Decision tree complexity and Betti numbers

@inproceedings{Yao1994DecisionTC,
  title={Decision tree complexity and Betti numbers},
  author={Andrew Chi-Chih Yao},
  booktitle={STOC},
  year={1994}
}
We show that any algebraic computation tree or any fixed-degree algebraic tree for solving the membership question of a compact set S ~ R“ must have height greater than Cl(log(@i(S))) – cn for each i, where pi(S) is the i-th Betti number. This generalizes a well-known result by Ben-Or [Be83] who proved this lower bound for the case i = O, and a recent result by Bjorner and Lovtisz [BL92] who proved this lower bound for all i for linear decision trees. 

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