Message Reduction in the LOCAL Model Is a Free Lunch

A new spanner construction algorithm is presented, working under the LOCAL model with unique edge IDs. Given an n-node communication graph, a spanner with a constant stretch and O(n^{1 + epsilon}) edges (for an arbitrarily small constant epsilon > 0) is constructed in a constant number of rounds sending O(n^{1 + epsilon}) messages whp. Consequently, we conclude that every t-round LOCAL algorithm can be transformed into an O(t)-round LOCAL algorithm that sends O(t * n^{1 + epsilon}) messages whp. This improves upon all previous message-reduction schemes for LOCAL algorithms that incur a log^{Omega (1)} n blow-up of the round complexity

An FPT Algorithm for Minimum Additive Spanner Problem

For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor ? and an additive approximation parameter ?, which we call (?, ?)-spanners

Improved Approximation for the Directed Spanner Problem

We prove that the size of the sparsest directed k-spanner of a graph can be approximated in polynomial time to within a factor of $\tilde{O}(\sqrt{n})$, for all k >= 3. This improves the $\tilde{O}(n^{2/3})$-approximation recently shown by Dinitz and Krauthgamer

Fast Partial Distance Estimation and Applications

We study approximate distributed solutions to the weighted {\it all-pairs-shortest-paths} (APSP) problem in the CONGEST model. We obtain the following results. $1.$ A deterministic $(1+o(1))$-approximation to APSP in $\tilde{O}(n)$ rounds. This improves over the best previously known algorithm, by both derandomizing it and by reducing the running time by a $\Theta(\log n)$ factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and require that these names are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of $o(n/\log n)$. In the relabeling model, we obtain the following results. $2.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$, running in $\tilde{O}(n^{1/2+1/k}+D)$ rounds, where $D$ is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approximation ratio from $O(k\log k)$ to $O(k)$. Also, the new algorithm uses uses labels of asymptotically optimal size, namely $O(\log n)$ bits. $3.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$, running in time $\tilde{O}((nD)^{1/2}\cdot n^{1/k}+D)$ and producing {\it compact routing tables} of size $\tilde{O}(n^{1/k})$. The node lables consist of $O(k\log n)$ bits. This improves on the approximation ratio of $\Theta(k^2)$ for tables of that size achieved by the best previously known algorithm, which terminates faster, in $\tilde{O}(n^{1/2+1/k}+D)$ rounds