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