Almost optimal distribution-free sample-based testing of k-modality

For an integer k ≥ 0, a sequence σ = σ1,..., σn over a fully ordered set is k-modal, if there exist indices 1 = a0 < a1 < · · · < ak+1 = n such that for each i, the subsequence σai,..., σai₊₁ is either monotonically non-decreasing or monotonically non-increasing. The property of k-modality is a natural extension of monotonicity, which has been studied extensively in the area of property testing. We study one-sided error property testing of k-modality in the distribution-free sample-based model. We prove an upper bound of¹ O (√kn log k/ε) on the sample complexity, and an almost matching lower bound of Ω (√kn/ε). When the underlying distribution is uniform, we obtain a completely tight bound of Θ (√kn/ε), which generalizes what is known for sample-based testing of monotonicity under the uniform distribution.