Boundary-Optimal Triangulation Flooding

Given a planar triangulation all of whose faces are initially white, we study the problem of colouring the faces black one by one so that the boundary between black and white faces as well as the number of connected black and white regions are small at all times. We call such a colouring sequence of the triangles a flooding. We prove that a flooding with boundary size O( √ n) and O(log n) regions can be computed in O(n logn) time. We also prove that it is in general impossible to guarantee boundary size O(n1−ǫ), for any ǫ > 0, and a number of regions that is o(logn), where n is the number of faces of the triangulation.