David P. Dobkin

A lower bound of 12n2 on linear search programs for the Knapsack problem

Previously the best known lower bound on this problem was 71 log 71 [I]. The result presented here is the first lower bound of better than n log n given for an NP-complete problem for a model that is actually used in practice. Previous non-linear lower bounds have been for computations involving only monotone circuits [8] or fanout limited to one. Our theorem is derived by combining results on linear search tree complexity [4] with results from threshold logic [Ill. In Section 2, we begin by presenting the results on linear search trees and threshold logic. Section 3 is devoted to using these results to obtain our main theorem.

Inclusion complete tally languages and the Hartmanis-Berman conjecture

Stimulated by the work of Hartmanis and Berman [5], we study the question of the existence of a tally languageL inNP such thatL is inP if and only ifP = NP. The method used is applicable to questions regarding the comparison of a wide range of pairs of classes of formal languages specified by machines whose computational resources are bounded in time or space.

On some generalizations of binary search

Classic binary search is extended to multidimensional search problems. These new search methods can efficiently solve several important problems of computer science. Applications of these results to an open problem in the theory of computation are discussed yielding new insight into the Lba problem.

A nonlinear lower bound on linear search tree programs for solving knapsack problems

By the method of region counting, a lower bound of n log2nn queries is obtained onlinear search tree programs which solve the n-dimensional knapsack problem. The region counting involves studying the structure of a subset of the hyperplanes defined by the problem. For this subset of hyperplanes, the result is shown to be tight.

Time and Space Bounds for Selection Problems

The complexity of a number of selection problems is considered. An algorithm is given to determine the mode of a multiset in a number of comparisons differing from the lower bound by only a lower order term. The problems of finding the kth largest element in a set in minimal and near minimal space are also discussed. A time space tradeoff is demonstrated for these problems.

Complexity measures and hierarchies for the evaluation of integers and polynomials

Abstract The complexity of evaluating integers and polynomials is studied. A new model is proposed for studying such complexities. This model differs from previous models by requiring the construction of constant to be used in the computation. This construction is given a cost which is dependent upon the size of the constant. Previous models used a uniform cost, of either 0 or 1, for operations involving constants. Using this model, proper hierarchies are shown to exist for both integers and polynomials with respect to evaluation cost. Furthmore, it is shown that almost all integers (polynomials) are as difficult to evaluate as the hardest integer (polynomial). These results remain true even if the underlying basis of binary operations which the algorithm performs are varied.

A lower bound of 1/2n 2 on linear search programs for the knapsack problem

Previously the best known lower bound on this problem was 71 log 71 [I]. The result presented here is the first lower bound of better than n log n given for an NP-complete problem for a model that is actually used in practice. Previous non-linear lower bounds have been for computations involving only monotone circuits [8] or fanout limited to one. Our theorem is derived by combining results on linear search tree complexity [4] with results from threshold logic [Ill. In Section 2, we begin by presenting the results on linear search trees and threshold logic. Section 3 is devoted to using these results to obtain our main theorem.