Sorting Networks: The Final Countdown

In this paper we extend the knowledge on the problem of empirically searching for sorting networks of minimal depth. We present new search space pruning techniques for the last four levels of a candidate sorting network by considering only the output set representation of a network. We present an algorithm for checking whether an $n$-input sorting network of depth $d$ exists by considering the minimal up to permutation and reflection itemsets at each level and using the pruning at the last four levels. We experimentally evaluated this algorithm to find the optimal depth sorting networks for all $n \leq 12$.Comment: arXiv admin note: substantial text overlap with arXiv:1502.0474

Joint Size and Depth Optimization of Sorting Networks

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 $n\leq9$ 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 $39$

Sorting Networks: to the End and Back Again

This paper studies new properties of the front and back ends of a sorting network, and illustrates the utility of these in the search for new bounds on optimal sorting networks. Search focuses first on the "outsides" of the network and then on the inner part. All previous works focus only on properties of the front end of networks and on how to apply these to break symmetries in the search. The new, out-side-in, properties help shed understanding on how sorting networks sort, and facilitate the computation of new bounds on optimal sorting networks. We present new parallel sorting networks for 17 to 20 inputs. For 17, 19, and 20 inputs these networks are faster than the previously known best networks. For 17 inputs, the new sorting network is shown optimal in the sense that no sorting network using less layers exists.Comment: IMADA-preprint-cs. arXiv admin note: text overlap with arXiv:1411.640