• DOI:10.4230/LIPIcs.ICALP.2022.98
• Corpus ID: 235377157

Strongly Sublinear Algorithms for Testing Pattern Freeness

@inproceedings{Newman2022StronglySA,
  title={Strongly Sublinear Algorithms for Testing Pattern Freeness},
  author={Ilan Newman and Nithin M. Varma},
  booktitle={ICALP},
  year={2022}
}

• I. Newman, Nithin M. Varma
• Published in ICALP 9 June 2021
• Mathematics

For a permutation π : [k] → [k], a function f : [n] → R contains a π-appearance if there exists 1 ≤ i1 < i2 < · · · < ik ≤ n such that for all s, t ∈ [k], it holds that f(is) < f(it) if and only if π(s) < π(t). The function is π-free if it has no π-appearances. In this paper, we investigate the problem of testing whether an input function f is π-free or whether at least εn values in f need to be changed in order to make it π-free. This problem is a generalization of the well-studied monotonicity testing and was first studied by Newman, Rabinovich, Rajendraprasad and Sohler [NRRS19]. We show that for all constants k ∈ N, ε ∈ (0, 1), and permutation π : [k] → [k], there is a one-sided error ε-testing algorithm for πfreeness of functions f : [n] → R that makes Õ(n) queries. We improve significantly upon the previous best upper bound O(n) by Ben-Eliezer and Canonne [BC18]. Our algorithm is adaptive, while the earlier best upper bound is known to be tight for nonadaptive algorithms.