Optimal Hitting Sets for Combinatorial Shapes

  • Aditya Bhaskara
  • Devendra Desai
  • Srikanth Srinivasan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)


We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests, including symmetric functions and combinatorial rectangles. Generalizing results of Linial, Luby, Saks, and Zuckerman (Combinatorica 1997) and Rabani and Shpilka (SICOMP 2010), we construct hitting sets for Combinatorial Shapes of size polynomial in the alphabet, dimension, and the inverse of the error parameter. This is optimal up to polynomial factors. The best previous hitting sets came from the Pseudorandom Generator construction of Gopalan et al., and in particular had size that was quasipolynomial in the inverse of the error parameter.

Our construction builds on natural variants of the constructions of Linial et al. and Rabani and Shpilka. In the process, we construct fractional perfect hash families and hitting sets for combinatorial rectangles with stronger guarantees. These might be of independent interest.


Hash Function Seed Length Pseudorandom Generator Polynomial Factor Hash Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aditya Bhaskara
    • 1
  • Devendra Desai
    • 2
  • Srikanth Srinivasan
    • 3
  1. 1.Department of Computer SciencePrinceton UniversityUSA
  2. 2.Department of Computer ScienceRutgers UniversityUSA
  3. 3.DIMACSRutgers UniversityUSA

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