Computational Geometry,Generalized Intersection Searching:Geometric Intersection Searching Problems

Geometric Intersection Searching Problems

Problems arising in diverse areas, such as VLSI layout design (Chapter 52), database querying (Chapter 60), robotics, and computer graphics (Chapter 54) can often be formulated as geometric intersection searching problems. In a generic instance of such a problem, a set, S, of geometric objects is to be preprocessed into a suitable data structure so that given a query object, q, we can answer efficiently questions regarding the intersection of q with the objects in S. The problem comes in four versions, depending on whether we want to report the intersected objects or simply count their number—the reporting version and the counting version, respectively—and whether S remains fixed or changes through insertion and deletion of objects—the static version and the dynamic version, respectively. In the dynamic version, which arises very often owing to the highly interactive nature of the above-mentioned applications, we wish to perform the updates more efficiently than simply recomputing the data structure from scratch after each update, while simultaneously maintaining fast query response times. We call these problems standard intersection searching problems in order to distinguish them from the generalized intersection searching problems that are the focus of this chapter. Due to their numerous applications, standard intersection searching problems have been the subject of much study and efficient solutions have been devised for many of them (see, for instance, [4, 13] and the references therein).

The efficiency of a standard intersection searching algorithm is measured by the space used by the data structure, the query time, and, in the dynamic setting, the update time. In a counting problem, these are expressed as a function of the input size n (i.e., the size of S); in a reporting problem, the space and update time are expressed as a function of n, whereas the query time is expressed as a function of both n and the output size k (i.e., the number of intersected objects) and is typically of the form O(f (n)+ k) or O(f (n)+ k · g(n)), for some functions f and g. Such a query time is called output-sensitive.

Generalized Intersection Searching

In many applications, a more general form of intersection searching arises: Here the objects in S come aggregated in disjoint groups and of interest are questions regarding the intersection of q with the groups rather than with the objects. (q intersects a group if and only if it intersects some object in the group.) In our discussion, it will be convenient to associate with each group a different color and imagine that all the objects in the group have that color. Then, in the generalized reporting (resp., generalized counting) problem, we want to report (resp., count) the distinct colors intersected by q; in the dynamic setting, an object of some (possibly new) color is inserted in S or an object in S is deleted. Note that the generalized problem reduces to the standard one when each color class has cardinality 1.

We give two examples of such generalized problems: Consider a database of mutual funds which contains for each fund its annual total return and its beta (a real number measuring the fund’s volatility). Thus each fund can be represented as a point in two dimensions. Moreover, funds are aggregated into groups according to the fund family they belong to. A typical query is to determine the families that offer funds whose total return is between, say, 15% and 20%, and whose beta is between, say, 0.9 and 1.1. This is an instance of the generalized 2-dimensional range searching problem. The output of this query enables a potential investor to initially narrow his/her search to a few families instead of having to plow through dozens of individual funds (all from the same small set of families) that meet these criteria. As another example, in the Manhattan layout of a VLSI chip, the wires (line segments) can be grouped naturally according to the circuits they belong to. A problem of interest to the designer is determining which circuits (rather than wires) become electrically connected when a new wire is added. This is an instance of the generalized orthogonal segment intersection searching problem.

One approach to solving a generalized problem is to try to take advantage of solutions known for the corresponding standard problem. For instance, we can solve a generalized reporting problem by first determining the objects intersected by q (a standard reporting problem) and then reading off the distinct colors. However, the query time can be very high since q could intersect Ω(n) objects but only O(1) distinct colors. For a generalized reporting problem, we seek query times that are sensitive to the number, i, of distinct colors intersected, typically of the form O(f (n) + i) or O(f (n) + i · g(n)), where f and g are polylogarithmic. (This is attainable using the approach just described if each color class has cardinality O(1). On the other hand, if there are only O(1) different color classes, we could simply run a standard algorithm on each color class in turn, stopping as soon as an intersection is found and reporting the corresponding color. The real challenge is when the number of color classes and the cardinalities of the color classes are not constants, but rather are (unknown) functions of n; throughout, we will assume this to be the case.) For a generalized counting problem, the situation is worse; it is not even clear how one can extract the answer for such a problem from the answer (a mere count) to the corresponding standard problem. One could, of course, solve the corresponding reporting problem and then count the colors, but this is not efficient. Thus it is clear that different techniques are needed.

In this chapter, we describe the research that has been conducted over the past few years on generalized intersection searching problems. We begin with a brief review of known results and then discuss a variety of techniques for these problems. For each technique, we give illustrative examples and provide pointers to related work. We conclude with a discussion of possible directions for further research.