56 research outputs found
Linear transformation distance for bichromatic matchings
Let be a set of points in general position, where is a
set of blue points and a set of red points. A \emph{-matching}
is a plane geometric perfect matching on such that each edge has one red
endpoint and one blue endpoint. Two -matchings are compatible if their
union is also plane.
The \emph{transformation graph of -matchings} contains one node for each
-matching and an edge joining two such nodes if and only if the
corresponding two -matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of for its diameter, which is asymptotically
tight
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
Bichromatic compatible matchings
Abstract For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BRmatchings are compatible if their union is also non-crossing. We prove that, for any two distinct BRmatchings M and M , there exists a sequence of BR-matchings M = M 1 , . . . , M k = M such that M i−1 is compatible with M i . This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [6]
Bichromatic compatible matchings
ABSTRACT For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BR-matchings are compatible if their union is also noncrossing. We prove that, for any two distinct BR-matchings M and M , there exists a sequence of BR-matchings M = M1, . . . , M k = M such that Mi−1 is compatible with Mi. This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [5]
Disjoint compatibility graph of non-crossing matchings of points in convex position
Let be a set of labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
. Two such matchings, and , are disjoint compatible if they do
not have common edges, and no edge of crosses an edge of . Denote by
the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each , the connected components of
form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.Comment: 46 pages, 30 figure
Flip Distance to some Plane Configurations
We study an old geometric optimization problem in the plane. Given a perfect matching M on a set of n points in the plane, we can transform it to a non-crossing perfect matching by a finite sequence of flip operations. The flip operation removes two crossing edges from M and adds two non-crossing edges. Let f(M) and F(M) denote the minimum and maximum lengths of a flip sequence on M, respectively. It has been proved by Bonnet and Miltzow (2016) that f(M)=O(n^2) and by van Leeuwen and Schoone (1980) that F(M)=O(n^3). We prove that f(M)=O(n Delta) where Delta is the spread of the point set, which is defined as the ratio between the longest and the shortest pairwise distances. This improves the previous bound for point sets with sublinear spread. For a matching M on n points in convex position we prove that f(M)=n/2-1 and F(M)={{n/2} choose 2}; these bounds are tight.
Any bound on F(*) carries over to the bichromatic setting, while this is not necessarily true for f(*). Let M\u27 be a bichromatic matching. The best known upper bound for f(M\u27) is the same as for F(M\u27), which is essentially O(n^3). We prove that f(M\u27)<=slant n-2 for points in convex position, and f(M\u27)= O(n^2) for semi-collinear points.
The flip operation can also be defined on spanning trees. For a spanning tree T on a convex point set we show that f(T)=O(n log n)
Flip Distance to a Non-crossing Perfect Matching
A perfect straight-line matching on a finite set of points in the
plane is a set of segments such that each point in is an endpoint of
exactly one segment. is non-crossing if no two segments in cross each
other. Given a perfect straight-line matching with at least one crossing,
we can remove this crossing by a flip operation. The flip operation removes two
crossing segments on a point set and adds two non-crossing segments to
attain a new perfect matching . It is well known that after a finite number
of flips, a non-crossing matching is attained and no further flip is possible.
However, prior to this work, no non-trivial upper bound on the number of flips
was known. If (resp.~) is the maximum length of the longest
(resp.~shortest) sequence of flips starting from any matching of size , we
show that and (resp.~ and
)
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