Nearly Tight Bounds for Discrete Search under Outlier Noise

Abstract

Binary search is one of the most fundamental search routines, exploiting the hidden structure of the search space. In particular, it cuts down exponentially on the complexity of the search assuming that the search space is monotone. This paper is prompted by a basic question – how does the query complexity of the search problem change if the data has corruption? In particular, we study the powerful outlier noise model and assuming a bound on the fraction of such corruptions, establish nearly matching upper and lower bounds for the following problems: (i) binary search on an ordered set of size [ n ] ; (ii) search on the posets { 0 , 1 } d ; and (iii) search on the posets [ n ] d . In all three cases, we use randomization to create robust versions of classical algorithms for these problems that handle corrupted data with relatively small performance penalties, specified as a function of the amount of corruption K . We complement these algorithmic results with almost matching lower bounds that show that no randomized algorithm can solve these problems with a smaller performance hit on the query complexity as a function of K .