Accessibility setting

Section: New Results

Compact Routing

On the Communication Complexity of Distributed Name-Independent Routing Schemes

Participants : Cyril Gavoille, Nicolas Hanusse, David Ilcinkas.

In [38] , we present a distributed asynchronous algorithm that, for every undirected weighted $n$-node graph $G$, constructs name-independent routing tables for $G$. The size of each table is $\tilde{O}\left(\sqrt{n}\right)$, whereas the length of any route is stretched by a factor of at most 7 w.r.t. the shortest path. At any step, the memory space of each node is $\tilde{O}\left(\sqrt{n}\right)$. The algorithm terminates in time $O\left(D\right)$, where $D$ is the hop-diameter of $G$. In synchronous scenarios and with uniform weights, it consumes $\tilde{O}(m\sqrt{n}+n^{3/2}minD,\sqrt{n})$ messages, where $m$ is the number of edges of $G$.

In the realistic case of sparse networks of poly-logarithmic diameter, the communication complexity of our scheme, that is $\tilde{O}\left(n^{3/2}\right)$, improves by a factor of $\sqrt{n}$ the communication complexity of any shortest-path routing scheme on the same family of networks. This factor is provable thanks to a new lower bound of independent interest.

There are Plane Spanners of Maximum Degree 4

Participant : Nicolas Bonichon.

Let $ℰ$ be the complete Euclidean graph on a set of points embedded in the plane. Given a fixed constant $t\ge 1$, a spanning subgraph $G$ of $ℰ$ is said to be a $t$-spanner of $ℰ$ if for any pair of vertices $u,v$ in $ℰ$ the distance between $u$ and $v$ in $G$ is at most $t$ times their distance in $ℰ$. A spanner is plane if its edges do not cross.

We consider the question: “What is the smallest maximum degree that can be achieved for a plane spanner of $ℰ$?” Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this work we show that there is a constant $t\ge 1$ such that the complete Euclidean graph always contains a plane $t$-spanner of maximum degree 4 and make a big step toward closing the question. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's $L_1$-Delaunay triangulation.