An Exponential Lower Bound on the Size of Algebraic Decision Trees for MAX

TitleAn Exponential Lower Bound on the Size of Algebraic Decision Trees for MAX
Publication TypeTechnical Report
Year of Publication1995
AuthorsGrigoriev DYu., Karpinski M, Yao AC
Other Numbers1006
KeywordsAlgebraic Decision Trees, Hypergraphs, Lower Bounds, MAX Problem, Minimal Cutsets, Selection Problems
Abstract

We prove an exponential lower bound on the size of any fixed -degree algebraic decision tree for solving MAX, the problem of finding the maximum of n real numbers. This complements the n-1 lower bound of Rabin on the depth of algebraic decision trees for this problem. The proof in fact gives an exponential lower bound on size for the polyhedral decision problem MAX= of testing whether the j-th number is the maximum among a list of n real numbers. Previously, except for linear decision trees, no nontrivial lower bounds on the size of algebraic decision trees for any familiar problems are known. We also establish an interesting connection between our lower bound and the maximum number of minimal cutsets for any rank-d hypergraphs on n vertices.

URLhttp://www.icsi.berkeley.edu/ftp/global/pub/techreports/1995/tr-95-066.pdf
Bibliographic Notes

ICSI Technical Report TR-95-066

Abbreviated Authors

D. Grigoriev, M. Karpinski, and A. C. Yao

ICSI Publication Type

Technical Report