7 research outputs found
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËťos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Smoothed analysis of probabilistic roadmaps
The probabilistic roadmap algorithm is a leading heuristic for robot motion planning. It is extremely efficient in practice, yet its worst case convergence time is unbounded as a function of the input’s combinatorial complexity. We prove a smoothed polynomial upper bound on the number of samples required to produce an accurate probabilistic roadmap, and thus on the running time of the algorithm, in an environment of simplices. This sheds light on its widespread empirical success.