On the structure of sets minimizing the rectilinear crossing number - PowerPoint PPT Presentation

Goal Minimizing the rectilinear crossing number j -facets and halving edges ≤ j -facets On the structure of sets minimizing the rectilinear crossing number O. Aichholzer, D. Orden, P. Ramos Crete, August 2005 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number j -facets and halving edges ≤ j -facets Goal ◮ Rectilinear crossing number problem: Determine minimum number of crossings of a straight-edge drawing of K n (vertices in general position). O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number j -facets and halving edges ≤ j -facets Goal ◮ Rectilinear crossing number problem: Determine minimum number of crossings of a straight-edge drawing of K n (vertices in general position). ◮ Structural properties of point sets minimizing crossings? O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number j -facets and halving edges ≤ j -facets Goal ◮ Rectilinear crossing number problem: Determine minimum number of crossings of a straight-edge drawing of K n (vertices in general position). ◮ Structural properties of point sets minimizing crossings? O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number j -facets and halving edges ≤ j -facets Goal ◮ Rectilinear crossing number problem: Determine minimum number of crossings of a straight-edge drawing of K n (vertices in general position). ◮ Structural properties of point sets minimizing crossings? O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Order type flip events ◮ Consider a set S of n points and move a point p 1 along a line: O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Order type flip events ◮ Consider a set S of n points and move a point p 1 along a line: The order type changes precisely when p 1 passes over a line spanned by some p 2 p 3 . p 3 p 2 p 1 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Order type flip events ◮ Consider a set S of n points and move a point p 1 along a line: The order type changes precisely when p 1 passes over a line spanned by some p 2 p 3 . p 3 p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Order type flip events ◮ Consider a set S of n points and move a point p 1 along a line: The order type changes precisely when p 1 passes over a line spanned by some p 2 p 3 . p 3 p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Order type flip events ◮ Consider a set S of n points and move a point p 1 along a line: The order type changes precisely when p 1 passes over a line spanned by some p 2 p 3 . ◮ We call this a ( k , l ) -flip if p 1 passes from the side of p 2 p 3 containing k points ( p 1 excluded) to the side with l points. p 3 l k p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l k p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l k p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l k p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l k p 1 p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l p 1 k p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l p 1 k p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets How flips affect the crossing number Lemma 1 A ( k , l ) -flip increases the rectilinear crossing number of S by k − l. p 3 l p 1 k p 2 O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Why halving rays are useful Halving ray: oriented line ℓ such that ◮ ℓ p O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Why halving rays are useful Halving ray: oriented line ℓ such that ◮ ◮ Passes trough exactly one extreme point p ∈ S . ◮ Splits S \ { p } into subsets of cardinalities ⌊ n − 1 2 ⌋ and ⌈ n − 1 2 ⌉ . ◮ Is oriented “away” from S . ℓ p O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Why halving rays are useful Halving ray: oriented line ℓ such that ◮ ◮ Passes trough exactly one extreme point p ∈ S . ◮ Splits S \ { p } into subsets of cardinalities ⌊ n − 1 2 ⌋ and ⌈ n − 1 2 ⌉ . ◮ Is oriented “away” from S . ℓ p O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Why halving rays are useful Halving ray: oriented line ℓ such that ◮ ◮ Passes trough exactly one extreme point p ∈ S . ◮ Splits S \ { p } into subsets of cardinalities ⌊ n − 1 2 ⌋ and ⌈ n − 1 2 ⌉ . ◮ Is oriented “away” from S . ℓ p O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Why halving rays are useful Halving ray: oriented line ℓ such that ◮ ◮ Passes trough exactly one extreme point p ∈ S . ◮ Splits S \ { p } into subsets of cardinalities ⌊ n − 1 2 ⌋ and ⌈ n − 1 2 ⌉ . ◮ Is oriented “away” from S . Lemma 2 ℓ r Let p be an extreme point of S and ℓ a halving ray for it. When moving p along ℓ in the given orientation, every flip event decreases the rectilinear crossing q number of S. p O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number

Goal Minimizing the rectilinear crossing number Flips j -facets and halving edges Halving rays ≤ j -facets Why halving rays are useful Halving ray: oriented line ℓ such that ◮ ◮ Passes trough exactly one extreme point p ∈ S . ◮ Splits S \ { p } into subsets of cardinalities ⌊ n − 1 2 ⌋ and ⌈ n − 1 2 ⌉ . ◮ Is oriented “away” from S . Lemma 2 ℓ r Let p be an extreme point of S and ℓ a halving ray for it. When moving p along ℓ in the given l ≥ ⌊ n − 1 2 ⌋ orientation, every flip event decreases the rectilinear crossing q number of S. p O. Aichholzer, D. Orden, P. Ramos On the structure of sets minimizing the crossing number