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The Complexity of Unary Subset Sum

  • Nutan Limaye
  • Meena Mahajan
  • Karteek Sreenivasaiah
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)

Abstract

Given a stream of n numbers and a number B, the subset sum problem deals with checking whether there exists a subset of the stream that adds to exactly B. The unary subset sum problem, USS, is the same problem when the input is encoded in unary. We prove that any p-pass randomized algorithm computing USS with error at most 1/3 must use space \(\Omega(\frac{B}{p})\). For p ≤ B, we give a randomized p-pass algorithm that computes USS with error at most 1/3 using space \(\tilde{O}(\frac{nB}{p})\). We give a deterministic one-pass algorithm which given an input stream and two parameters B, ε, decides whether there exist a subset of the input stream that adds to a value in the range \(\left[(1-\epsilon)B,(1+\epsilon)B\right]\) using space \(O\left(\frac{\log B}{\epsilon}\right)\). We observe that USS is monotone (under a suitable encoding) and give a monotone NC2 circuit for USS. We also show that any circuit using ε-approximator gates for USS under this encoding needs Ω(n/logn) gates to compute the Disjointness function.

Keywords

Communication Complexity Input Stream Polynomial Time Approximation Scheme Monotone Formula Unary Subset 
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

  • Nutan Limaye
    • 1
  • Meena Mahajan
    • 2
  • Karteek Sreenivasaiah
    • 2
  1. 1.Indian Institute of TechnologyBombayIndia
  2. 2.The Institute of Mathematical SciencesChennaiIndia

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