Samuel R. Buss

Spherical averages and applications to spherical splines and interpolation

This article introduces a method for computing weighted averages on spheres based on least squares minimization that respects spherical distance. We prove existence and uniqueness properties of the weighted averages, and give fast iterative algorithms with linear and quadratic convergence rates. Our methods are appropriate to problems involving averages of spherical data in meteorological, geophysical, and astronomical applications. One simple application is a method for smooth averaging of quaternions, which generalizes Shoemakes spherical linear interpolation.The weighted averages methods allow a novel method of defining Bézier and spline curves on spheres, which provides direct generalization of Bézier and B-spline curves to spherical spline curves. We present a fast algorithm for spline interpolation on spheres. Our spherical splines allow the use of arbitrary knot positions; potential applications of spherical splines include smooth quaternion curves for applications in graphics, animation, robotics, and motion planning.

The Boolean formula value problem is in ALOGTIME

The Boolean formula value problem is to determine the truth value of a variable-free Boolean formula, or equivalently, to recognize the true Boolean sentences. N. Lynch [ll] gave log space algorithms for the Boolean formula value problem and for the more general problem of recognizing a parenthesis context-free grammar. This paper shows that these problems have alternating log time algorithms. This answers the question of Cook [5] of whether the Boolean formula value problem is log space complete it is not, unless log space and alternating log time are identical. Our results are optimal since, for an appropriately defined notion of fog time reductions, the Boolean formula value problem is complete for alternating log time under deterministic log time reductions; consequently, it is al30 complete for alternating log time under AC0 reductions. It follows that the Boolean formula value problem is not in the log time hierarchy. There are two reasons why the Boolean formula value problem is interesting. First, a Boolean (or propositional) formula is a very fundamental concept of logic. The computational complexity of evaluating a Boolean formula is therefore of interest. Indeed, the results below will give a precise characterisation of the computational complexity of determining the truth value of a Boolean formula. Second, the existence of an alternating log time algorithm for the Boolean formula problem implies the existence of log depth, polynomial size circuits for this problem and hence there are (at least theoretically) good parallel algorithms for determining the value of a Boolean sentence. As mentioned above, N. Lynch [ll] first studied the complexity of the Boolean formula problem. It follows from Lynch’s work that the Boolean formula value problem is in NC’, since Borodin [l] showed that LOCSPACE C NCZ. Another early significant result on this problem was due to Spira [17] who showed that for every formula of size n, there is an equivalent formula of size O(n2) and depth O(log n). An improved construction, which also applied to the evaluation of rational expressions, was obtained by Brent [z]. Spira’s result WAS significant in part because because it implied that there might be a family of polynomial size, log depth circuits for recognizing true Boolean formulas. In other words, that the Boolean formula value problem might be in (non-uniform) NC’. However, it was not known if the transformations of formulas defined by Brent and Spira could

Selectively Damped Least Squares for Inverse Kinematics

We introduce two methods for the inverse kinematics of multibodies with multiple end effectors. The first method clamps the distance of the target positions. Experiments show this is effective in reducing oscillation when target positions are unreachable. The second method is an extension of damped least squares called selectively damped least squares (SDLS), which adjusts the damping factor separately for each singular vector of the Jacobian singular value decomposition based on the difficulty of reaching the target positions. SDLS has advantages in converging in fewer iterations and in not requiring ad-hoc damping constants. Source code is available online.

Polynomial size proofs of the propositional pigeonhole principle

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquharts theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

An optimal parallel algorithm for formula evaluation

A new approach to Buss’s

On truth-table reducibility to SAT

{\textbf{NC}}^1

Resolution proofs of generalized pigeonhole principles

algorithm [Proc. 19th ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1987, pp. 123–131] for evaluation of Boolean formulas is presented. This problem is shown to be complete for

A Switching Lemma for Small Restrictions and Lower Bounds for k -DNF Resolution

{\textbf{NC}}^1

Resolution and the Weak Pigeonhole Principle

over

Proof complexity in algebraic systems and bounded depth Frege systems with modular counting

{\textbf{AC}}^0