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A simple property of number representations yields a unit cross section relation between points and interval representations. Applied to product regions in a vector space, one obtains simple, practical and flexible algorithms for dynamic range search and related queries.
We investigate two lowest common ancestor (LCA) problems on trees. We give a linear time algorithm for the off-line problem, on a random access machine (RAM). The half-line problem is one in which LCA queries on a fixed tree are arriving on line. We extend our RAM algorithm to answer each individual query in 0(1) time, with 0(n) preprocessing time. Tarjan observed that this result helps to explicate...
We employ elementary results from the theory of several complex variables to obtain a quadratic lower bound on the complexity of computing the mean distance between points in the plane. This problem has 2N inputs and a single output and we show that exactly N(N-1)/2 square roots must be computed by any program over +, -, ×, ÷,) √, log and comparisons, even allowing an arbitrary field of constants...
Decision procedures for validity in intuitionistic propositional calculus and modal propositional calculus are given which require a running time proportional to a polynomial in the length of the formula on a nondeterministic Turing machine. Using a theorem of Cook's and well-known transformations from intuitionistic to classical and modal to intuitionistic logics, the validity problem for intuitionistic...
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