Distributed computing of efficient routing schemes in generalized chordal graphs

Description

Efficient 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 $d$.

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