Membership to a Polyhedron by Algebraic Decision Trees DIMA GRIGORIEV * MAREK KARPINSKI t NICOLAI VOROBJOV $ We describe a new method of proving lower bounds on the depth of algebraic decision trees and apply it to prove a lower bound C!(log N ) for testing membership to a convex polyhedron having IV facets of all dimensions, provided that N is large enough. This bound apparently does not follow from the methods developed by M. Ben-Or, A, Bjorner, L. Lovasz, and A. Yao ([B 83], [BLY 92]) because the topological invariants used in these methods become trivial for a convex polyhedra. *Departments of Computer Science and Mathematics, Penn State University, University Park, PA 16802, Email: firstname.lastname@example.org. Supported in part by the VolkswagenStiftung. t Department of Computer Science, University of Bonn, 53117 Bonn, and the International Computer Science Institute, Berkeley, Cfllfornia. Research supported in part by DFG Grant KA 673/4–1, by the ESPRIT BR Grants 7097 and ECUS030, and by the Volkswagen-Stiftung. Email: email@example.com. edu t Departments of Computer Science and Mathematics, Penn State University, University Park, PA 16802, Email: vorobjov@theory. cse.psu.edu Permission to copy without fee all or part of this material 1s granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy otherwise, or to republish, requires a fee ardor speoific permission. Introduction A problem of testing membership to a semialgebraic set Z was considered by many authors (see, e.g., [B 83], [B 92], [BKL 92], [BL 92], [BLY 92], [MH 85], [Y 92], [Y 94], [YR 80] and the references there). Here we consider a problem of testing membership to a convex polyhedron Pin n-dimensional space lRn. Let P have N facets of all the dimensions, In [MH 85] it was shown, in particular, that for this problem O(log N)n”(l) upper bound is valid for the depth of linear decision trees, in [YR 80] a lower bound ft(log iV) was obtained. A similar question was open for algebraic decision trees. In the present paper we prove a lower bound fl(log IV) for the depth of algebraic decision trees testing membership to P (see the theorem below). Several topological methods were introduced for obtaining lower bounds for the complexity of testing membership to Z by linear decision trees, algebraic decision trees , algebraic computation trees (the definitions one can find in, e.g., [B 83]). STOC 945/94 Montreal, Quebec, Canada @ 1994 ACM 0-89791 -663-6/9410005..$3.50
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