Some Lower-Bound Results on Interval Routing in Planar Graphs

Interval routing is a space-efficient routing method for computer networks. For the method to be practical, the routes it generates must be either shortest paths or not too much longer than the shortest paths. We answer the question ofwhat is the lower bound on the longest path that any interval routing scheme (IRS)may be able to generate for arbitrary planar graphs. In general, theworstcase performance of an algorithm is dependent upon the length of the longest path. We consider the generalized form of IRS in which a link is allowed to have up toM ( 1) interval labels. The results we derive cover the range ofM values from 1 to (pn): (1) a lower bound of 2M+1 2M D 1 on the longest path for M = 1; : : : ; ( 3 pn), and (2) a lower bound of 4M+1 4M D 1 on the longest path forM up to (pn), whereD is the diameter and n the number of nodes. The second result also implies a lower bound of (pn) on the number of labels needed by any optimal IRS (one that generates optimally-short paths).