Abstract
We introduce a new powerful method for provinglower bounds onrandomized anddeterministic analytic decision trees, and give direct applications of our results towards some concrete geometric problems. We design alsorandomized algebraic decision trees for recognizing thepositive octant in ℝn or computing MAX in ℝn in depth logO(1) n. Both problems are known to have linear lower bounds for the depth of any deterministic analytic decision tree recognizing them. The mainnew (andunifying) proof idea of the paper is in the reduction technique of the signs oftesting functions in a decision tree to the signs of theirleading terms at the specially chosen points. This allows us to reduce the complexity of adecision tree to the complexity of a certainBoolean circuit.
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Grigoriev, D., Karpinski, M. & Smolensky, R. Randomization and the computational power of analytic and algebraic decision trees. Comput Complexity 6, 376–388 (1996). https://doi.org/10.1007/BF01270388
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DOI: https://doi.org/10.1007/BF01270388