Empty Monochromatic Simplices Oswin Aichholzer 1 , Ruy Fabila-Monroy - PowerPoint PPT Presentation

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • k -colored sets S ⊂ R d of n points in general position • How many empty monochromatic d -simplices exist? ◦ Urrutia, 2003: d = 3 , k = 4 : there always exists an empty monochromatic d -simplex (tetrahedron) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • k -colored sets S ⊂ R d of n points in general position • How many empty monochromatic d -simplices exist? ◦ Urrutia, 2003: d = 3 , k = 4 : there always exists an empty monochromatic d -simplex (tetrahedron) ◦ by proving that every S can be triangulated with more than 3 n tetrahedra Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • k -colored sets S ⊂ R d of n points in general position • How many empty monochromatic d -simplices exist? ◦ Urrutia, 2003: d = 3 , k = 4 : there always exists an empty monochromatic d -simplex (tetrahedron) ◦ by proving that every S can be triangulated with more than 3 n tetrahedra • Problem: triangulate S with many d -simplices Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • k -colored sets S ⊂ R d of n points in general position • How many empty monochromatic d -simplices exist? ◦ Urrutia, 2003: d = 3 , k = 4 : there always exists an empty monochromatic d -simplex (tetrahedron) ◦ by proving that every S can be triangulated with more than 3 n tetrahedra • Problem: triangulate S with many d -simplices (”minmax”) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • sets S ⊂ R d of n points in general position • Large sized triangulations: Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 8

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • sets S ⊂ R d of n points in general position • Large sized triangulations: ◦ Edelsbrunner, Preparata, and West, 1990: 15 n 2 + O ( n ) tetrahedra R 3 : upper bound of 7 ◦ Brass, 2005: R 3 : ∃ sets of points where every triangulation has 5 / O ( n 3 ) tetrahedra Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 8

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • sets S ⊂ R d of n points in general position • Large sized triangulations: ◦ Edelsbrunner, Preparata, and West, 1990: 15 n 2 + O ( n ) tetrahedra R 3 : upper bound of 7 ◦ Brass, 2005: R 3 : ∃ sets of points where every triangulation has 5 / O ( n 3 ) tetrahedra R d : ∃ sets of points where every triangulation has d + d − 1 1 ⌈ d 2 ⌉ ) d -simplices O ( n d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 8

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • sets S ⊂ R d of n points in general position • Large sized triangulations: ◦ Rothschild and Straus, 1985: R d : all triangulations have at least ( n − d ) d -simplices Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 9

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • sets S ⊂ R d of n points in general position • Large sized triangulations: ◦ Rothschild and Straus, 1985: R d : all triangulations have at least ( n − d ) d -simplices ◦ Urrutia, 2003: R 3 : ∃ triangulation with more than 3 n tetrahedra Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 9

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • sets S ⊂ R d of n points in general position • Large sized triangulations: ◦ Rothschild and Straus, 1985: R d : all triangulations have at least ( n − d ) d -simplices ◦ Urrutia, 2003: R 3 : ∃ triangulation with more than 3 n tetrahedra • we generalize / improve to: ◦ ∃ triangulation with at least ( dn +Ω(log n )) d -simplices Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 9

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap Literature Theorem 1 ”Lower Bound Theorem” Theorem 2 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap Literature Theorem 1 ”Lower Bound Theorem” Theorem 2 T1, T2 Large sized triangulations c d = d 3 + d 2 + d Lemmata 3, 4 Theorem 5: d> 2 � � h, log2( n ) ∃T , |T |≥ dn +max − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap T1 Literature Pulling complexes Theorem 1 T5 Lemmata 6 – 13 ”Lower Bound Theorem” Theorem 2 Lemma 9 Lemma 10 T1, T2 Large sized triangulations c d = d 3 + d 2 + d Lemmata 3, 4 Theorem 5: d> 2 � � h, log2( n ) ∃T , |T |≥ dn +max − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap T1 Literature Pulling complexes Theorem 1 T5 Lemmata 6 – 13 ”Lower Bound Theorem” Theorem 2 Lemma 9 Lemma 10 T1, T2 Large sized triangulations T2 L6 c d = d 3 + d 2 + d Lemmata 3, 4 L3 Theorem 5: d> 2 Order lemma � � Lemmata 14, 15 h, log2( n ) ∃T , |T |≥ dn +max − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap T1 Literature Pulling complexes Theorem 1 T5 Lemmata 6 – 13 ”Lower Bound Theorem” Theorem 2 Lemma 9 Lemma 10 T1, T2 Large sized triangulations T1 L8, L11–13 T2 L6 c d = d 3 + d 2 + d Lemmata 3, 4 Discrepancy lemma L3 Theorem 5: d> 2 Order lemma Lemmata 16, 18 – 23 � � Lemmata 14, 15 h, log2( n ) ∃T , |T |≥ dn +max − c d 2 d Corollary 17 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap T1 Literature Pulling complexes Theorem 1 T5 Lemmata 6 – 13 ”Lower Bound Theorem” Theorem 2 Lemma 9 Lemma 10 T1, T2 Large sized triangulations T1 L8, L11–13 T2 L6 c d = d 3 + d 2 + d Lemmata 3, 4 Discrepancy lemma L3 Theorem 5: d> 2 Order lemma Lemmata 16, 18 – 23 � � Lemmata 14, 15 h, log2( n ) ∃T , |T |≥ dn +max − c d 2 d Corollary 17 L19, 21–23 C17 , L16 T5 L15 Empty Monochromatic Simplices in k -Colored Point Sets Theorems 24, 27 – 29 Corollaries 25, 26 Theorem 29: d ≥ k ≥ 3 Corollary 25: d> 2 , k = d +1 Ω( n d − k +1+2 − d ) EMS ∃ linear number of EMS Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Roadmap T1 Literature Pulling complexes Theorem 1 T5 Lemmata 6 – 13 ”Lower Bound Theorem” Theorem 2 Lemma 9 Lemma 10 T1, T2 Large sized triangulations T1 L8, L11–13 T2 L6 c d = d 3 + d 2 + d Lemmata 3, 4 Discrepancy lemma L3 Theorem 5: d> 2 Order lemma Lemmata 16, 18 – 23 � � Lemmata 14, 15 h, log2( n ) ∃T , |T |≥ dn +max − c d 2 d Corollary 17 L19, 21–23 C17 , L16 T5 L15 L14 L18, 20 Empty Monochromatic Simplices Empty Monochromatic Simplices in k -Colored Point Sets in 2 -Colored Point Sets Theorems 24, 27 – 29 Observation 30 Corollaries 25, 26 Theorems 31 – 33 Theorem 29: d ≥ k ≥ 3 Corollary 25: d> 2 , k = d +1 Theorem 33: d ≥ 2 , k =2 Ω( n d − k +1+2 − d ) EMS ∃ linear number of EMS Ω( n d − 2 / 3 ) EMS Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 10

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) , c d = d 3 + d 2 + d . . . constant ∃ triangulation of size at least ( d +1) n − c d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) , c d = d 3 + d 2 + d . . . constant ∃ triangulation of size at least ( d +1) n − c d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) , c d = d 3 + d 2 + d . . . constant ∃ triangulation of size at least ( d +1) n − c d Proof: By Theorem 1: CH( S ) has at least dn − d ( d +1) edges 2 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) , c d = d 3 + d 2 + d . . . constant ∃ triangulation of size at least ( d +1) n − c d Proof: By Theorem 1: CH( S ) has at least dn − d ( d +1) edges 2 ⇒ ∃ point p ∈ S with degree at least 2 d in CH( S ) as long as n>d ( d +1) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) , c d = d 3 + d 2 + d . . . constant ∃ triangulation of size at least ( d +1) n − c d Proof: By Theorem 1: CH( S ) has at least dn − d ( d +1) edges 2 ⇒ ∃ point p ∈ S with degree at least 2 d in CH( S ) as long as n>d ( d +1) • successively remove such points p from S until d ( d +1) points left • arbitrary triangulation T d ( d +1) of size at least d ( d +1) − d = d 2 • insert points p in reversed order: ≥ 2 d − ( d − 1) = d +1 d -simplices each Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations: convex set Lemma 3: ∀ S ⊂ R d of n points in convex position d> 2 , n>d ( d +1) , c d = d 3 + d 2 + d . . . constant ∃ triangulation of size at least ( d +1) n − c d Proof: By Theorem 1: CH( S ) has at least dn − d ( d +1) edges 2 ⇒ ∃ point p ∈ S with degree at least 2 d in CH( S ) as long as n>d ( d +1) • successively remove such points p from S until d ( d +1) points left • arbitrary triangulation T d ( d +1) of size at least d ( d +1) − d = d 2 • insert points p in reversed order: ≥ 2 d − ( d − 1) = d +1 d -simplices each ⇒ triangulation of size at least d 2 +( d +1)( n − d ( d +1)) = ( d +1) n − c d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 11

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h > log 2 ( n ) / (2 d ) > d ( d + 1) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h > log 2 ( n ) / (2 d ) > d ( d + 1) ◦ ∃ triangulation of P of size at least ( d +1) h − c d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h > log 2 ( n ) / (2 d ) > d ( d + 1) ◦ ∃ triangulation of P of size at least ( d +1) h − c d ◦ insert the remaining n − h points ⇒ d additional d -simplices each Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h > log 2 ( n ) / (2 d ) > d ( d + 1) ◦ ∃ triangulation of P of size at least ( d +1) h − c d ◦ insert the remaining n − h points ⇒ d additional d -simplices each ◦ ⇒ resulting triangulation has size at least dn + h − c d > dn + log 2 ( n ) − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h ≤ log 2 ( n ) / (2 d ) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h ≤ log 2 ( n ) / (2 d ) os-Szekeres: ∃ convex set Q ⊂ S , | Q | > log 2 ( n ) ◦ Erd˝ >d ( d +1) 2 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h ≤ log 2 ( n ) / (2 d ) os-Szekeres: ∃ convex set Q ⊂ S , | Q | > log 2 ( n ) ◦ Erd˝ >d ( d +1) 2 ◦ P ′ = P \ Q : ∃ triangulation of P ′ ∪ Q of size at least ( d +1) | Q |− c d + | P ′ | Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h ≤ log 2 ( n ) / (2 d ) os-Szekeres: ∃ convex set Q ⊂ S , | Q | > log 2 ( n ) ◦ Erd˝ >d ( d +1) 2 ◦ P ′ = P \ Q : ∃ triangulation of P ′ ∪ Q of size at least ( d +1) | Q | + | P ′ |− c d ◦ insert the remaining points ⇒ d additional d -simplices each Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn +max − c d Proof: 2 d Two cases: • | P | = h ≤ log 2 ( n ) / (2 d ) os-Szekeres: ∃ convex set Q ⊂ S , | Q | > log 2 ( n ) ◦ Erd˝ >d ( d +1) 2 ◦ P ′ = P \ Q : ∃ triangulation of P ′ ∪ Q of size at least ( d +1) | Q | + | P ′ |− c d ◦ insert the remaining points ⇒ d additional d -simplices each ◦ ⇒ resulting triangulation has size at least ( d +1) | Q | + | P ′ |− c d + d ( n −| Q |−| P ′ | ) > dn + log 2 ( n ) − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 12

Institute for Software Technology Graz University of Technology P 23629–N18 Note on Theorem 5 • The constant c d in Lemma 3 can be improved to d 3 2 + 13 d 2 12 + 7 d ◦ 12 . . . equals 25 for d = 3 • For d = 3 Theorem 5 improves to � � h, log 2 n ◦ 3 n + max − 25 6 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 13

Institute for Software Technology Graz University of Technology P 23629–N18 Note on Theorem 5 • The constant c d in Lemma 3 can be improved to d 3 2 + 13 d 2 12 + 7 d ◦ 12 . . . equals 25 for d = 3 • For d = 3 Theorem 5 improves to � � h, log 2 n ◦ 3 n + max − 25 6 • [ EPW ] : Every set of n points in general position in R 3 , with h convex hull points, has a tetrahedrization of size at least 3( n − h ) + 4 h − 25 for h ≥ 13 . [ EWP ] H. Edelsbrunner, F.P. Preparata, and D.B. West. Tetrahedrizing point sets in three dimensions. 1990. Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 13

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least � � h, log 2 ( n ) dn + max − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 14

Institute for Software Technology Graz University of Technology P 23629–N18 Large sized triangulations Theorem 5: ∀ S ⊂ R d of n points in general position d> 2 , n> 4 d 2 ( d + 1) , h . . . number of convex hull points ∃ triangulation of size at least dn + log 2 ( n ) − c d 2 d Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 14

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set p Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set } � � n − 1 2 p Π } � � n − 1 2 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set Π ′ } � � n − 1 2 p Π } � � n − 1 2 Π ′′ Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set Π ′ p Π ′′ Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes • d -simplicial complex K of S ⊂ R d such that • for a predefined subset X ⊂ S , ( 1 ≤ | X | ≤ d − 1 ) • each d -simplex contains X in its vertex set Π ′ p Π ′′ Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 15

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes Lemma 9: ◦ ∀ S ⊂ R d ( d> 3 ) of n> 4 d 2 ( d +1) points in general position ◦ ∀ point p ∈ S ⇒ ∃ d -dimensional simplicial complex of size at least ( d − 1) n + log 2 n 2( d − 1) − 2 c d − 1 all whose d -simplices have p as a vertex Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 16

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 17

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes Π ( r − 1) -dimensional X r := | X | Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 17

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes ( d − ( r − 1)) -dimensional Π ′ Π ( r − 1) -dimensional X r := | X | Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 17

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes ( d − ( r − 1)) -dimensional Π ′ Π ( r − 1) -dimensional X r := | X | Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 17

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes ( d − ( r − 1)) -dimensional Π ′ Π ( r − 1) -dimensional X r := | X | Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 17

Institute for Software Technology Graz University of Technology P 23629–N18 Pulling complexes Lemma 10: ◦ ∀ S ⊂ R d ( d> 3 ) of n> 4 d 2 ( d +1) points in general position ◦ ∀ X ⊂ S and 1 ≤ | X | ≤ d − 3 ⇒ ∃ d -dimensional simplicial complex of size at least log 2 n ( d −| X | ) n + 2( d −| X | ) − 2 c d − 1 all whose d -simplices contain X in their vertex set Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 18

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Order Lemma” • ”Generalized Order Lemma” (Lemma 15) ◦ S ⊂ R d set of n ≥ d +1 points in general position ◦ d > 2 , h := | CH( S ) ∩ S | ⇒ ∃ d -dimensional simplicial complex with at least ( d − 1) n + ( n − h ) (2 (1 − d ) ) +2 h − c d d -simplices, each having at least one of their vertices in CH( S ) ∩ S Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 19

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Order Lemma” • ”Generalized Order Lemma” (Lemma 15) ◦ S ⊂ R d set of n ≥ d +1 points in general position ◦ d > 2 , h := | CH( S ) ∩ S | ⇒ ∃ d -dimensional simplicial complex with at least ( d − 1) n + ( n − h ) (2 (1 − d ) ) +2 h − c d d -simplices, each having at least one of their vertices in CH( S ) ∩ S } � . � . ( d − 1) times � . � ( n − h ) (2 (1 − d ) ) ⇔ �� � ( n − h ) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 19

Institute for Software Technology Graz University of Technology P 23629–N18 Discrepancy • k -colored set S ⊂ R d of n points in general position ◦ k . . . constant, d ≥ 2 ◦ ( S 1 , . . . , S k ) . . . color classes of S ◦ S max . . . biggest color class Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 20

Institute for Software Technology Graz University of Technology P 23629–N18 Discrepancy • k -colored set S ⊂ R d of n points in general position ◦ k . . . constant, d ≥ 2 ◦ ( S 1 , . . . , S k ) . . . color classes of S ◦ S max . . . biggest color class • discrepancy δ ( S ) : Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 20

Institute for Software Technology Graz University of Technology P 23629–N18 Discrepancy • k -colored set S ⊂ R d of n points in general position ◦ k . . . constant, d ≥ 2 ◦ ( S 1 , . . . , S k ) . . . color classes of S ◦ S max . . . biggest color class • discrepancy δ ( S ) : ◦ bichromatic ( k =2 ): δ ( S ) := | S max |−| S \ S max | Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 20

Institute for Software Technology Graz University of Technology P 23629–N18 Discrepancy • k -colored set S ⊂ R d of n points in general position ◦ k . . . constant, d ≥ 2 ◦ ( S 1 , . . . , S k ) . . . color classes of S ◦ S max . . . biggest color class • discrepancy δ ( S ) : ◦ δ ( S ) := � ( | S max |−| S i | ) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 20

Institute for Software Technology Graz University of Technology P 23629–N18 Discrepancy • k -colored set S ⊂ R d of n points in general position ◦ k . . . constant, d ≥ 2 ◦ ( S 1 , . . . , S k ) . . . color classes of S ◦ S max . . . biggest color class • discrepancy δ ( S ) : ◦ δ ( S ) := � ( | S max |−| S i | ) = ( k − 1) | S max |−| S \ S max | = k | S max |− n Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 20

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → apply Lemma 10 to S max and X - ∃ d -dimensional simplicial complex, K X ( S max ) - |K X ( S max ) | ≥ ( d −| X | ) | S max | + log 2 | S max | 2( d −| X | ) − 2 c d − 1 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → |K X ( S max ) | ≥ ( d −| X | ) | S max | + log 2 | S max | 2( d −| X | ) − 2 c d − 1 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → |K X ( S max ) | ≥ ( d −| X | ) | S max | + log 2 | S max | 2( d −| X | ) − 2 c d − 1 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → |K X ( S max ) | ≥ ( k − 1) | S max | + log 2 | S max | − 2 c d − 1 2( k − 1) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → |K X ( S max ) | ≥ ( k − 1) | S max | + log 2 | S max | − 2 c d − 1 of which at least 2( k − 1) ◦ ( k − 1) | S max | + log 2 | S max | − 2 c d − 1 −| S \ S max | are empty 2( k − 1) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → |K X ( S max ) | ≥ ( k − 1) | S max | + log 2 | S max | − 2 c d − 1 of which at least 2( k − 1) ◦ ( k − 1) | S max | + log 2 | S max | − 2 c d − 1 −| S \ S max | are empty 2( k − 1) ( k − 1) | S max |−| S \ S max | = δ ( S ) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → K X ( S max ) : ≥ δ ( S )+ log 2 | S max | − 2 c d − 1 empty monochr. d -simplices 2( k − 1) Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21

Institute for Software Technology Graz University of Technology P 23629–N18 ”Generalized Discrepancy Lemma” • ”Generalized Discrepancy Lemma” (Lemma 19) ◦ k -colored set S ⊂ R d of n points in general position n > k · 4 d 2 ( d +1) ◦ d ≥ k > 3 , n d − k +1 · ( δ ( S ) + log n ) � � ⇒ S determines Ω empty monochromatic d -simplices • Proof: ◦ choose a set X ⊂ S max of d − k +1 points → K X ( S max ) : ≥ δ ( S )+ log 2 | S max | − 2 c d − 1 empty monochr. d -simplices 2( k − 1) � | S max | � ◦ many subsets X d − k +1 Thomas Hackl: Eurogiga Midterm Conference, July 9 th – 13 th , 2012 21