Optimum Multi-Dimensional Interval Routing Schemes on Networks with Dynamic Cost Links

One of the fundamental tasks in any distributed computing system is routing messages between pairs of nodes. An Interval Routing Scheme (IRS) is a~space efficient way of routing messages in a network. The problem of characterizing graphs that support an IRS is a well-known problem and has been studied for some variants of IRS. It is natural to assume that the costs of links may vary over time (dynamic cost links) and to try to find an IRS which routes all messages on shortest paths (optimum IRS). In this paper, we study this problem for a~variant of IRS in which the labels assigned to the vertices are $d$-ary integer tuples ($d$-dimensional IRS). The only known results in this case are for specific graphs like hypercubes, $n$-dimensional grids, or for the 1-dimensional case. We give a complete characterization for the class of networks supporting multi-dimensional strict and linear (no cyclic intervals) interval routing schemes with dynamic cost links

Distributed computing of efficient routing schemes in generalized chordal graphs

International audienceEfficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. Two of them are of particular interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of $k$-chordal graphs, i.e., graphs with no induced cycles of length more than $k$. In the class of $k$-chordal graphs, our routing scheme achieves an additive stretch of at most $k-1$, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus $k-1$. In order to compute the routing tables of any $n$-node graph with diameter $D$ we propose a distributed algorithm which uses messages of size $O(\log n)$ and takes $O(D)$ time. The corresponding routing scheme achieves the stretch of $k-1$ on $k$-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of $1$ in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, the distributed computation of the routing tables takes $O(\min\{\Delta D , n\})$ time, where $\Delta$ is the maximum degree of the graph. Our routing schemes use addresses of size $\log n$ bits and local memory of size $2(d-1) \log n$ bits per node of degree $d$

Interval Routing on k-trees

A graph has an optimal `-interval routing scheme if it is possible to direct messages along shortest paths by labeling each edge with at most ` pairwise-disjoint subintervals of the cyclic interval [1 : : : n] (where each node of the graph is labeled by an integer in the range). Although much progress has been made for ` = 1, there is as yet no general tight characterization of the classes of graphs associated with larger `. Bodlaender et al. have shown that under the assumption of dynamic cost links each graph with an optimal `-interval routing scheme has treewidth at most 4`. For the setting without dynamic cost links, this paper addresses the complementary question of the number of intervals required to label classes of graphs of treewidth k. Although it has been shown that there exist graphs of treewidth two that require an arbitrarily large number of intervals, our work demonstrates a class of graphs of treewidth two, namely 2-trees, that are guaranteed to allow 3-interval routing..