The compactness of interval routing

The compactness of a graph measures the space complexity of its shortest path routing tables. Each outgoing edge of a node x is assigned a (pairwise disjoint) set of addresses, such that the unique outgoing edge containing the address of a node y is the first edge of a shortest path from x to y. The complexity measure used in the context of interval routing is the minimum number of intervals of consecutive addresses needed to represent each such set. minimized over all possible choices of addresses and all choices of shortest paths. This paper establishes asymptotically tight bounds of n/4 on the compactness of an n-node graph. More specifically, it is shown that every n-node graph has compactness at most n/4+o(n), and conversely, there exists an n-node graph whose compactness is n/4 - o(n). Both bounds improve upon known results. (A preliminary version of the lower bound has been partially published in Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 1300, pp. 259-268, 1997.).