Non-Circleability Proof of N △ 6 Proof. C 1 , . . . , C 6 . . . circles (on S 2 ) E 1 , . . . , E 6 . . . planes (in R 3 ) move planes away from the origin no great-circle arr. ⇒ events occur not all planes contain the origin 15
Non-Circleability Proof of N △ 6 Proof. C 1 , . . . , C 6 . . . circles (on S 2 ) E 1 , . . . , E 6 . . . planes (in R 3 ) move planes away from the origin no great-circle arr. ⇒ events occur first event is triangle flip ( ∄ digons) 15
Non-Circleability Proof of N △ 6 Proof. C 1 , . . . , C 6 . . . circles (on S 2 ) E 1 , . . . , E 6 . . . planes (in R 3 ) move planes away from the origin no great-circle arr. ⇒ events occur first event is triangle flip ( ∄ digons) but triangle flip not possible because all triangles in NonKrupp. Contradiction. 15
Non-Circularizability Proof of N 2 6 Proof. (similar) C 1 , . . . , C 6 . . . circles E 1 , . . . , E 6 . . . planes 16
Non-Circularizability Proof of N 2 6 Proof. (similar) C 1 , . . . , C 6 . . . circles E 1 , . . . , E 6 . . . planes move planes towards the origin 16
Non-Circularizability Proof of N 2 6 Proof. (similar) C 1 , . . . , C 6 . . . circles E 1 , . . . , E 6 . . . planes move planes towards the origin ∃ NonKrupp subarr. ⇒ events occur ∃ point of intersection outside the unit-sphere (will move inside) 16
Non-Circularizability Proof of N 2 6 Proof. (similar) C 1 , . . . , C 6 . . . circles E 1 , . . . , E 6 . . . planes move planes towards the origin ∃ NonKrupp subarr. ⇒ events occur first event is triangle flip ( ∄ digons) 16
Non-Circularizability Proof of N 2 6 Proof. (similar) C 1 , . . . , C 6 . . . circles E 1 , . . . , E 6 . . . planes move planes towards the origin ∃ NonKrupp subarr. ⇒ events occur first event is triangle flip ( ∄ digons) but triangle flip not possible because all triangles in Krupp. Contradiction. 16
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Proof. • C 1 , . . . , C n . . . circles E 1 , . . . , E n . . . planes 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Proof. • C 1 , . . . , C n . . . circles E 1 , . . . , E n . . . planes • move planes towards the origin 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Proof. • C 1 , . . . , C n . . . circles E 1 , . . . , E n . . . planes • move planes towards the origin • all triples Krupp ⇒ all intersections remain inside ⇒ no events 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Proof. • C 1 , . . . , C n . . . circles E 1 , . . . , E n . . . planes • move planes towards the origin • all triples Krupp ⇒ all intersections remain inside ⇒ no events • we obtain a great-circle arrangement 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Corollaries: • ∀ non-stretchable arr. of pseudolines ∃ corresponding non-circleable arr. of pseudocircles 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Corollaries: • ∀ non-stretchable arr. of pseudolines ∃ corresponding non-circleable arr. of pseudocircles • deciding circleability is ∃ R -complete ( NP ⊆ ∃ R ⊆ PSPACE ) 17
Great-(Pseudo)Circles Great-Circle Theorem: An arrangement of great-pcs. is circleable (i.e., has a circle representation) if and only if it has a great-circle repr. Corollaries: • ∀ non-stretchable arr. of pseudolines ∃ corresponding non-circleable arr. of pseudocircles • deciding circleability is ∃ R -complete • ∃ infinite families of minimal non-circ. arrangements • ∃ arr with a disconnected realization space • . . . 17
Computational Part • find circle representations heuristically • hard instances by hand 18
Computational Part • enumeration via recursive search on flip graph △ -flip digon-flip 19
Computational Part • connected arrangements encoded via primal-dual graph • intersecting arrangements encoded via dual graph arrangement primal-dual gr. primal graph dual graph 20
Part II: Triangles in Arrangements 21
Part II: Triangles in Arrangements assumption throughout part II: intersecting . . . any 2 pseudocircles cross twice 21
Cells in Arrangements digon , triangle , quadrangle , pentagon , . . . , k -cell p k . . . # of k -cells p 2 = 6 p 3 = 4 p 4 = 8 p 5 = 0 p 6 = 4 22
Triangles in Digon-free Arrangements Gr¨ unbaum’s Conjecture (’72): • p 3 ≥ 2 n − 4 ? 23
Triangles in Digon-free Arrangements Gr¨ unbaum’s Conjecture (’72): • p 3 ≥ 2 n − 4 ? Known: • p 3 ≥ 4 n/ 3 [Hershberger and Snoeyink ’91] • p 3 ≥ 4 n/ 3 for non-simple arrangements, tight for infinite family [Felsner and Kriegel ’98] 23
Triangles in Digon-free Arrangements Gr¨ unbaum’s Conjecture (’72): • p 3 ≥ 2 n − 4 ? Known: • p 3 ≥ 4 n/ 3 [Hershberger and Snoeyink ’91] • p 3 ≥ 4 n/ 3 for non-simple arrangements, tight for infinite family [Felsner and Kriegel ’98] Our Contribution: • disprove Gr¨ unbaum’s Conjecture • p 3 < 1 . 45 n • New Conjecture: 4 n/ 3 is tight 23
Triangles in Digon-free Arrangements Theorem. The minimum number of triangles in digon-free arrangements of n pseudocircles is (i) 8 for 3 ≤ n ≤ 6 . (ii) ⌈ 4 3 n ⌉ for 6 ≤ n ≤ 14 . (iii) < 1 . 45 n for all n = 11 k + 1 with k ∈ N . 24
Figure: Arrangement of n = 12 pcs with p 3 = 16 triangles.
Figure: Arrangement of n = 12 pcs with p 3 = 16 triangles.
26
• traverses 1 triangle • forms 2 triangles 26
Proof of the Theorem 27
Proof of the Theorem 27
Proof of the Theorem 27
Proof of the Theorem • start with C 1 := A 12 • merge C k and A 12 − → C k +1 • n ( C k ) = 11 k + 1 , p 3 ( C k ) = 16 k 11 k +1 increases as k increases with limit 16 16 k • 11 = 1 . 45 28
Proof of the Theorem • start with C 1 := A 12 • merge C k and A 12 − → C k +1 maintain the path! • n ( C k ) = 11 k + 1 , p 3 ( C k ) = 16 k 11 k +1 increases as k increases with limit 16 16 k • 11 = 1 . 45 28
Triangles in Digon-free Arrangements Theorem. The minimum number of triangles in digon-free arrangements of n pseudocircles is (i) 8 for 3 ≤ n ≤ 6 . (ii) ⌈ 4 3 n ⌉ for 6 ≤ n ≤ 14 . (iii) < 1 . 45 n for all n = 11 k + 1 with k ∈ N . Conjecture. ⌈ 4 n/ 3 ⌉ is tight for infinitely many n . 29
Triangles in Digon-free Arrangements • ∃ unique arrangement N △ with n = 6 , p 3 = 8 6 • N △ appears as a subarrangement of every arr. with 6 p 3 < 2 n − 4 for n = 7 , 8 , 9 • N △ is non-circularizable 6 30
Triangles in Digon-free Arrangements • ∃ unique arrangement N △ with n = 6 , p 3 = 8 6 • N △ appears as a subarrangement of every arr. with 6 p 3 < 2 n − 4 for n = 7 , 8 , 9 • N △ is non-circularizable 6 • ⇒ Gr¨ unbaum’s Conjecture might still be true for arrangements of circles! 30
Triangles in Arrangements with Digons Theorem. p 3 ≥ 2 n/ 3 31
Triangles in Arrangements with Digons Theorem. p 3 ≥ 2 n/ 3 intersecting Proof. • C . . . pseudocircle in A • All incident digons lie on the same side of C . digon digon C 31
Triangles in Arrangements with Digons Theorem. p 3 ≥ 2 n/ 3 intersecting Proof. • C . . . pseudocircle in A • All incident digons lie on the same side of C . no red-blue intersection possible! digon digon C 31
Triangles in Arrangements with Digons Theorem. p 3 ≥ 2 n/ 3 Proof. • C . . . pseudocircle in A • All incident digons lie on the same side of C . • ∃ two digons or triangles on each side of C [Hershberger and Snoeyink ’91] . 31
Triangles in Arrangements with Digons Theorem. p 3 ≥ 2 n/ 3 Conjecture. p 3 ≥ n − 1 31
Maximum Number of Triangles 3 n 2 + O ( n ) Theorem. p 3 ≤ 2 32
Maximum Number of Triangles 3 n 2 + O ( n ) Theorem. p 3 ≤ 2 � n 4 � • construction for infinitely many values of n , 3 2 based on pseudoline arrangements [Blanc ’11] 32
Maximum Number of Triangles 3 n 2 + O ( n ) Theorem. p 3 ≤ 2 � n 4 � • construction for infinitely many values of n , 3 2 based on pseudoline arrangements [Blanc ’11] � n • Question: p 3 ≤ 4 � + O (1) ? 3 2 32
Maximum Number of Triangles 3 n 2 + O ( n ) Theorem. p 3 ≤ 2 � n 4 � • construction for infinitely many values of n , 3 2 based on pseudoline arrangements [Blanc ’11] � n • Question: p 3 ≤ 4 � + O (1) ? 3 2 2 3 4 5 6 7 8 9 10 n ≥ 37 ≥ 48 ≥ 60 simple 0 8 8 13 20 29 ≥ 37 ≥ 48 ≥ 60 +digon-free - 8 8 12 20 29 ⌊ 4 � n � ⌋ 1 4 8 13 20 28 37 48 60 3 2 32
Recommend
More recommend
