Lower Bounds for One-to-one Packet Routing on Trees using Hot-Potato Algorithms

In this paper, we consider hot-potato packet routing of one-to-one routing patterns on n-node trees. By applying a ‘charging argument’, we show that any greedy hot-potato algorithm routes a oneto-one routing pattern within 2(n − 1) steps. On the other hand, a trivial lower bound suggests that at least 3n/2 steps are required by any oblivious greedy algorithm. As the main contribution of the paper, we tighten the 2(n − 1) upper bound by constructing (for all sufficiently large n) an elaborate one-to-one packet routing problem on an n-node tree for which an oblivious greedy hotpotato algorithm requires at least 2n − o(n) steps. This improved lower bound is also shown to be valid for the minimum-distance heuristic. For trees of maximum degree d , we establish a lower bound of 2((d − 3)/(d − 2))n − o(n) routing steps.