Sorting nine inputs requires twenty-five comparisons

Michael Codish, Luís Cruz-Filipe, Michael Frank, Peter Schneider-Kamp

Research output: Contribution to journalArticlepeer-review


This paper describes a computer-assisted non-existence proof of 9-input sorting networks consisting of 24 comparators, hence showing that the 25-comparator sorting network found by Floyd in 1964 is optimal. As a corollary, the 29-comparator network found by Waksman in 1969 is optimal when sorting 10 inputs. This closes the two smallest open instances of the optimal-size sorting network problem, which have been open since the results of Floyd and Knuth from 1966 proving optimality for sorting networks of up to 8 inputs.

Original languageAmerican English
Pages (from-to)551-563
Number of pages13
JournalJournal of Computer and System Sciences
Issue number3
StatePublished - 1 Jan 2016


  • Computer-assisted proofs
  • SAT solving
  • Sorting networks
  • Symmetry breaking

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science
  • Applied Mathematics
  • Computer Networks and Communications
  • Computational Theory and Mathematics


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