Endhost-Based Shortest Path Routing in Dynamic Networks: Supporting Materials

Proof. The proof is based on a similar idea as in the proof of Theorem 3.1 in [2]: if we modify the link weights so that a competitive link becomes part of the optimal path and thus must be used most of the time by any good algorithm, then this link must be used sufficiently often (Ω(log T ) in T steps) by the same algorithm under the original link weights because we do not know which weight configuration is in effect, thus introducing a minimum learning cost. Specifically, consider a competitive link i ∈ E. For any ǫ > 0, select a new link weight distribution Li such that a suboptimal path p ∈ P ′ containing link i becomes optimal and |D(Li||L ′ i)−Di| ≤ ǫDi. Consider any algorithm achieving O(log T ) regret under arbitrary link configurations. Let UT (i) (U ′ T (i)) denote the number of times link i is used up to time T under the original (new) configuration. The O(log T )