Testing and Searching Pattern Avoiding Sequences

A sequence avoids a pattern π ∈ R if there is no subsequence that shares the same relative ordering as π. We develop techniques to understand how many queries it takes to test whether a sequence is π-avoiding or -far from π-avoiding. We initiate the study of searching in π-avoiding or nearly π-avoiding sequences, and show that it is possible to efficiently find most elements in a nearly (2, 1)-avoiding, i.e. sorted, sequence. Pattern avoidance is a specific instance of hereditary properties, which are closed under taking subsequences. We prove the existence of general properties that require linear queries to test, implying the existence of hereditary properties that require linear queries to test.