Compatible Geometric Matchings Elizabeth Kupin May 12th, 2011 24th - PowerPoint PPT Presentation

Compatible Geometric Matchings Elizabeth Kupin May 12th, 2011 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing at the University of Louisville ekupin@math.rutgers.edu

Geometric Graph Theory Def : A geometric graph has a fixed embedding into the plane, where all the edges are embedded as straight line segments.

Geometric Graph Theory Def : A geometric graph has a fixed embedding into the plane, where all the edges are embedded as straight line segments. Def : A perfect geometric matching is a geometric graph of a perfect matching. In this talk, we also require that the matching is non-crossing, and that the vertices are in general position (i.e. no 3 collinear).

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings.

Compatible Matchings Def : Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing. Proposition 1 : Every even set of points in general position admits a pair of compatible perfect geometric matchings. c e a b f d a < c + d b < e + f From the triangle inequality, we see that the sum of the lengths of the edges in the matchings decreases with every step.

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching?

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No! Question (revised): If an adversary gives us an even perfect geometric matching, can we always find a second compatible one?

Main Question Question : If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No! Question (revised): If an adversary gives us an even perfect geometric matching, can we always find a second compatible one? In some special cases yes, but the full question remains open.

Overview Compatible Matching Conjecture: given an even perfect geometric matching we can find a second, compatible perfect geometric matching. This conjecture grew out of the Workshop on Combinatorial Geometry in 2006, and was officially introduced by Aichholzer et al. in 2008.

Overview Compatible Matching Conjecture: given an even perfect geometric matching we can find a second, compatible perfect geometric matching. This conjecture grew out of the Workshop on Combinatorial Geometry in 2006, and was officially introduced by Aichholzer et al. in 2008. Their paper proves the conjecture when the edges in the original matching are convex hull connected. Based on the proof of the convex hull connected case, we get the following new result:

Overview Compatible Matching Conjecture: given an even perfect geometric matching we can find a second, compatible perfect geometric matching. This conjecture grew out of the Workshop on Combinatorial Geometry in 2006, and was officially introduced by Aichholzer et al. in 2008. Their paper proves the conjecture when the edges in the original matching are convex hull connected. Based on the proof of the convex hull connected case, we get the following new result: Main Theorem : (K, 2010) For any even perfect geometric matching there is a compatible perfect matching, whose edges are piecewise linear paths with at most two bends.

Convex Hull Connected Matchings Def : A matching is convex hull connected if every edge has at least one endpoint on the convex hull.

Convex Hull Connected Matchings Def : A matching is convex hull connected if every edge has at least one endpoint on the convex hull. Theorem 1 : (Aichholzer et al., ‘08) If a perfect geometric matching is even and convex hull connected, we can find a compatible perfect geometric matching.

Convex Hull Connected Matchings Def : A matching is convex hull connected if every edge has at least one endpoint on the convex hull. Theorem 1 : (Aichholzer et al., ‘08) If a perfect geometric matching is even and convex hull connected, we can find a compatible perfect geometric matching. Proof : (algorithm) Step 1: Cast out splitters, i.e. edges with both endpoints on the convex hull but not adjacent.

Convex Hull Connected Matchings Def : A matching is convex hull connected if every edge has at least one endpoint on the convex hull. Theorem 1 : (Aichholzer et al., ‘08) If a perfect geometric matching is even and convex hull connected, we can find a compatible perfect geometric matching. Proof : (algorithm) Step 1: Cast out splitters, i.e. edges with both endpoints on the convex hull but not adjacent.

Convex Hull Connected Matchings Def : A matching is convex hull connected if every edge has at least one endpoint on the convex hull. Theorem 1 : (Aichholzer et al., ‘08) If a perfect geometric matching is even and convex hull connected, we can find a compatible perfect geometric matching. Proof : (algorithm) Step 1: Cast out splitters, i.e. edges with both endpoints on the convex hull but not adjacent. Step 2: Take alternating gaps along the perimeter.

Convex Hull Connected Matchings Def : A matching is convex hull connected if every edge has at least one endpoint on the convex hull. Theorem 1 : (Aichholzer et al., ‘08) If a perfect geometric matching is even and convex hull connected, we can find a compatible perfect geometric matching. Proof : (algorithm) Step 1: Cast out splitters, i.e. edges with both endpoints on the convex hull but not adjacent. Step 2: Take alternating gaps along the perimeter. To find the second half of the matching, we will create a polygon and then apply an earlier result: the Polygon Lemma.

Polygon Lemma Polygon Lemma : (Abellanas et al., ‘05) For any polygon P and even set S of points on the perimeter of P , there is a perfect geometric matching of S that is contained in P . 1 1 Technically this is only true with an additional (but uninteresting) condition on S , that all the sets we consider will satisfy.

Polygon Lemma Polygon Lemma : (Abellanas et al., ‘05) For any polygon P and even set S of points on the perimeter of P , there is a perfect geometric matching of S that is contained in P . 1 Step 3: Open a wedge around every edge that has an endpoint not on the convex hull, to create a polygon. 1 Technically this is only true with an additional (but uninteresting) condition on S , that all the sets we consider will satisfy.

Polygon Lemma Polygon Lemma : (Abellanas et al., ‘05) For any polygon P and even set S of points on the perimeter of P , there is a perfect geometric matching of S that is contained in P . 1 Step 3: Open a wedge around every edge that has an endpoint not on the convex hull, to create a polygon. 1 Technically this is only true with an additional (but uninteresting) condition on S , that all the sets we consider will satisfy.