Abstract

this paper one direction, namely, allowing the bound k to grow with the number of inputs. When k = n, the number of inputs, then the information-theoretic bound is correct. In general, one would like to establish trade-offs between the widths of the queries and the height of the decision tree. In this context, note that Theorem 4.6 can be extended to show that for every r [0.5, 1] there is an nr-bounded decision tree of height O(n2-29 that recognizes EU. The number of distinct input values (i.e., the size of S) has to be very large, especially in the general case, owing to the repeated use of Ramsey's theorem. In fact, it seems that 2(n log n) steps are needed to solve EU, even when the number of distinct values is O(n). Is it possible to avoid the use of Ramsey's theorem, and give a combinatorial proof of the 2(n 1ogn) lower bound, when the domain has size O(n)? The results of this paper can be interpreted as closure theorems, in the following sense. Given a problem that is defined using the order structure of its domain S, turn an 01oumm amuuon eaata tniat uses on the [JlU:l 5truttUlc, imposing additional structure (i.e., defining additional predicates) does not help. Note that the element-uniqueness problem is defined in terms of the equality relation. 1towever, a decision tree that uses only tests for equality requires (n 2) steps to solve EU. Adding a (arbitrary) total-order structure on S helps to solve the problem. The same question, namely, finding a minimal extension of the structure for which a computational problem is defined, such that an optimal solution exists, can be raised for other structures. For example, can one show that if a problem is defined in R" using polynomial inequalities of degree k, then an optimal solution exists that uses only compariso...

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