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Journal of the ACM (JACM)
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ACM Computing Surveys (CSUR)
A lower bound for randomized algebraic decision trees
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Randomized Ω(n2) lower bound for knapsack
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
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STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
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Optimal randomized comparison based algorithms for collision
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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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.