An Answer to the Bose-Nelson Sorting Problem for 11 and 12 Channels

Jannis Harder

An Answer to the Bose-Nelson Sorting Problem for 11 and 12 Channels

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arxiv 2012.04400 v3 pith:QFVSSMZC submitted 2020-12-08 cs.DS

Jannis Harder This is my paper · ORCID

classification cs.DS

keywords sortingnetworksalgorithmbose-nelsonchannelchannelscomparatorscorresponding

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We show that 11-channel sorting networks have at least 35 comparators and that 12-channel sorting networks have at least 39 comparators. This positively settles the optimality of the corresponding sorting networks given in The Art of Computer Programming vol. 3 and closes the two smallest open instances of the Bose-Nelson sorting problem. We obtain these bounds by generalizing a result of Van Voorhis from sorting networks to a more general class of comparator networks. From this we derive a dynamic programming algorithm that computes the optimal size for a sorting network with a given number of channels. From an execution of this algorithm we construct a certificate containing a derivation of the corresponding lower size bound, which we check using a program formally verified using the Isabelle/HOL proof assistant.

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