4 Conclusions and Open Questions

arose naturally is: Is the bi diam always bounded by O(D 2) for graphs with diameter D ? A negative answer could imply a better lower bound. However, we have recently shown ((EMZ]) that bi diam(G) = (D 2 G), where G is any 2-edge connected graph, and thus the answer to the question is positive (this result was independently proved by D]). It implies that, using the same technique we used, one cannot hope to improve the lower bound in more than a multipative constant. Another question that remains open is that of nding a non-trivial upper bound (on the length of a path a message traverses) for Linear Interval Routing. In this paper we study the performance of Linear Interval Routing in terms of the longest path traversed by a message, relative to the diameter D of the network. There are two advantages of Linear Interval Routing over Interval Routing. The rst one is its simplicity in implementation. This is important especially in fast networks, where the decision function is often the bottleneck of the communication. The second and main advantage is that it can be used for routing in dynamic networks. Until now it was not clear what the eeciency of Linear Interval Routing is. Whereas an upper bound of 2D and a lower bound of 1:75D ? 1 were known for Interval Routing , there is still no non-trivial upper bound for Linear Interval Routing , while our result shows a lower bound of (D 2). We thus conclude that by restricting ourselves to linear intervals, we might need to pay a signiicant price in time. Following are some open questions which arise in connection with our result. Let an edge-simple cycle be a cycle that contains each edge at most once (no restriction on the number of occurrences of vertices). In a 2-edge-connected graph G = (V; E) every two vertices u and v have an edge-simple cycle in G which includes them. Let bi dist G (u; v) denote the length of the shortest edge-simple cycle that includes u and v, and bi diam G = max u;v2V bi dist G (u; v). Since the lower bound presented in the paper is derived by the sum of two edge-disjoint paths that are used by messages, it is essentially a lower bound of bi diam G for 2-edge-connected graphs.