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Compatible Geometric Matchings Elizabeth Kupin May 12th, 2011 24th - - PowerPoint PPT Presentation
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
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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).
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Compatible Matchings
Def: Two geometric matchings are compatible if they are disjoint and (mutually) non-crossing.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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. a b c d e f 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.
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Main Question
Question: If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching?
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Main Question
Question: If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!
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Main Question
Question: If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!
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Main Question
Question: If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!
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Main Question
Question: If an adversary gives us a perfect geometric matching, can we always find a second, compatible matching? No!
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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?
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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.
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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.
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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:
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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.
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Convex Hull Connected Matchings
Def: A matching is convex hull connected if every edge has at least one endpoint on the convex hull.
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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.
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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.
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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.
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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.
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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.
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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
1Technically this is only true with an additional (but uninteresting)
condition on S, that all the sets we consider will satisfy.
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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.
1Technically this is only true with an additional (but uninteresting)
condition on S, that all the sets we consider will satisfy.
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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.
1Technically this is only true with an additional (but uninteresting)
condition on S, that all the sets we consider will satisfy.
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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. Apply the Polygon Lemma to match up the remaining points.
1Technically this is only true with an additional (but uninteresting)
condition on S, that all the sets we consider will satisfy.
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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. Apply the Polygon Lemma to match up the remaining points. Step 4: Combine the edges from steps 2 and 3 to obtain the second, compatible perfect matching.
1Technically this is only true with an additional (but uninteresting)
condition on S, that all the sets we consider will satisfy.
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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. Apply the Polygon Lemma to match up the remaining points. Step 4: Combine the edges from steps 2 and 3 to obtain the second, compatible perfect matching.
1Technically this is only true with an additional (but uninteresting)
condition on S, that all the sets we consider will satisfy.
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Main Result
Theorem 2: (K, ‘10) For every even perfect geometric matching, there is a compatible matching that uses piecewise linear paths with at most two bends in place of line segments.
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Main Result
Theorem 2: (K, ‘10) For every even perfect geometric matching, there is a compatible matching that uses piecewise linear paths with at most two bends in place of line segments. Proof: First break down the region into smaller pieces:
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Main Result
Theorem 2: (K, ‘10) For every even perfect geometric matching, there is a compatible matching that uses piecewise linear paths with at most two bends in place of line segments. Proof: First break down the region into smaller pieces:
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Main Result
Theorem 2: (K, ‘10) For every even perfect geometric matching, there is a compatible matching that uses piecewise linear paths with at most two bends in place of line segments. Proof: First break down the region into smaller pieces: Forces every edge to have at least one endpoint
- n the perimeter of a
new, smaller region. We call these new regions perimeter closed.
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Main Result
Theorem 2: (K, ‘10) For every even perfect geometric matching, there is a compatible matching that uses piecewise linear paths with at most two bends in place of line segments. Proof: First break down the region into smaller pieces: Forces every edge to have at least one endpoint
- n the perimeter of a
new, smaller region. We call these new regions perimeter closed. Within each region, apply the same proof technique as for the convex hull connected case. In some places we shift things apart by ǫ, so we don’t have strict overlap.
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Main Result
Theorem 2: (K, ‘10) For every even perfect geometric matching, there is a compatible matching that uses piecewise linear paths with at most two bends in place of line segments. Proof: First break down the region into smaller pieces: Forces every edge to have at least one endpoint
- n the perimeter of a
new, smaller region. We call these new regions perimeter closed. Within each region, apply the same proof technique as for the convex hull connected case. In some places we shift things apart by ǫ, so we don’t have strict overlap.
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Two Bend Limit
Consider a piecewise linear edge with many bends. Idea: Replace long portions of the path with single line segments, while remaining in the interior of the region. Exterior of Region Interior of Region
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Two Bend Limit
Consider a piecewise linear edge with many bends. Idea: Replace long portions of the path with single line segments, while remaining in the interior of the region. Exterior of Region Interior of Region
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Two Bend Limit
Consider a piecewise linear edge with many bends. Idea: Replace long portions of the path with single line segments, while remaining in the interior of the region. Exterior of Region Interior of Region
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Two Bend Limit
Consider a piecewise linear edge with many bends. Idea: Replace long portions of the path with single line segments, while remaining in the interior of the region. Exterior of Region Interior of Region
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Two Bend Limit
Consider a piecewise linear edge with many bends. Idea: Replace long portions of the path with single line segments, while remaining in the interior of the region. Exterior of Region Interior of Region One piecewise linear path ends up with fewer segments and/or broken up over more points; both decrease the number of bends.
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Two Bend Limit
Consider a piecewise linear edge with many bends. Idea: Replace long portions of the path with single line segments, while remaining in the interior of the region. Exterior of Region Interior of Region One piecewise linear path ends up with fewer segments and/or broken up over more points; both decrease the number of bends. A small amount of further analysis shows that at the end of this process there can be at most two bends between adjacent points
- n the perimeter, and that there are cases where two is achieved.
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References:
◮ Aichholzer, O.; Bereg, S.; Dumitrescu, A.; Garc´
ıa, A.; Huemer, C.; Hurtado, F.; Kano, M.; M´ arquez, A.; Rappaport, D.; Smorodinsky, S.; Souvaine, D.; Urrutia, J.; Wood, D. Compatible geometric matchings. Comput. Geom. 42 (2009), 617- 626.
◮ Abellanas, M.; Garc´
ıa, A.; Hurtado, F.; Tejel, J.; Urrutia, J. Augmenting the connectivity of geometric graphs. Comput.
- Geom. 40 (2008), 220-230. (includes the Polygon Lemma)