A lower bound for nearly minimal adaptive and hot potato algorithms

Recently, Chinn et al. [10] presented lower bounds for store-and-forward permutation routing algorithms on the n x n mesh with bounded buffer size and where a packet must take a shortest (or minimal) path to its destination. We extend their analysis to algorithms that are nearly minimal. We also apply this technique to the domain of hot potato algorithms, where there is no storage of packets and the shortest path to a destination is not assumed (and is in general impossible). We show that "natural" variants and "improvements" of several algorithms in the literature perform poorly in the worst case. As a result, we identify algorithmic features that are undesirable for worst-case hot potato permutation routing. Recent works in hot potato routing have tried to define simple and greedy classes of algorithms. We show that when an algorithm is too simple and too greedy, its performance in routing permutations is poor in the worst case. Specifically the technique of [10] is also applicable to algorithms that do not necessarily send packets in minimal or even nearly minimal paths: it may be enough that they naively attempt to do so when possible. In particular, our results show that a certain class of greedy algorithms that was suggested recently by Ben-Dor et al. [6] contains algorithms that have poor performance in routing worst-case permutations.