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<p><b>Abstract</b>—Interval routing is a space-efficient method for point-to-point networks. It is based on labeling the edges of a network with intervals of vertex numbers (called interval labels). An <tmath>$M$</tmath>-label scheme allows up to <tmath>$M$</tmath> labels to be attached on an edge. For arbitrary graphs of size <tmath>$n$</tmath>, <tmath>$n$</tmath> the number of vertices, the problem is to determine the minimum <tmath>$M$</tmath> necessary for achieving optimality in the length of the longest routing path. The longest routing path resulted from a labeling is an important indicator of the performance of any algorithm that runs on the network. We prove that there exists a graph with <tmath>$D=\Omega(n^{\frac{1}{3}})$</tmath> such that if <tmath>$M\leq {\frac{n}{18D}}-O(\sqrt{\frac{n}{D}})$</tmath>, the longest path is no shorter than <tmath>$D+\Theta({\frac{D}{\sqrt{M}}})$</tmath>. As a result, for any <tmath>$M$</tmath>-label IRS, if the longest path is to be shorter than <tmath>$D+\Theta({\frac{D}{\sqrt{M}}})$</tmath>, at least <tmath>$M=\Omega({\frac{n}{D}})$</tmath> labels per edge would be necessary.</p>
Compact routing, computational complexity, computer networks, distributed systems, graph theory, interval routing, optimization, shortest paths.

F. C. Lau and S. S. Tse, "On the Space Requirement of Interval Routing," in IEEE Transactions on Computers, vol. 48, no. , pp. 752-757, 1999.
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