The Gram-Schmidt walk: a cure for the Banaszczyk blues

  title={The Gram-Schmidt walk: a cure for the Banaszczyk blues},
  author={Nikhil Bansal and Daniel Dadush and Shashwat Garg and Shachar Lovett},
  journal={Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing},
An important result in discrepancy due to Banaszczyk states that for any set of n vectors in ℝm of ℓ2 norm at most 1 and any convex body K in ℝm of Gaussian measure at least half, there exists a ± 1 combination of these vectors which lies in 5K. This result implies the best known bounds for several problems in discrepancy. Banaszczyk’s proof of this result is non-constructive and an open problem has been to give an efficient algorithm to find such a ± 1 combination of the vectors. In this paper… 

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Constructive Discrepancy Minimization for Convex Sets

  • T. Rothvoss
  • Mathematics
    2014 IEEE 55th Annual Symposium on Foundations of Computer Science
  • 2014
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Roth’s estimate of the discrepancy of integer sequences is nearly sharp

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  • Computer Science, Mathematics
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
  • 2016
An efficient algorithm is given that finds a coloring with discrepancy O((t log n)1/2), matching the best known non-constructive bound for the problem due to Banaszczyk, and gives an algorithmic O(log 1/2 n) bound.

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