Property-Preserving data reconstruction

  • Authors:

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

  • Affiliations:

  • Dept Comp Sci., Princeton University, Princeton, NJ;Dept Comp Sci., Princeton University, Princeton, NJ;Dept Comp Sci., Princeton University, Princeton, NJ;Dept Comp Sci., Princeton University, Princeton, NJ

  • Venue:
  • ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
  • Year:
  • 2004

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Abstract

We initiate a new line of investigation into online property-preserving data reconstruction. Consider a dataset which is assumed to satisfy various (known) structural properties; eg, it may consist of sorted numbers, or points on a manifold, or vectors in a polyhedral cone, or codewords from an error-correcting code Because of noise and errors, however, an (unknown) fraction of the data is deemed unsound, ie, in violation with the expected structural properties Can one still query into the dataset in an online fashion and be provided data that is always sound? In other words, can one design a filter which, when given a query to any item I in the dataset, returns a sound item J that, although not necessarily in the dataset, differs from I as infrequently as possible No preprocessing should be allowed and queries should be answered online We consider the case of a monotone function Specifically, the dataset encodes a function f:{1,...n} ↦R that is at (unknown) distance ε from monotone, meaning that f can—and must—be modified at εn places to become monotone. Our main result is a randomized filter that can answer any query in O(log2n log log n) time while modifying the function f at only O(εn) places The amortized time over n function evaluations is O(log n) The filter works as stated with probability arbitrarily close to 1 We also provide an alternative filter with O(log n) worst case query time and O(εn log n) function modifications.