Hypergraphs and decision trees

  • Andrew C. Yao
Conference paper

DOI: 10.1007/3-540-62559-3_1

Part of the Lecture Notes in Computer Science book series (LNCS, volume 1197)
Cite this paper as:
Yao A.C. (1997) Hypergraphs and decision trees. In: d'Amore F., Franciosa P.G., Marchetti-Spaccamela A. (eds) Graph-Theoretic Concepts in Computer Science. WG 1996. Lecture Notes in Computer Science, vol 1197. Springer, Berlin, Heidelberg

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\(\subseteq\)Rn defined by a finite set of constraints fi(x)=0, gj(x) > 0 where fi, gj are polynomials of degrees not exceeding d. For any set S\(\subseteq\)Rn, let κd(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 κd(S). Thus, any lower bound to κd(S) yields also a lower bound the size complexity for the corresponding membership problem.

Let An={(x1,x2,...,xn) ¦ xi ≥ 0}. A well-known result of Rabin states that any algebraic decision tree for the membership question of An 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 κd(An) 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 κd(An) 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.

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

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

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