Decision tree complexity and betti numbers

Abstract

We show that any algebraic computation tree or any fixed-degree algebraic tree for solving the membership question of a compact set S C Rn must have height greater than Ω(log(/β1-(S))) ⊆ rn 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 t = 0, and a recent result by Bjorner and Lovasz [BL92] who proved this lower bound for all i for linear decision trees.