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Journal Article

#### Decision trees: old and new results

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##### Citation

Fleischer, R. (1999). Decision trees: old and new results.*
Information and Computation,* *152*, 44-61.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-35B6-3

##### Abstract

In this paper, we prove two general lower bounds for algebraic
decision trees which test membership in a set $S\subseteq R^n$ which is
defined by linear inequalities.
Let $rank(S)$ be
the maximal dimension of a linear subspace contained in the closure of
$S$ {in the Euclidean topology}.
First we show that any decision tree for $S$ which uses
products of linear functions (we call such functions
{\em 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.
Out 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 $n-rank(S)$.