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TR95-063 | 19th December 1995 00:00

A Lower Bound for Randomized Algebraic Decision Trees



We extend the lower bounds on the depth of algebraic decision trees
to the case of {\em randomized} algebraic decision trees (with
two-sided error) for languages being finite unions of hyperplanes
and the intersections of halfspaces, solving a long standing open
problem. As an application, among other things, we derive, for the
first time, an $\Omega(n^2)$ {\em randomized} lower bound for the
{\em Knapsack Problem} which was previously only known for
deterministic algebraic decision trees. It is worth noting that for
the languages being finite unions of hyperplanes our proof method
yields also a new elementary technique for deterministic algebraic
decision trees without making use of Milnor's bound on Betti number
of algebraic varieties.

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