decomposition with diameter bound O((log n)/?) such that each edge e ? E is an inter-cluster edge with probability at most ?? w(e) in O((log n)/?) rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA\u2713]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results

Optimal Vertex Fault Tolerant Spanners (for fixed stretch)

A $k$-spanner of a graph $G$ is a sparse subgraph $H$ whose shortest path distances match those of $G$ up to a multiplicative error $k$. In this paper we study spanners that are resistant to faults. A subgraph $H \subseteq G$ is an $f$ vertex fault tolerant (VFT) $k$-spanner if $H \setminus F$ is a $k$-spanner of $G \setminus F$ for any small set $F$ of $f$ vertices that might "fail." One of the main questions in the area is: what is the minimum size of an $f$ fault tolerant $k$-spanner that holds for all $n$ node graphs (as a function of $f$, $k$ and $n$)? This question was first studied in the context of geometric graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more recently been considered in general undirected graphs [Chechik et al. STOC '09, Dinitz and Krauthgamer PODC '11]. In this paper, we settle the question of the optimal size of a VFT spanner, in the setting where the stretch factor $k$ is fixed. Specifically, we prove that every (undirected, possibly weighted) $n$-node graph $G$ has a $(2k-1)$-spanner resilient to $f$ vertex faults with $O_k(f^{1 - 1/k} n^{1 + 1/k})$ edges, and this is fully optimal (unless the famous Erdos Girth Conjecture is false). Our lower bound even generalizes to imply that no data structure capable of approximating $dist_{G \setminus F}(s, t)$ similarly can beat the space usage of our spanner in the worst case. We also consider the edge fault tolerant (EFT) model, defined analogously with edge failures rather than vertex failures. We show that the same spanner upper bound applies in this setting. Our data structure lower bound extends to the case $k=2$ (and hence we close the EFT problem for $3$-approximations), but it falls to $\Omega(f^{1/2 - 1/(2k)} \cdot n^{1 + 1/k})$ for $k \ge 3$. We leave it as an open problem to close this gap.Comment: To appear in SODA 201

Average Size of Unstretched Remote-Spanners

International audienceMotivated by the optimization of link state routing in ad hoc networks, and the concept of multipoint relays, we introduce the notion of remote-spanner. Given an unweighted graph $G$, a remote spanner is a set of links $H$ such that for any pair of nodes $(u,v)$ there exists a shortest path in $G$ for which all links in the path that are not adjacent to $u$ belong to $H$. The remote spanner is a kind of minimal topology information beyond its neighborhood that any node would need in order to compute its shortest paths in a distributed way. This can be extended to $k$-connected graphs by considering minimum length sum over $k$ disjoint paths as distance. In this paper, we give distributed algorithms for computing remote-spanners in order to obtain sparse remote-spanners with various properties. We provide a polynomial distributed algorithm that computes a $k$-connecting unstretched remote-spanner whose number of edges is at a factor $2(1+\log \Delta)$ from optimal where $\Delta$ is the maximum degree of a node. Interestingly, its expected compression ratio in number of edges is O(\frackn\log n) in Erdös-Rényi graph model and O((\frackn)^\frac23) in the unit disk graph model with a uniform Poisson distribution of nodes

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 - Directed Multiplicative Spanner admits a polynomial kernel of size ?(k?t?) and can be solved in randomized (4t)^k? n^?(1) time, - Directed Additive Spanner is W[1]-hard when parameterized by k even if t = 1 and 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 FPT when parameterized by t and k

Remote spanners: what to know beyond neighbors

International audienceMotivated by the fact that neighbors are generally known in practical routing algorithms, we introduce the notion of remote-spanner. Given an unweighted graph $G$, a sub-graph $H$ with vertex set $V(H)=V(G)$ is an \emph{(\a,\b)-remote-spanner} if for each pair of points $u$ and $v$ the distance between $u$ and $v$ in $H_u$, the graph $H$ augmented by all the edges between $u$ and its neighbors in $G$, is at most \a times the distance between $u$ and $v$ in $G$ plus \b. We extend this definition to $k$-connected graphs by considering minimum length sum over $k$ disjoint paths as distance. We then say that an (\a,\b)-remote-spanner is \emph{$k$-connecting }. In this paper, we give distributed algorithms for computing (1+\eps,1-2\eps)-remote-spanners for any \eps>0, $k$-connecting $(1,0)$-remote-spanners for any $k\ge 1$ (yielding $(1,0)$-remote-spanners for $k=1$) and $2$-connecting $(2,-1)$-remote-spanners. All these algorithms run in constant time for any unweighted input graph. The number of edges obtained for $k$-connecting $(1,0)$-remote-spanner is within a logarithmic factor from optimal (compared to the best $k$-connecting $(1,0)$-remote-spanner of the input graph). Interestingly, sparse $(1,0)$-remote-spanners (i.e. preserving exact distances) with $O(n^4/3)$ edges exist in random unit disk graphs. The number of edges obtained for (1+\eps,1-2\eps)-remote-spanners and $2$-connecting $(2,-1)$-remote-spanners is linear if the input graph is the unit ball graph of a doubling metric (distances between nodes are unknown). Our methodology consists in characterizing remote-spanners as sub-graphs containing the union of small depth tree sub-graphs dominating nearby nodes. This leads to simple local distributed algorithms