Let R and B be sets of n red and n blue points in the plane, respectively, with P = R \cup B. Let M be a perfect matching between points from R and B, using n straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segmentin M with bn(M), which we also call the%DIFDELCMD < \intro[~]{value, bn} %%% of M. We aim to find a matching under given constraints that minimizes bn(M). Any such matching is called a%DIFDELCMD < \intro[~]{bottleneck matching} %%% of P .
Given a set of n red and n blue points in the plane, we are interested in matching red points with blue points by straight line segments so that the segments do not cross. Bottleneck matching is such a matching that minimizes the length of %DIFDELCMD < \intro[~]{value, bn} %%% %DIFDELCMD < \intro[~]{bottleneck matching} %%% the longest segment. We develop a range of tools for dealing with non-crossing matchings of points in convex position, which enable us to solve the problem of finding bottleneck matchings of points in convex position in O(n^2) time. Combining the same set of tools with a geometric analysis, we design an O(n)-time algorithm for the case where the given points lie on a circle. Previously best known results were O(n^3) for points in convex position, and O(n \log n) for points on a circle .
R
Let R and B be sets
Given a set
R
respectively, with P = R \cup B. Let M be a perfect matching between points from R and B, using n
we are interested in matching red points with blue points by
R
to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote
so that the segments do not cross. Bottleneck matching is such a matching that minimizes
D
a longest line segmentin M with bn(M), which we also call the
D
of M. We aim to find a matching under given constraints that minimizes bn(M). Any such matching is called a
R
of P
the longest segment. We develop a range of tools for dealing with non-crossing matchings of points in convex position, which enable us to solve the problem of finding bottleneck matchings of points in convex position in O(n^2) time. Combining the same set of tools with a geometric analysis, we design an O(n)-time algorithm for the case where the given points lie on a circle. Previously best known results were O(n^3) for points in convex position, and O(n \log n) for points on a circle