Estimating the distance to a monotone function

  • Authors:

  • Nir Ailon;Bernard Chazelle;Seshadhri Comandur;Ding Liu

  • Affiliations:

  • Department Computer Science, Princeton University, Princeton, New Jersey;Department Computer Science, Princeton University, Princeton, New Jersey;Department Computer Science, Princeton University, Princeton, New Jersey;Department Computer Science, Princeton University, Princeton, New Jersey

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2007

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Abstract

In standard property testing, the task is to distinguish betweenobjects that have a property ℘ and those that areε-far from ℘, for some ε 0. In thissetting, it is perfectly acceptable for the tester to provide anegative answer for every input object that does not satisfy℘. This implies that property testing in and of itselfcannot be expected to yield any information whatsoever about thedistance from the object to the property. We address this problemin this paper, restricting our attention to monotonicity testing. Afunction f : {1,…,n} ➝ R is atdistance εf from being monotone if it can(and must) be modified at εfnplaces to become monotone. For any fixed δ 0, wecompute, with probability at least 2/3, an interval [(1/2-δ)ε,ε] that enclosesεf. The running time of our algorithm isO(εf-1 log logεf- 1 log n), which isoptimal within a factor of loglogεf-1 and represents asubstantial improvement over previous work. We give a secondalgorithm with an expected running time ofO(εf-1 log nloglog log n). Finally, we extend our results to multivariatefunctions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg.,2007