Equivalence of Binary and Ternary Algebraic Decision Trees

Abstract.

Two models of algebraic decision trees have been studied. The binary model allows queries of the form ``is p(x ₁ ,... , x _n ) ≤ 0?'', while in the ternary model the value of the query polynomial is compared with zero and one of three possible answers ( < ,=, > ) is returned, each answer leading to a distinct subtree.

It is straightforward to simulate a ternary tree with a binary tree at a cost of doubling the size and depth of the tree. However, no simulation in the other direction was previously known. Indeed, Grigoriev et al. [4] have recently shown that if the degree of the query polynomials is bounded by a constant, then there exist binary decision trees for which such a simulation necessarily entails an exponential increase in size.

We show that a binary algebraic decision tree T can be converted to a ternary algebraic decision tree T' at the expense of a quadratic increase in size and a linear increase in depth.