probabilistic, nondeterministic, and alternating decision trees (preliminary version)
This work generalizes decision trees in order to model algorithms which allow probabilistic, nondeterministic, or alternating control. Two geometric techniques for proving lower bounds on the time required by ordinary decision trees (Dobkin and Lipton's -&-ldquo;region-counting-&-rdquo; technique as applied to the knapsack and element uniqueness problems [1], and Reingold's technique as applied to set equality [4]) are shown to be special cases of one unified technique, which in fact applies to nondeterministic decision trees as well. This technique is applied to yield tight upper and lower bounds on the nondeterministic time for solving element uniqueness, set disjointness, set membership, set equality, -&-egr;-closeness [2], and knapsack problems, as well as many of these problems complements. In section 3 we present evidence that probabilistic decision trees have lower bounds matching the deterministic upper bounds for many of the problems mentioned above, and that they are not substantially more powerful for any similar problems. In section 4 it is shown that non-logarithmic lower bounds on the time required by alternating decision trees will not be easy to demonstrate, because such lower bounds would also apply to time on the seemingly more general alternating Turing machine model.
decision tree version general order model algorithm control lower bound timing yield evidence matching demonstrate