Introduction to Combinatorial...
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![Page 1: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/1.jpg)
Introduction to Combinatorial Geometry
Sathish Govindarajan
Department of Computer Science and AutomationIndian Institute of Science, Bangalore
Research promotion workshop on Graphs and GeometryIndian Institute of Technology, Roorkee
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![Page 2: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/2.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
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![Page 3: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/3.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]
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![Page 4: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/4.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]Different proofs
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![Page 5: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/5.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]Different proofs
Radon’s theorem [1921]
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![Page 6: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/6.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]Different proofs
Radon’s theorem [1921]Induction
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![Page 7: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/7.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]Different proofs
Radon’s theorem [1921]InductionShrinking ball technique
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![Page 8: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/8.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]Different proofs
Radon’s theorem [1921]InductionShrinking ball techniqueBrouwer’s theorem
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![Page 9: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/9.jpg)
Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Generalized in different directions [survey by Eckhoff ’93]Different proofs
Radon’s theorem [1921]InductionShrinking ball techniqueBrouwer’s theoremExtremal proof [Mustafa and Ray, 2007]
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![Page 10: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/10.jpg)
Extremal proof for Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
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![Page 11: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/11.jpg)
Extremal proof for Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Extremal proof [Mustafa and Ray ’07]Construct a point p that is contained in all the objects
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![Page 12: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/12.jpg)
Extremal proof for Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Extremal proof [Mustafa and Ray ’07]Construct a point p that is contained in all the objects
d = 1 : Intervals in 1D
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![Page 13: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/13.jpg)
Extremal proof for Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Extremal proof [Mustafa and Ray ’07]Construct a point p that is contained in all the objects
d = 1 : Intervals in 1D
Extend to d = 2
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![Page 14: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/14.jpg)
Extremal proof for Helly’s Theorem
Theorem
Let C be a collection of convex objects in Rd . If every d + 1 objects inC have a common intersection, then all the objects in C have acommon intersection.
Extremal proof [Mustafa and Ray ’07]Construct a point p that is contained in all the objects
d = 1 : Intervals in 1D
Extend to d = 2Proof generalizes to d dimensions.
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Intervals in 1D
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Intervals in 1D
S - set of intervals on the real line
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Intervals in 1D
S - set of intervals on the real line
Every 2 intervals in S intersect
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![Page 18: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/18.jpg)
Intervals in 1D
S - set of intervals on the real line
Every 2 intervals in S intersect
Claim: All the intervals have a common intersection
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![Page 19: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/19.jpg)
Intervals in 1D
S - set of intervals on the real line
Every 2 intervals in S intersect
Claim: All the intervals have a common intersection
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![Page 20: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/20.jpg)
Intervals in 1D
S - set of intervals on the real line
Every 2 intervals in S intersect
Claim: All the intervals have a common intersection
Extremal proof
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![Page 21: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/21.jpg)
Intervals in 1D
S - set of intervals on the real line
Every 2 intervals in S intersect
Claim: All the intervals have a common intersection
Extremal proofConstruct a point p that is contained in all the intervals
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![Page 22: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/22.jpg)
Intervals in 1D
S - set of intervals on the real line
Every 2 intervals intersect
Extremal proofConstruct a point p that is contained in all the intervals
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Intervals in 1D
S - set of intervals on the real line
Every 2 intervals intersect
Extremal proofConstruct a point p that is contained in all the intervals
p : Leftmost right endpoint
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![Page 24: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/24.jpg)
Intervals in 1D
S - set of intervals on the real line
Every 2 intervals intersect
Extremal proofConstruct a point p that is contained in all the intervals
p : Leftmost right endpoint
Claim: All the intervals contain p
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Intervals in 1D
Construct a point p that is contained in all the intervals
p : Leftmost right endpoint
Claim: All the intervals contain p
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![Page 26: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/26.jpg)
Intervals in 1D
Construct a point p that is contained in all the intervals
p : Leftmost right endpoint
Claim: All the intervals contain p
Proof by contradiction
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![Page 27: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/27.jpg)
Interval Graphs
S - set of intervals on the line
a b c
ed
f
a b c
ed
f
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Interval Graphs
S - set of intervals on the line
a b c
ed
f
a b c
ed
f
V - set of intervals si
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Interval Graphs
S - set of intervals on the line
a b c
ed
f
a b c
ed
f
V - set of intervals si
(si , sj ) ∈ E if intervals si and sj intersect
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobs
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)Connection between this problem and interval graphs?
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![Page 36: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/36.jpg)
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)Connection between this problem and interval graphs?Maximum independent set in Interval graph
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![Page 37: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/37.jpg)
Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)Connection between this problem and interval graphs?Maximum independent set in Interval graph
Greedy Algorithm to solve the problem (Exercise)with Proof of correctness
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Applications of Interval Graphs
Operations Research, Computational Biology, Mobile Networks
Consultant problem:Jobs: (6, 12), (8, 10), (7, 13), (9, 17), (11, 15), (12, 16), (15, 18)Choose the maximum number of (non-conflicting) jobsOptimal choice: (8, 10), (11, 15), (15, 18)Connection between this problem and interval graphs?Maximum independent set in Interval graph
Greedy Algorithm to solve the problem (Exercise)with Proof of correctness
Extension: What if jobs have different profits?(Use dynamic programming)
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Axis Parallel Rectangles in 2D
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Axis Parallel Rectangles in 2D
S - set of axis parallel rectangles
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Axis Parallel Rectangles in 2D
S - set of axis parallel rectangles
Every 2 rectangles intersect
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Axis Parallel Rectangles in 2D
S - set of axis parallel rectangles
Every 2 rectangles intersect
Claim: There exists a point p contained in all the rectangles
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Axis Parallel Rectangles in 2D
S - set of axis parallel rectangles
Every 2 rectangles intersect
Claim: There exists a point p contained in all the rectanglesIs it true?
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Circles in 2D
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Circles in 2D
S - set of circles
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Circles in 2D
S - set of circles
Every 2 circles intersect
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Circles in 2D
S - set of circles
Every 2 circles intersect
Claim: There exists a point p contained in all the circles
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Circles in 2D
S - set of circles
Every 2 circles intersect
Claim: There exists a point p contained in all the circles
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Circles in 2D
S - set of circles
Every 2 circles intersect
Claim: There exists a point p contained in all the circlesNot true
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Circles in 2D
S - set of circles
Every 2 circles intersect
Claim: There exists a point p contained in all the circlesNot true
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
Extremal proof [Mustafa and Ray ’07]Construct a point p that is contained in all the objects
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
Ca
CbPab
Cab
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
Ca
CbPab
Cab
pab : Lowest point in Cab = Ca ∩ Cb
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
Ca
CbPab
Cab
pab : Lowest point in Cab = Ca ∩ Cb
Choose the pair of objects (Ci ,Cj) such that pij is highest amongall pairs
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Helly’s Theorem in R2
Theorem (Helly’s Theorem in R2)
Let C be a collection of convex objects in R2. If every 3 objects in Chave a common intersection, then all the objects in C have a commonintersection
Ca
CbPab
Cab
pab : Lowest point in Cab = Ca ∩ Cb
Choose the pair of objects (Ci ,Cj) such that pij is highest amongall pairsClaim: pij is contained in all objects in C
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)
Ci
Pij
Cij
Cj
Ck
Pjk
Pik
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)
Ci
Pij
Cij
Cj
Ck
Pjk
Pik
If pij is not contained in Ck
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)
Ci
Pij
Cij
Cj
Ck
Pjk
Pik
If pij is not contained in Ck
pjk higher than pij - Contradiction
Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial GeometryResearch promotion workshop on Graphs and
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Ci
Pij
Cij
Cj
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Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Ci
Pij
Cij
Cj
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)
Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial GeometryResearch promotion workshop on Graphs and
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![Page 65: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/65.jpg)
Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Ci
Pij
Cij
Cj
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)Ck intersect both Ci and Cj below pij
pik and pjk must be lower than pij
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![Page 66: Introduction to Combinatorial Geometrycs.rkmvu.ac.in/~sghosh/public_html/iitr_igga/iitroorkee_sathish.pdf · Sathish Govindarajan (Indian Institute of Science)Introduction to Combinatorial](https://reader034.fdocuments.net/reader034/viewer/2022052613/5f1a90873c2f4f12985a2725/html5/thumbnails/66.jpg)
Helly’s Theorem in R2
Claim: pij is contained in Ck for all k
Ci
Pij
Cij
Cj
Cij ∩ Ck 6= ∅ (Every 3 objects intersect)Ck intersect both Ci and Cj below pij
pik and pjk must be lower than pij
By convexity, pij is contained in Ck
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
Helly’s theorem: For p = 3,q = 3, 1 point is sufficient
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
Helly’s theorem: For p = 3,q = 3, 1 point is sufficient
Theorem (Alon and Kleitman ’92)
C is pierced by constant (f (p,q,d)) number of points
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
Helly’s theorem: For p = 3,q = 3, 1 point is sufficient
Theorem (Alon and Kleitman ’92)
C is pierced by constant (f (p,q,d)) number of points
α fraction of d + 1-tuples intersect (counting argument)
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
Helly’s theorem: For p = 3,q = 3, 1 point is sufficient
Theorem (Alon and Kleitman ’92)
C is pierced by constant (f (p,q,d)) number of points
α fraction of d + 1-tuples intersect (counting argument)∃ a point contained in β-fraction of all convex objects(by Fractional Helly)
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
Helly’s theorem: For p = 3,q = 3, 1 point is sufficient
Theorem (Alon and Kleitman ’92)
C is pierced by constant (f (p,q,d)) number of points
α fraction of d + 1-tuples intersect (counting argument)∃ a point contained in β-fraction of all convex objects(by Fractional Helly)Add points iteratively such that all convex objects have a largefraction of points contained in them (by Iterative re-weighting)
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Hadwiger-Debrunner (p, q) problem
Definition
For any positive integers, p,q, let C be a family of convex objects C inR
d with [p,q]-property. How many points are needed to pierce C?
Helly’s theorem: For p = 3,q = 3, 1 point is sufficient
Theorem (Alon and Kleitman ’92)
C is pierced by constant (f (p,q,d)) number of points
α fraction of d + 1-tuples intersect (counting argument)∃ a point contained in β-fraction of all convex objects(by Fractional Helly)Add points iteratively such that all convex objects have a largefraction of points contained in them (by Iterative re-weighting)Constant number of points pierce all objects (Weak ǫ-nets)
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Centerpoint Theorem
Theorem (Centerpoint Theorem)
Let P be a set of n points in the plane. There exists a point p in theplane that is contained in every convex object containing > 2
3n pointsof P.
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Centerpoint Theorem
Theorem (Centerpoint Theorem)
Let P be a set of n points in the plane. There exists a point p in theplane that is contained in every convex object containing > 2
3n pointsof P.
Take any 3 convex objects Ci ,Cj ,Ck containing > 23n points
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Centerpoint Theorem
Theorem (Centerpoint Theorem)
Let P be a set of n points in the plane. There exists a point p in theplane that is contained in every convex object containing > 2
3n pointsof P.
Take any 3 convex objects Ci ,Cj ,Ck containing > 23n points
Ci ∩ Cj ∩ Ck 6= ∅ (Counting argument)
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Centerpoint Theorem
Theorem (Centerpoint Theorem)
Let P be a set of n points in the plane. There exists a point p in theplane that is contained in every convex object containing > 2
3n pointsof P.
Take any 3 convex objects Ci ,Cj ,Ck containing > 23n points
Ci ∩ Cj ∩ Ck 6= ∅ (Counting argument)
Applying Helly theorem, there exists a point p contained in allsuch convex objects
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Centerpoint Theorem
Theorem (Centerpoint Theorem)
Let P be a set of n points in the plane. There exists a point p in theplane that is contained in every convex object containing > 2
3n pointsof P.
Take any 3 convex objects Ci ,Cj ,Ck containing > 23n points
Ci ∩ Cj ∩ Ck 6= ∅ (Counting argument)
Applying Helly theorem, there exists a point p contained in allsuch convex objects
The constant 23 is the best possible
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Strong Centerpoint
Can we restrict the centerpoint to belong to P?
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Strong Centerpoint
Can we restrict the centerpoint to belong to P?NONo, even for halfspaces
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Strong Centerpoint for axis parallel rectangles
Theorem (Strong Centerpoint Theorem (Ashok, Azmi, G. ’14))
Let P be a set of n points in the plane. There exists a point p ∈ P thatis contained in every rectangle containing > 3
4n points of P.
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Strong Centerpoint for axis parallel rectangles
Theorem (Strong Centerpoint Theorem (Ashok, Azmi, G. ’14))
Let P be a set of n points in the plane. There exists a point p ∈ P thatis contained in every rectangle containing > 3
4n points of P.
The constant 34 is the best possible
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Axis-Parallel Rectangles
n/2 + 2
n/4 − 1
n/4 − 1
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Axis-Parallel Rectangles
n/2 + 2
n/4 − 1
n/4 − 1 The second column containsn2 + 2 points.
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Axis-Parallel Rectangles
n/2 + 2
n/4 − 1
n/4 − 1 The second column containsn2 + 2 points.
Since regions (1,2) and (3,2)contain at most n
4 − 1 pointseach, the region (2,2) is notempty
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Axis-Parallel Rectangles
n/2 + 2
n/4 − 1
n/4 − 1 Select any point from region(2,2) as the ǫ-net.
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Axis-Parallel Rectangles
n/2 + 2
n/4 − 1
n/4 − 1 Select any point from region(2,2) as the ǫ-net.
Any axis-parallel rectanglethat does not contain thechosen point will have ≤ 3n
4points.
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Small Weak Epsilon Nets
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Small Weak Epsilon Nets
Select many points instead of just one
Theorem (Generalized Centerpoints)
Let P be a set of n points in the plane. There exists a set of i points Qin the plane such that c ∩ Q 6= ∅ for any convex object c containing> ǫin points of P.
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Small Weak Epsilon Nets
Select many points instead of just one
Theorem (Generalized Centerpoints)
Let P be a set of n points in the plane. There exists a set of i points Qin the plane such that c ∩ Q 6= ∅ for any convex object c containing> ǫin points of P.
Bounds for ǫi?
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Small Weak Epsilon Nets
Select many points instead of just one
Theorem (Generalized Centerpoints)
Let P be a set of n points in the plane. There exists a set of i points Qin the plane such that c ∩ Q 6= ∅ for any convex object c containing> ǫin points of P.
Bounds for ǫi?
Centerpoint Theorem: ǫ1 = 2/3
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Small Weak Epsilon Nets
Select many points instead of just one
Theorem (Generalized Centerpoints)
Let P be a set of n points in the plane. There exists a set of i points Qin the plane such that c ∩ Q 6= ∅ for any convex object c containing> ǫin points of P.
Bounds for ǫi?
Centerpoint Theorem: ǫ1 = 2/3
Extension: ǫ2 = 4/7
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Small Weak Epsilon Nets
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Small Weak Epsilon Nets
Select many points instead of just one
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Small Weak Epsilon Nets
Select many points instead of just one
Special convex objects - rectangles, circles, halfspaces, . . .
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Small Weak Epsilon Nets
Select many points instead of just one
Special convex objects - rectangles, circles, halfspaces, . . .
Rectangles Halfspaces Disks Convex setsLB UB LB UB LB UB LB UB
ǫ1 1/2 2/3 2/3 2/3ǫ2 2/5 1/2 1/2 4/7 4/7ǫ3 1/3 0 1/4 8/15 5/11 8/15
Table: Summary of bounds [Aronov et al ’09, MR ’07]
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Small Weak Epsilon Nets
Select many points instead of just one
Special convex objects - rectangles, circles, halfspaces, . . .
Rectangles Halfspaces Disks Convex setsLB UB LB UB LB UB LB UB
ǫ1 1/2 2/3 2/3 2/3ǫ2 2/5 1/2 1/2 4/7 4/7ǫ3 1/3 0 1/4 8/15 5/11 8/15
Table: Summary of bounds [Aronov et al ’09, MR ’07]
Open problem: Find exact value of ǫi for small i?
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Small Strong epsilon nets
Restrict Q ⊆ P
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Small Strong epsilon nets
Restrict Q ⊆ P
Theorem (Generalized Strong Centerpoints)
Let P be a set of n points in the plane. There exists a set of i pointsQ ⊆ P such that c∩Q 6= ∅ for any object c containing > ǫin points of P.
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Small Strong epsilon nets
Restrict Q ⊆ P
Theorem (Generalized Strong Centerpoints)
Let P be a set of n points in the plane. There exists a set of i pointsQ ⊆ P such that c∩Q 6= ∅ for any object c containing > ǫin points of P.
Rectangles Halfspaces DisksLB UB LB UB LB UB
ǫ1 3/4 1 1ǫ2 5/9 5/8 3/5 2/3 3/5 2/3ǫ3 9/20 5/9 1/2 1/2 2/3
Table: Summary of bounds [AAG ’10]
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Small Strong epsilon nets
Restrict Q ⊆ P
Theorem (Generalized Strong Centerpoints)
Let P be a set of n points in the plane. There exists a set of i pointsQ ⊆ P such that c∩Q 6= ∅ for any object c containing > ǫin points of P.
Rectangles Halfspaces DisksLB UB LB UB LB UB
ǫ1 3/4 1 1ǫ2 5/9 5/8 3/5 2/3 3/5 2/3ǫ3 9/20 5/9 1/2 1/2 2/3
Table: Summary of bounds [AAG ’10]
Open problem: Find exact value (for k = 2)
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First Selection Lemma (FSL)
For induced triangles in R2, Boros and Furedi (1984), showed thatthe centerpoint is present in n3
27 (constant fraction) trianglesinduced by P. This constant is tight.
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FSL for Axis-Parallel Rectangles in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
8 axis-parallelrectangles induced by P. This bound is tight.
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FSL for Axis-Parallel Rectangles in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
8 axis-parallelrectangles induced by P. This bound is tight.
The tightness of the bound - P distributed around the boundary of acircle.
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FSL for Axis-Parallel Rectangles in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
8 axis-parallelrectangles induced by P. This bound is tight.
The tightness of the bound - P distributed around the boundary of acircle.
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p ∈ P such that p is contained in at least n2
16induced rectangles. This bound is tight.
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FSL for Axis-Parallel Rectangles in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
8 axis-parallelrectangles induced by P. This bound is tight.
The tightness of the bound - P distributed around the boundary of acircle.
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p ∈ P such that p is contained in at least n2
16induced rectangles. This bound is tight.
Proved using weak and strong centerpoint w.r.t axis parallel rectangles
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FSL for Axis-Parallel Rectangles in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
8 axis-parallelrectangles induced by P. This bound is tight.
The tightness of the bound - P distributed around the boundary of acircle.
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p ∈ P such that p is contained in at least n2
16induced rectangles. This bound is tight.
Proved using weak and strong centerpoint w.r.t axis parallel rectangles
Open problem: FSL for boxes in higher dimension
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FSL for Disks in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
6 disksinduced by P.
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FSL for Disks in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
6 disksinduced by P.
Proof uses centerpoint
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FSL for Disks in R2
Theorem (Ashok, G., Mishra, Rajgopal ’13)
There exists a point p in R2, which is present in at least n2
6 disksinduced by P.
Proof uses centerpoint
Open problem: Obtain tight bounds for disks
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Questions?
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