Testing for forbidden order patterns in an array

In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : f1 ng R of length n contains a pattern 2 Sk (Sk is the group of permutations of k elements), i there are indices i1 < i2 ik, such that f(ix) > f(iy) whenever π (x) > π (y). If f does not containwe say f is free. For example, for (2; 1), the property of being free is equivalent to being non-decreasing, i.e. monotone. The property of being (k; k 1 1) free is equivalent to the property of having a partition into at most k 1 non-decreasing subsequences. Let 2 Sk, k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that is free from the case that f diffiers in more than en places from any free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones: For monotone patterns of length k, i.e., (k; k 1 1) and (1; 2k), we design non-adaptive one-sided error tests of (1 log n)O(k2) query complexity. For non-monotone patterns, we show that for any size-k non-monotone any non-adaptive one-sided error test requires at least (p n) queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to (n12=(k+1)). On the other hand, there always exists a non- Adaptive one-sided error test for 2 Sk with O(1=kn11=k) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. for Department of Computer Science, University of Haifa, Israel. Department of Computer Science, University of Haifa, Israel. Department of Computer Science, Indian Institute of Technology Palakkad, India. (Work done while visiting the Caesarea Rothschild Institute, University of Haifa.) xDepartment of Computer Science, TU Dortmund, Germany. (The author acknowledges the support of ERC grant 307696.) (1; 3; 2), we describe an test with (almost tight) query complexity of eO( p n). Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any 2 S3, tests freeness by making (1 log n)O(1) queries. For all algorithms presented here, the running times are linear in their query complexity.