Joint Size and Depth Optimization of Sorting Networks
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Sorting networks are oblivious sorting algorithms with many interesting theoretical properties and practical applications. One of the related classical challenges is the search of optimal networks respect to size (number of comparators) of depth (number of layers). However, up to our knowledge, the joint size-depth optimality of small sorting networks has not been addressed before. This paper presents size-depth optimality results for networks up to 12 channels. Our results show that there are sorting networks for n≤9 inputs that are optimal in both size and depth, but this is not the case for 10 and 12 channels. For n=10 inputs, we were able to proof that optimal-depth optimal sorting networks with 7 layers require 31 comparators while optimal-size networks with 29 comparators need 8 layers. For n=11 inputs we show that networks with 8 or 9 layers require at least 35 comparators (the best known upper bound for the minimal size). And for networks with n=12 inputs and 8 layers we need 40 comparators, while for 9 layers the best known size is 39.
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