A (fairly) Simple Circuit that (usually) Sorts

This paper provides an analysis of a natural k-round tournament over n = 2' players, and demonstrates that the tournament possesses a surprisingly strong ranking property. The ranking property of this tournament is exploited by using it as a building block for efficient parallel sorting algorithms under a variety of different models of computation. Three important applications are provided. First, a sorting circuit of depth 7.44logn is defined that sorts all but a superpolynomially small fraction of the n! possible input permutations. Second, a randomized sorting algorithm is given for the hypercube and related parallel computers (the butterfly, cube-connected cycles and shuffle-exchange) that runs in O(1og n) word steps with very high probability. Third, a randomized algorithm is given for sorting n O(m)-bit records on an n log n node butterfly that runs in O(m + log n) bit steps with very high probability. 1 Introduction Consider the following k-round tournament defined over n = 2k players. In the first round, n/2 matches are played according to a random pairing of the n players. The next k-1 rounds are defined by recursively running a tournament amongst the n/2 winners, and (in parallel) a separate tournament amongst the n/2 losers. Note that the depth k comparator circuit corresponding to this tournament is an n-input butterfly network in which the input is a random permutation and the two outputs of each comparator gate are oriented in the same direction. Hence, this tournament will be referred to as the butterfly toumament of order k. After the tournament has been completed, each player has achieved a unique sequence of match outcomes (wins and losses, 1's and 0's) of length k. Let player i be the player that achieves a W-L sequence corresponding to the k-bit number i , that is, the player " routed " to the ith output of the n-input butterfly com-of all matches are determined by an underlying total order. Further assume that the tournament has available n distinct amounts of prize money to be assigned to the n possible outcome sequences. How should these amounts be assigned? Clearly the largest amount of money should be assigned to player n-1 = W k , who is guaranteed to be the best player. Similarly, the smallest prize should be awarded to player 0 = Lk. On the other hand, it is not clear how to rank all of the remaining n-2 W-L …