Decision trees: old and new results

Abstract In this paper, we prove two general lower bounds for algebraic decision trees which test membership in a set S ⊆ R n which is defined by linear inequalities. Let rank( S ) be the maximal dimension of a linear sub- space contained in the closure of S (in Euclidean topology). First we show that any decision tree for S which uses products of linear functions (we call such functions mlf- functions ) must have depth at least n −rank( S ). This solves an open question raised by A. C. Yao and can be used to show that mlf-functions are not really more powerful than simple comparisons between the input variables when computing the largest k out of n elements. Yao proved this result in the special case when products of at most two linear functions are allowed. Our proof also shows that any decision tree for this problem must have exponential size. Using the same methods, we can give an alternative proof of Rabin's theorem, namely that the depth of any decision tree for S using arbitrary analytic functions is at least n −rank( S ).