We discuss applications of quantum computation to geometric data processing. Especially, we give efficient algorithms for intersection problems and proximity problems. Our algorithms are based on Brassard et al. 's amplitude amplification method, and analogous to Buhrman et al. 's algorithm for element distinctness. Revealing these applications is useful for classifying geometric problems, and also emphasizing potential usefulness of quantum computation in geometric data processing. Thus, the results will promote research and development of quantum computers and algorithms.

In this paper, we investigate inverse problems of the interval query problem in application to data mining. Let I be the set of all intervals on U = {1, 2, , n}. Consider an objective function f(I), conditional functions ui(I) on I, and define an optimization problem of finding the interval I maximizing f(I) subject to ui(I) > Ki for given real numbers Ki (i = 1, 2, , h). We propose efficient alogorithms to solve the above optimization problem if the objective function is either additive or quotient, and the conditional functions are additive, where a function f is additive if f(I) = ΣiIf^(i) extending a function f^ on U, and quotient if it is represented as a quotient of two additive functions. We use computational-geometric methods such as convex hull, range searching, and multidimensional divide-and-conquer.

This paper addresses the problem of detecting digital line components in a given binary image consisting of n black dots arranged over N N integer grids. The most popular method in computer vision for this purpose is the one called Hough Transform which transforms each black point to a sinusoidal curve to detect digital line components by voting on the dual plane. We start with a definition of a line component to be detected and present several different algorithms based on the definition. The one extreme is the conventional algorithm based on voting on the subdivided dual plane while the other is the one based on topological walk on an arrangement of sinusoidal curves defined by the Hough transform. Some intermediate algorithm based on half-planar range counting is also presented. Finally, we discuss how to incorporate several practical conditions associated with minimum density and restricted maximality.

In [5], the following pyramid construction problem was proposed: Given nonnegative valued functions ρ and µ in d variables, we consider the optimal pyramid maximizing the total parametric gain of ρ against µ. The pyramid can be considered as the optimal unimodal approximation of ρ relative to µ, and can be applied to hierarchical data segmentation. In this paper, we give efficient algorithms for a couple of two-dimensional pyramid construction problems.

This paper presents an efficient algorithm for constructing at-most-k levels of an arrangement of n lines in the plane in time O(nk+n log n), which is optimal since Ω(nk) line segments are included there. The algorithm can sweep the at-most-k levels of the arrangement using O(n) space. Although Everett et al. recently gave an algorithm for constructing the at-most-k levels with the same time complexity independently, our algorithm is superior with respect to the space complexity as a sweep algorithm. Then, we apply the algorithm to a bipartitioning problem of a bichromatic point set： For r red points and b blue points in the plane and a directed line L, the figure of demerit fd(L) associated with L is defined to be the sum of the number of blue points below L and that of red ones above L. The problem we are going to consider is to find an optimal partitioning line to minimize the figure of demerit. Given a number k, our algorithm first determines whether there is a line whose figure of demerit is at most k, and further finds an optimal bipartitioning line if there is one. It runs in O(kn+n log n) time (n=r+b), which is subquadratic if k is sublinear.

Topological Walk is an algorithm that can sweep an arrangement of n lines in O(n2) time and O(n) space. This paper revisits Topological Walk to give its new interpretation in contrast with Topological Sweep. We also survey applications of Topological Walk to make the distinction clearer.

We give a unified view to greedy geometric routing algorithms in ad hoc networks. For this, we first present a general form of greedy routing algorithm using a class of objective functions which are invariant under congruent transformations of a point set. We show that several known greedy routing algorithms such as Greedy Routing, Compass Routing, and Midpoint Routing can be regarded as special cases of the generalized greedy routing algorithm. In addition, inspired by the unified view of greedy routing, we propose three new greedy routing algorithms. We then derive a sufficient condition for our generalized greedy routing algorithm to guarantee packet delivery on every Delaunay graph. This condition makes it easier to check whether a given routing algorithm guarantees packet delivery, and it is closed under convex linear combination of objective functions. It is shown that Greedy Routing, Midpoint Routing, and the three new greedy routing algorithms proposed in this paper satisfy the sufficient condition, i.e., they guarantee packet delivery on Delaunay graphs. We also discuss merits and demerits of these methods.

Given a combinatorial problem on a set of weighted elements, if we change the weight using a parameter, we obtain a parametric version of the problem, which is often used as a tool for solving mathematical programming problems. One interesting question is how to describe and analyze the trajectory of the solution. If we consider the trajectory of each weight function as a curve in a plane, we have a set of curves from the problem instance. The curves induces a cell complex called an arrangement, which is a popular research target in computational geometry. Especially, for the parametric version of the problem of computing the minimum weight base of a matroid or polymatroid, the trajectory of the solution becomes a subcomplex in an arrangement. We introduce the interaction between the two research areas, combinatorial optimization and computational geometry, through this bridge.