Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal...

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Problem session – Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran. Some problems are classic, and solved or partially solved, some other are still open. All of them are to be seen as a small sample of the well-known ability of Ferran in asking interesting questions, and proposing nice open problems. Oriol Serra: Midpoints problem Prosenjit Bose: The last open problem Ferran asked me Bernardo ´ Abrego: Crossings in proximity graphs Jean Cardinal: Alternating cell paths in bichromatic arrangements David Orden (on behalf of the Madrid group): Art gallery problems (variants) Alberto M´ arquez: Heterochromatic triangulations Oswin Aichholzer: Blocking Delaunay triangulations David Rappaport: Coins problems Alfredo Garc´ ıa: Open problems on compatible graphs Joseph Mitchell: Reflexity of point sets

Transcript of Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal...

Page 1: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Problem session – Ferran homage

This is an informal recollection of the slides used at EGC, during the problem session devotedto honor Ferran. Some problems are classic, and solved or partially solved, some other arestill open. All of them are to be seen as a small sample of the well-known ability of Ferranin asking interesting questions, and proposing nice open problems.

• Oriol Serra: Midpoints problem

• Prosenjit Bose: The last open problem Ferran asked me

• Bernardo Abrego: Crossings in proximity graphs

• Jean Cardinal: Alternating cell paths in bichromatic arrangements

• David Orden (on behalf of the Madrid group): Art gallery problems (variants)

• Alberto Marquez: Heterochromatic triangulations

• Oswin Aichholzer: Blocking Delaunay triangulations

• David Rappaport: Coins problems

• Alfredo Garcıa: Open problems on compatible graphs

• Joseph Mitchell: Reflexity of point sets

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Midpoints ProblemP a set of points in the plane in general positionM(P) set of midpoints of the

(n2

)segments determined by the points.

Question (Ferran): Determine µ(n) = min{|M(P)| : |P| = n}.

O. Serra (UPC) Midpoints problem Barcelona 2015 1 / 1

Page 3: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Midpoints ProblemP a set of points in the plane in general positionM(P) set of midpoints of the

(n2

)segments determined by the points.

Question (Ferran): Determine µ(n) = min{|M(P)| : |P| = n}.

M(P) =1

2|A+ A|

Hurtado, Urrutia [??] µ(n) = (nlog2 3).

Pach (2003) µ(n) ≤ nec log n and lim supn→∞ µ(n)/n =∞(Behrend construction+Freiman Theorem: general position→ sets with no3–term arithmetic progressions.)

Sanders (2010) µ′(n) ≥ n(log n)1/3

log log n

Problem 8.3.(Sanders 2010) Find a constant an absolute constant c > 2/3 suchthat

µ(n) ≥ n logc n.

O. Serra (UPC) Midpoints problem Barcelona 2015 1 / 1

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Introduction Results

The last open problem Ferran asked me

Prosenjit Bose

School of Computer ScienceCarleton University

Canada

The last open problem Ferran asked me

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Introduction Results

Recent Area of Interest

Biplane Graph

A straight-line embedding of a graph on a point set is biplaneprovided that the graph is the disjoint union of two plane graphs.

The last open problem Ferran asked me

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Introduction Results

Open Problem

Question

Given a set of points in the plane, can we build a biplane graphwhose spanning ratio is better than any plane graph?

The last open problem Ferran asked me

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Introduction Results

Introduction

Geometric t-spanner

The last open problem Ferran asked me

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Introduction Results

Introduction

Geometric t-spanner

The last open problem Ferran asked me

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Introduction Results

Introduction

Geometric t-spanner

The last open problem Ferran asked me

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Introduction Results

Introduction

Geometric t-spanner

Given a set P of points in the plane, a graph G is a t-spanner witht ≥ 1 provided that

∀x , y ∈ P, dG (x , y) ≤ t · |xy |

The last open problem Ferran asked me

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Introduction Results

Introduction

Current Best Result

Xia (2012) showed that the spanning ratio of the Delaunaytriangulation is at most 1.998? Can we do better for biplanegraphs?

The last open problem Ferran asked me

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Introduction Results

Biplane graph with lower spanning ratio

Two rotated copies of the empty equilateral triangle Delaunaytriangulation has better spanning ratio.

Spanning ratio of half-θ6-graph is√3 · cosα+ sinα,

where α is angle between uw and closest bisector

The last open problem Ferran asked me

Page 13: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Introduction Results

Biplane graph with lower spanning ratio

Two rotated copies of the empty equilateral triangle Delaunaytriangulation has better spanning ratio.

Spanning ratio of half-θ6-graph is√3 · cosα+ sinα,

where α is angle between uw and closest bisector

The last open problem Ferran asked me

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Introduction Results

Rotating the Cones

Two rotated copies of the empty equilateral triangle Delaunaytriangulation has better spanning ratio.

Spanning ratio of half-θ6-graph is√3 · cosα+ sinα,

where α is angle between uw and closest bisector

The last open problem Ferran asked me

Page 15: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Introduction Results

Rotating the Cones

Two rotated copies of the empty equilateral triangle Delaunaytriangulation has better spanning ratio.

Spanning ratio of half-θ6-graph is√3 · cosα+ sinα,

where α is angle between uw and closest bisector

The last open problem Ferran asked me

Page 16: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Introduction Results

Rotating the Cones

Two rotated copies of the empty equilateral triangle Delaunaytriangulation has better spanning ratio.

Spanning ratio of half-θ6-graph is√3 · cosα+ sinα,

where α is angle between uw and closest bisector

The last open problem Ferran asked me

Page 17: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Introduction Results

Rotating the Cones

Two rotated copies of the empty equilateral triangle Delaunaytriangulation has better spanning ratio.

Spanning ratio of half-θ6-graph is√3 · cosα+ sinα,

where α is angle between uw and closest bisectorRotating by π/6 gives spanning ratio of roughly 1.932

The last open problem Ferran asked me

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(2008) What is the list number of crossings in a geometric proximity graph of n points in the plane?

k-nearest neighbor graph

each point is adjacent to its k nearest neighbors

Problem. If k=9 or 10, what is the least number of crossings in the k-nearest neighbor graph of an n-element point set?

Crossings of proximity graphs

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Crossings of proximity graphs

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Cell-Paths

1 / 6

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Alternating Cell-Paths in Bichromatic Arrangements

Ferran, Sevilla, february 2013

Analogy: long plane alternating paths in red-blue points in convexposition (famous poison problem – cf. Delia’s talk this morning)

2 / 6

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Questions

• Maximum value p(n) such that every bichromatic linearrangement in the plane has an alternating cell-path oflength at least p(n)

• Maximum value f (n) such that every line arrangement in theplane has a cell-path of length at least f (n)

3 / 6

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Results

Two main papers:

1. Aichholzer, C., Hackl, Hurtado, Korman, Pilz, Silveira,Uehara, Vogtenhuber, Welzl (Sevilla ComPoSe Workshop,CCCG’13)

2. Hoffmann, Kleist, Miltzow (to appear in MFCS’15)

Results:

• f (n) ∼ n2/3.

• p(n) ≥ n.

• p(n) ≤ 2n − 1 – only one red line!

• Some “near-balanced” bichromatic arrangements do not havealternating paths longer than 2.8n

4 / 6

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Improved Upper Bound

• Hoffmann et al.: For arbitrary k, there exists an arrangementof 3k blue lines and 2k red lines with no alternating pathlonger than 14k

5 / 6

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Art Gallery Problem (variants)The guards are watched by another guards (summer of 1992)

P polygon

G set of guards

Conditions on Vis(P, G)

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Art Gallery Problem (variants)The guards are watched by another guards (summer of 1992)

Vis(P,G) is connected COOPERATIVE GUARDSVis(P,G) no isolated vertices GUARDED GUARDSConsecutive vertices CONSECUTIVE GUARDS…………………………………………..

90’s Hernández, García00’s Michael, Pinciu, Zylinski, Kosowski, Malafiejski

Orthogonal polygons, grids, coloring proofs, etc.

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Heterochromatic triangulations

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Heterochromatic triangulations

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Heterochromatic triangulations

Given two point sets (red and blue), does it admit a heterochromatictriangulation?

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Heterochromatic triangulations

Given two point sets (red and blue), does it admit a heterochromatictriangulation?

Minimum number of edges to guarantee a heterochromatic triangulation

Algorithm?

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Blocking Delaunay triangulations

Given is a set B of n blue points in general position in the plane.Add a set R of red points, such that the Delaunay triangulation ofB ∪R does not contain an edge between two blue points. We saythat the set R blocks the Delaunay edges of B. What is theminimal cardinality of R?

There are sets which require n points to be blocked [1], and it canbe shown that |R| ≤ (3/2)n is sufficient [2].

Close the gap!Conjecture [2]: n points are always necessary and sufficient.

Algorithmic question: How fast can a minimal blocking set R becomputed?

This problem was stated in [1] and also presented by Ferran duringthe open problem session in Dagstuhl 2011.

[1] B. Aronov, M. Dulieu, and F. Hurtado. Witness (delaunay) graphs. CG:TA 44(6-7):329-344, 2011.[2] O. Aichholzer, R. Fabila-Monroy, T. Hackl, M. van Kreveld, A. Pilz, P. Ramos, and B. Vogtenhuber. BlockingDelaunay Triangulations. CG:TA 46(2):154-159, 2013.

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Problem Statement

• Given a two sets of n non-overlapping unit discs (coins) determine the distance between them.

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Earth Movers Distance

• The Earth movers distance is the amount of work (moves) needed to transform one set to another.

• Work is measured by the number of straight line collision free translations of a coin (without lifting it) along a line.

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3n/2 moves are sometimes necessary to move congruentcoins from one configuration to another.

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At least two moves are needed to get the firstcoin in place.

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At least two moves are needed to get the firstcoin in place.

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At least two moves are needed to get the firstcoin in place.

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One more for the second coin.

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And so on.

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Summary

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2. A. Dumitrescu and M. Jiang, On reconfiguration of disks in the plane and related problems. Comput. Geom.. 2013, 191-202.

1. M. Abellanas, S. Bereg, F. Hurtado, A.G. Olaverri, D. Rappaport, and J. Tejel, Moving coins. Comput. Geom.. 2006, 35-48.

References

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Some open problems on

Compatible Graphs

Alfredo García and Javier Tejel

Universidad de Zaragoza, Spain

XVI EGC, Barcelona, July 1-3, 2015

Page 43: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible graphs

Given a set S of n points on the plane, two plane geometric graphs on S, G1 = (S, E1)

and G2 = (S, E2), are compatible if G1 U G2 = (S, E1 U E2) is also a plane geometric

graph.

They are disjoint if E1 ∩ E2 is empty.

Page 44: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible graphs

Given a set S of n points on the plane, two plane geometric graphs on S, G1 = (S, E1)

and G2 = (S, E2), are compatible if G1 U G2 = (S, E1 U E2) is also a plane geometric

graph.

They are disjoint if E1 ∩ E2 is empty.

Page 45: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible matchings

Conjecture: Given a perfect matching M on S, |S| = 2n and n even, is there another

perfect matching M’, disjoint with M ?

Page 46: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible matchings

Conjecture: Given a perfect matching M on S, |S| = 2n and n even, is there another

perfect matching M’, disjoint with M ?

Solved in “Disjoint Compatible Geometric Matchings”, by M. Ishaque, D. Souvaine,

C.D. Tóth, Discrete and Computational Geometry, 49, 89-131,2013.

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Compatible spanning trees

Given a plane geometric spanning tree T1, is there a compatible tree T2 disjoint with T1 ?

dS(T1) = Min. number of common edges between T1 and any other compatible tree T2

What is the value of dn= Min dS(T1) ? (|S| = n, T1 spanning tree)

Page 48: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible spanning trees

Given a plane geometric spanning tree T1, is there a compatible tree T2 disjoint with T1 ?

dS(T1) = Min. number of common edges between T1 and any other compatible tree T2

What is the value of dn= Min dS(T1) ? (|S| = n, T1 spanning tree)

Page 49: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible spanning trees

Given a plane geometric spanning tree T1, is there a compatible tree T2 disjoint with T1 ?

dS(T1) = Min. number of common edges between T1 and any other compatible tree T2

What is the value of dn= Min dS(T1) ? (|S| = n, T1 spanning tree)

Page 50: Problem session { Ferran homage€¦ · Problem session { Ferran homage This is an informal recollection of the slides used at EGC, during the problem session devoted to honor Ferran.

Compatible spanning trees

4

3

5

2 nd

nn

Conjecture: Given a plane geometric

spanning tree T1, there is another

compatible tree T2 having in common

with T1 at most n/5 + O(1) edges.

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Compatible graphs: path and matching

common edges 4

n

20

9n

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Compatible tree and matching

common edges 4

n

20

9n

Conjecture: Given a plane Hamiltonian path P1 there exists a compatible perfect

matching M1 having in common with P1 at most n/4 edges.

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Reflexivity of Point Sets• Given a set S of n points in the plane, the

reflexivity, r(S), is the minimum number of reflex vertices in a polygonalization of S

• Compute r(S) exactly or approximately?• Let R(n) = max r(S), for |S|=n• What is R(n)?– [Arkin et al]: floor(n/4) ≤ R(n) ≤ ceil(n/2)– [Ackerman,Aichholzer,Keszegh’08]: R(n) ≤ (5/12)n+o(1)

• Reflexivity of imprecise point sets?

Conjecture: floor(n/4) ≤ R(n) ≤ ceil(n/2)Conjecture: R(n) ≤ R(n+1)

r(S) is a measure of “convexity”

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7 reflex 3 reflex

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