How can geometric optimization improve your algorithm?

1 Dimensionality reduction

One way to improve your algorithm is to reduce the dimensionality of the problem, which means working with fewer variables or coordinates. This can make the problem easier to solve, as it reduces the complexity and the memory requirements of the algorithm. For example, if you want to find the closest pair of points among a set of points in the plane, you can use a divide-and-conquer algorithm that splits the plane into smaller regions and recursively compares the distances within and across the regions. This algorithm takes O(n log n) time, where n is the number of points. However, if you want to find the closest pair of points among a set of points in the three-dimensional space, the same algorithm would take O(n log^2 n) time, which is worse. To reduce the dimensionality, you can use a technique called random projection, which maps the points from the three-dimensional space to a lower-dimensional space, such as the plane, while preserving the distances between them with high probability. This way, you can use the same algorithm as before, but with a lower time complexity.

3 Structure exploitation

A third way to improve your algorithm is to exploit the structure of the problem, which means using the properties or the patterns of the geometric objects to guide the algorithm. This can make the problem easier to solve, as it reduces the search space and the number of operations that the algorithm needs to perform. For example, if you want to find the convex hull of a set of points in the plane, which is the smallest polygon that contains all the points, you can use a brute-force algorithm that checks every possible polygon and selects the smallest one. This algorithm takes O(n^3) time, where n is the number of points. However, if you exploit the structure of the problem, you can use a technique called Graham scan, which sorts the points by their polar angle and then scans them in a counterclockwise order, discarding the points that are not part of the convex hull. This algorithm takes O(n log n) time, which is much faster.

4 Approximation techniques

A fourth way to improve your algorithm is to use approximation techniques, which means finding solutions that are close enough to the optimal ones, but not necessarily exact. This can make the problem easier to solve, as it relaxes the constraints and the requirements of the algorithm. For example, if you want to find the smallest circle that contains a set of points in the plane, which is called the minimum enclosing circle, you can use an exact algorithm that iterates over all the possible circles and selects the smallest one. This algorithm takes O(n^4) time, where n is the number of points. However, if you use an approximation technique, you can use a randomized algorithm that selects a random point and grows a circle around it until it covers all the points, and then repeats this process until it finds a good enough circle. This algorithm takes O(n) expected time, which is much faster, but it may not find the exact minimum enclosing circle.