问题
I've researched several algorithms for determining whether a point lies in a convex hull, but I can't seem to find any algorithm which can do the trick in O(logn) time, nor can I come up with one myself.Let a[] be an array containing the vertices of the convex hull, can I pre-process this array in anyway, to make it possible to check if a new point lies inside the convex hull in O(logn) time.
回答1:
Looks like you can.
- Sort vertices in
a[]by polar angle relative to one of vertices (call it A). O(N log N), like convex hull computation. - Read point, determine its polar angle. O(1)
- Find two neighbor vertices, one of them should have polar angle less than point from step 2, and other should have angle bigger (B and C). O(log N), binary search
- Then simple geometry: draw the triangle between points from A, B, C and check if point from step 2 lies inside. O(1)
来源：https://stackoverflow.com/questions/28801832/determine-whether-a-point-lies-in-a-convex-hull-in-olog-n-time