Abstract
We describe two methods for estimating the size and depth of decision trees where a linear test is performed at each node. Both methods are applied to the question of deciding, by a linear decision tree, whether given n real numbers, some k of them are equal. We show that the minimum depth of a linear decision tree for this problem is Θ(n log(n/k)). The upper bound is easy; the lower bound can be established for k = O(n1/4−e) by a volume argument; for the whole range, however, our proof is more complicated and it involves the use of some topology as well as the theory of Mobius functions.
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