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Hypergraphs and decision trees

  • Andrew C. Yao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1197)

Abstract

We survey some recent results on a space decomposition problem that has a close relationship to the size of algebraic decision trees. These results are obtained by establishing connections between the decomposition problem and some extremal questions in hypergraphs.

Let d, n ≥ 1 be integers. A d-elementary cell is a set D\(\subseteq\)R n defined by a finite set of constraints f i (x)=0, g j (x) > 0 where f i , g j are polynomials of degrees not exceeding d. For any set S\(\subseteq\)R n , let κ d (S) be the smallest number of disjoint d-elementary cells that S can be decomposed into. It is easy to see that any degree-d (ternary) algebraic decision tree for solving the membership question of S must have size no less than κ d (S). Thus, any lower bound to κ d (S) yields also a lower bound the size complexity for the corresponding membership problem.

Let A n ={(x1,x2,...,xn) ¦ x i ≥ 0}. A well-known result of Rabin states that any algebraic decision tree for the membership question of A n must have height at least n. In this talk we discuss a recent result by Grigoriev, Karpinski and Yao [GKY], which gives an exponential lower bound to κ d (A n ) for any fixed d, and hence to the size of any fixed degree (ternary) algebraic decision tree for solving this problem. The proof utilizes a new connection between κ d (A n ) and the maximum number of minimal cutsets for any rank-d hypergraph on n vertices. We also discuss an improved lower bound by Wigderson and Yao [WY]. Open questions are presented.

References

  1. [GKY]
    D. Grigoriev, M. Karpinski and A. Yao, “An exponential lower bound on the size of algebraic decision trees for MAX,” preprint, December 1995.Google Scholar
  2. [WY]
    A. Wigderson and A. Yao, manuscript, March 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Andrew C. Yao
    • 1
  1. 1.Department of Computer SciencePrinceton UniversityPrincetonUSA

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