Improved Testing Algorithms for Monotonicity

We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f : Σ 7→ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f , and rejects f with high probability if it is -far from being monotone (i.e., every monotone function differs from f on more than an fraction of the domain). For any > 0, the query and time complexities of the test are O((n/ ) · log |Σ| · log |Ξ|). The previous best known bound was Õ((n/ ) · |Σ|2 · |Ξ|). We also present an alternative test for the boolean range Ξ = {0, 1} whose complexity is independent of alphabet size |Σ|. This test has query complexity O((n/ ) log(n/ )) and time complexity O((n/ ) log(n/ )). ∗Lab for Computer Science, MIT, 545 Technology Sq. Cambridge, MA 02139. email: [email protected]. †Dept. of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. e-mail: [email protected]. ‡Lab for Computer Science, MIT, 545 Technology Sq. Cambridge, MA 02139. email: e [email protected]. §Lab for Computer Science, MIT, 545 Technology Sq. Cambridge, MA 02139. email: [email protected]. ¶Dept. of EE – Systems, Tel Aviv University, Ramat Aviv, Israel. e-mail: [email protected]. ‖DIMACS Center, Rutgers University, Piscataway, NJ 08854. email: [email protected]. Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 17 (1999)