Geometric Approximation Algorithms

The Peace of Olivia. How sweat and peaceful it sounds! There the great powers noticed for the first time that the land of the Poles lends itself admirably to partition. – The tin drum, Gunter Grass In this chapter, we are going to discuss two basic geometric algorithms. The first one, computes the closest pair among a set of n points in linear time. This is a beautiful and surprising result that exposes the computational power of using grids for geometric computation. Next, we discuss a simple algorithm for approximating the smallest enclosing ball that contains k points of the input. This at first looks like a bizarre problem, but turns out to be a key ingridiant to our later discussion. For r a real positive number and a point p = (x, y) in R 2 , define G r (p) to be the point (x/r r, y/r r). We call r the width of the grid G r. Observe that G r partitions the plane into square regions, which we call grid cells. Formally, for any i, j ∈ Z, the intersection of the half-planes x ≥ ri, x < r(i + 1), y ≥ r j and y < r(j + 1) is said to be a grid cell. Further we define a grid cluster as a block of 3 × 3 contiguous grid cells. Note, that every grid cell C of G r , has a unique ID; indeed, let p = (x, y) be any point in C, and consider the pair of integer numbers id C = id(p) = (x/r , y/r). Clearly, only points inside C are going to be mapped to id C. This is very useful, since we store a set P of points inside a grid efficiently. Indeed, given a point p, compute its id(p). We associate with each unique id a data-structure that stores all the points falling into this grid cell (of course, we do not maintain such data-structures for grid cells which are empty). So, once we computed id(p), we fetch the data structure for this cell, by using hashing. Namely, we store pointers to all those data-structures in a hash table, where each such data-structure is indexed by its unique id. Since the ids are integer numbers, we can do the hashing in constant time. Assumption 1.1.1 Through out the discourse, we are going to assume …