Chan's algorithm

In computational geometry, Chan's algorithm, named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set of points, in 2- or 3-dimensional space. The algorithm takes time, where is the number of vertices of the output (the convex hull). In the planar case, the algorithm combines an algorithm (Graham scan, for example) with Jarvis march (), in order to obtain an optimal time. Chan's algorithm is notable because it is much simpler than the Kirkpatrick–Seidel algorithm, and it naturally extends to 3-dimensional space. This paradigm has been independently developed by Frank Nielsen in his Ph.D. thesis. A single pass of the algorithm requires a parameter which is between 0 and (number of points of our set ). Ideally, but , the number of vertices in the output convex hull, is not known at the start. Multiple passes with increasing values of are done which then terminates when (see below on choosing parameter ). The algorithm starts by arbitrarily partitioning the set of points into subsets with at most points each; notice that . For each subset , it computes the convex hull, , using an algorithm (for example, Graham scan), where is the number of points in the subset. As there are subsets of points each, this phase takes time. During the second phase, Jarvis's march is executed, making use of the precomputed (mini) convex hulls, . At each step in this Jarvis's march algorithm, we have a point in the convex hull (at the beginning, may be the point in with the lowest y coordinate, which is guaranteed to be in the convex hull of ), and need to find a point such that all other points of are to the right of the line , where the notation simply means that the next point, that is , is determined as a function of and . The convex hull of the set , , is known and contains at most points (listed in a clockwise or counter-clockwise order), which allows to compute in time by binary search. Hence, the computation of for all the subsets can be done in time.