Abstract

We prove an \Omega\Gamma/1 2 n= lg lg n) lower bound for the depth of n-input sorting networks based on the shuffle permutation. The best previously known lower bound was the trivial \Omega\Gammaiv n) bound, while the best upper bound is given by Batcher's \Theta(lg 2 n)-depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest. 1 Introduction A variety of different classes of sorting networks has been described in the literature. Of particular interest here are the so-called AKS network [1] discovered by Ajtai, Koml'os and Szemer'edi, and the sorting networks proposed by Batcher [2]. The AKS network is the only known sorting network with O(lg n) depth. However, the topology of the network is highly irregular, and the multiplicative constant hidden by the O-notation is impractically large [1, 11]. On the other hand, the networks proposed by Batcher have a relatively simple interconnection structure and a small constant...

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