Improved Routing and Sorting on Multibutter ies

This paper shows that an N -node AKS network (as described by Paterson) can be embedded in a 3N 2 -node degree-8 multibutter y network with load 1, congestion 1, and dilation 2. The result has several implications, including the rst deterministic algorithms for sorting and nding the median of n logn keys on an n-input multibutter y in O(logn) time, a work-e cient algorithm for nding the median of n log2 n log logn keys on an n-input multibutter y in O(logn log logn) time, and a three-dimensional VLSI layout for the ninput AKS network with volume O(n3=2). While these algorithms are not practical, they provide further evidence of the robustness of multibutter y networks. We also present a separate, and more practical, deterministic algorithm for routing h relations on an n-input multibutter y in O(h+logn) time. Previously, only algorithms for solving h one-to-one routing problems were known. Finally, we show that a 2-folded butter y, whose individual splitters do not exhibit expansion, can emulate a bounded-degree multibutter y with an ( ; ) expansion property, for any < 1=4.