A lower bound for nearly minimal adaptive and hot potato algorithms

Abstract

Recently, Chinn, Leighton, and Tompa [I0] presented lower bounds for store-and-forward permutation routing algorithms on the n × n mesh with bounded buffer size and where a packet must take a shortest (or minima 0 path to its destination. We extend their analysis to algorithms that are nearly minimal. (In their preliminary work, Chinn et al. [10] mention a similar result that seems, however, incorrect.) 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. Researchers in hot potato routing have defined 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 rn|nlrna.l 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-Dot, Halevi, and Schuster [6] contains algorithms that have poor performance in routing worst case permutations.