Drawing graphs with vertices and edges in convex position and large - - PowerPoint PPT Presentation

Drawing graphs with vertices and edges in convex position and large polygons in Minkowski sums

Kolja Knauer

LIF Marseille Graph Drawing, September 29, 2015

Ignacio Garc´ ıa-Marco

LIP ENS Lyon

P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in weakly convex position if P on the ”boundary”

• f the convex hull of P

consider graph drawings f : G ֒ → R2 such that:

• edges straight-line segments
• vertices and midpoints of edges on different

points P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in weakly convex position if P on the ”boundary”

• f the convex hull of P

consider graph drawings f : G ֒ → R2 such that:

• edges straight-line segments
• vertices and midpoints of edges on different

points P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in weakly convex position if P on the ”boundary”

• f the convex hull of P
• k

consider graph drawings f : G ֒ → R2 such that:

• edges straight-line segments
• vertices and midpoints of edges on different

points P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in weakly convex position if P on the ”boundary”

• f the convex hull of P
• k

not ok

consider graph drawings f : G ֒ → R2 such that:

• edges straight-line segments
• vertices and midpoints of edges on different

points P ⊆ R2 in strictly convex position if P is the set of vertices

• f the convex hull of P

P ⊆ R2 in weakly convex position if P on the ”boundary”

• f the convex hull of P
• k

not ok for i, j ∈ {s, w, a} define Gj

i

as class of graphs drawable s.th. midpoints position      strictly convex if j = s weakly convex if j = w arbitrary if j = a. vertex position      strictly convex if i = s weakly convex if i = w arbitrary if i = a.

for i, j ∈ {s, w, a} define Gj

i

as class of graphs drawable s.th. midpoints position      strictly convex if j = s weakly convex if j = w arbitrary if j = a. vertex position      strictly convex if i = s weakly convex if i = w arbitrary if i = a. Gs

s = Gs w

Gw

s

Gw

w = Gw a = Ga s = Ga w = Ga a

Gs

a

Theorem [G-M,K]:

for i, j ∈ {s, w, a} define Gj

i

as class of graphs drawable s.th. midpoints position      strictly convex if j = s weakly convex if j = w arbitrary if j = a. vertex position      strictly convex if i = s weakly convex if i = w arbitrary if i = a. Gs

s = Gs w

Gw

s

Gw

w = Gw a = Ga s = Ga w = Ga a

Gs

a

Theorem [G-M,K]:

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , ,

non-planar

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

, . . . ∈ Gs

s

, , , , , ,

so, this works...and I wont finish these drawings...

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v v doesn’t see its interior edges

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v v doesn’t see its interior edges v w every edge is seen at least once

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v v doesn’t see its interior edges v w every edge is seen at least once = ⇒ every edge is exterior at least once

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v v doesn’t see its interior edges v w every edge is seen at least once G ∈ Gw

s wlog δ > 1

= ⇒ every edge is exterior at least once

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

v w v sees vw and w does not v v has at most 2 exterior edges v v doesn’t see its interior edges v w every edge is seen at least once G ∈ Gw

s wlog δ > 1

= ⇒ every edge is exterior at least once |E(G)| = 2n−doubly exteriors

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many?

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? v w if I have an edge vw

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? v w if I have an edge vw = ⇒ ∃ doubly exterior in v ∪ H+(vw)

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? v w if I have an edge vw = ⇒ ∃ doubly exterior in v ∪ H+(vw)

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? if I have an edge vw = ⇒ ∃ doubly exterior in v ∪ H+(vw) = ⇒ ∃ 2 doubly exteriors

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? if I have an edge vw = ⇒ ∃ doubly exterior in v ∪ H+(vw) = ⇒ ∃ 2 doubly exteriors

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? if I have an edge vw = ⇒ ∃ doubly exterior in v ∪ H+(vw) = ⇒ ∃ 2 doubly exteriors

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≤ 2n − 3 :

G ∈ Gw

s wlog δ > 1

|E(G)| = 2n−doubly exteriors how many? if I have an edge vw = ⇒ ∃ doubly exterior in v ∪ H+(vw) = ⇒ ∃ 2 doubly exteriors = ⇒ ∃ 3 doubly exteriors

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≥ 2n − 3 :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≥ 2n − 3 :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≥ 2n − 3 :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≥ 2n − 3 :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gw

s (n) ≥ 2n − 3 :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gs

s(n) ≥ ⌊ 3 2n − 1⌋ :

gj

i (n) max number of edges n-vertex graph in Gj i

Conjecture [Halmann, Onn, Rothblum 07]: All graphs in Gs

s are planar and therefore gs s(n) ≤ 3n − 6.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

gs

s(n) ≥ ⌊ 3 2n − 1⌋ :

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

what is the truth?

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

what is the truth? Proposition [G-M,K]: There are cubic graphs in Gw

s .

,

, . . . CkK2

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

what is the truth? Proposition [G-M,K]: There are cubic graphs in Gw

s .

Conjecture [G-M,K]: Graphs in Gs

s are 2-degenerate.

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

what is the truth? Proposition [G-M,K]: There are cubic graphs in Gw

s .

Conjecture [G-M,K]: Graphs in Gs

s are 2-degenerate.

Proposition [G-M,K]: Gw

s is not closed under adding leafs.

?

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

what is the truth? Proposition [G-M,K]: There are cubic graphs in Gw

s .

Conjecture [G-M,K]: Graphs in Gs

s are 2-degenerate.

Proposition [G-M,K]: Gw

s is not closed under adding leafs.

Question: Is Gs

s closed under adding leafs?

Theorem [G-M,K]: We have ⌊ 3

2n − 1⌋ ≤ gs s(n) ≤ gw s (n) = 2n − 3.

what is the truth? Proposition [G-M,K]: There are cubic graphs in Gw

s .

Conjecture [G-M,K]: Graphs in Gs

s are 2-degenerate.

Proposition [G-M,K]: Gw

s is not closed under adding leafs.

Question: Is Gs

s closed under adding leafs?

Conjecture [G-M,K]: Every graph has a (multiple) subdivision in Gs

s.

What has all this to do with Minkowski sums?

What has all this to do with Minkowski sums?

A, B ⊆ Rd then A + B := {a + b | a ∈ A, b ∈ B}

What has all this to do with Minkowski sums?

A, B ⊆ Rd then A + B := {a + b | a ∈ A, b ∈ B} if G ⊂ R2 then midpoints ⊆ { 1

2(u + v) | u = v ∈ V } ⊂ 1 2(V + V )

What has all this to do with Minkowski sums?

A, B ⊆ Rd then A + B := {a + b | a ∈ A, b ∈ B} if G ⊂ R2 then midpoints ⊆ { 1

2(u + v) | u = v ∈ V } ⊂ 1 2(V + V )

largest number of convexly independent points in A + A for n-vertex convex set A ⊆ R2 is Θ(gs

s).

because only vertices can be added

What has all this to do with Minkowski sums?

A, B ⊆ Rd then A + B := {a + b | a ∈ A, b ∈ B} if G ⊂ R2 then midpoints ⊆ { 1

2(u + v) | u = v ∈ V } ⊂ 1 2(V + V )

largest number of convexly independent points in A + A for n-vertex convex set A ⊆ R2 is Θ(gs

s).

because only vertices can be added

• gj

i (n) := max n′ + m, such that G ∈ Gj i with |E(G)| = m, |V (G)| = n

and n′ of its vertices can be added to the set of midpoints, such that the resulting set is in      strictly convex if j = s weakly convex if j = w arbitrary if j = a. position. fighting for constants

• gj

i (n) := max n′ + m, such that G ∈ Gj i with |E(G)| = m, |V (G)| = n

and n′ of its vertices can be added to the set of midpoints, such that the resulting set is in      strictly convex if j = s weakly convex if j = w arbitrary if j = a. position. fighting for constants Theorem [G-M,K]: We have gw

s (n) = 2n.

• gj

i (n) := max n′ + m, such that G ∈ Gj i with |E(G)| = m, |V (G)| = n

and n′ of its vertices can be added to the set of midpoints, such that the resulting set is in      strictly convex if j = s weakly convex if j = w arbitrary if j = a. position. fighting for constants Theorem [G-M,K]: We have gw

s (n) = 2n.

Theorem [G-M,K]: We have ⌊ 3

2n⌋ ≤

gs

s(n) ≤ 2n − 2.

• gj

i (n) := max n′ + m, such that G ∈ Gj i with |E(G)| = m, |V (G)| = n

and n′ of its vertices can be added to the set of midpoints, such that the resulting set is in      strictly convex if j = s weakly convex if j = w arbitrary if j = a. position. fighting for constants Theorem [G-M,K]: We have gw

s (n) = 2n.

Theorem [G-M,K]: We have ⌊ 3

2n⌋ ≤

gs

s(n) ≤ 2n − 2.

Corollary: largest number of convexly independent points in A + A for n-vertex convex set A ⊆ R2 lies within ⌊ 3

2n⌋ and 2n − 2.

Large convexly independent sets in Minkowski sums

A, B ⊆ R2 or R3, |A| = m, |B| = n. A = B A convex B convex large convex in minkowski sum

R2

O(m

2 3 n 2 3 + m + n)

Ω(m

2 3 n 2 3 + m + n)

Ω(n

4 3 + n)

O((m + n) log(m + n))

2 3n ≤ · ≤ 2n − 2

Eisenbrand, Pach, Rothvoß, Sopher B´ ılka, Buchin, Fulek, Kiyomi, Tanigawa, T´

• th

Swanepoel, Valtr Tiwary

Large convexly independent sets in Minkowski sums

A, B ⊆ R2 or R3, |A| = m, |B| = n. A = B A convex B convex large convex in minkowski sum

R2

O(m

2 3 n 2 3 + m + n)

Ω(m

2 3 n 2 3 + m + n)

Ω(n

4 3 + n)

O((m + n) log(m + n))

2 3n ≤ · ≤ 2n − 2

Eisenbrand, Pach, Rothvoß, Sopher B´ ılka, Buchin, Fulek, Kiyomi, Tanigawa, T´

• th

Swanepoel, Valtr Tiwary linear expected

Large convexly independent sets in Minkowski sums

A, B ⊆ R2 or R3, |A| = m, |B| = n. A = B A convex B convex large convex in minkowski sum

R2 R3

O(m

2 3 n 2 3 + m + n)

Ω(m

2 3 n 2 3 + m + n)

Ω(n

4 3 + n)

O((m + n) log(m + n))

2 3n ≤ · ≤ 2n − 2

· ≤ mn

1 3n2 ≤ · ≤ 3 8n2 + O(n

3 2 )

mn ≤ ·

1 4n2 ≤ ·

Eisenbrand, Pach, Rothvoß, Sopher B´ ılka, Buchin, Fulek, Kiyomi, Tanigawa, T´

• th

Swanepoel, Valtr Tiwary Fukuda, Weibel Halman, Onn, Rothblum linear expected

· ≤ mn

Large convexly independent sets in Minkowski sums

A, B ⊆ R2 or R3, |A| = m, |B| = n. A = B A convex B convex large convex in minkowski sum

R2 R3

O(m

2 3 n 2 3 + m + n)

Ω(m

2 3 n 2 3 + m + n)

Ω(n

4 3 + n)

O((m + n) log(m + n))

2 3n ≤ · ≤ 2n − 2

· ≤ mn

1 3n2 ≤ · ≤ 3 8n2 + O(n

3 2 )

mn ≤ ·

1 4n2 ≤ ·

Eisenbrand, Pach, Rothvoß, Sopher B´ ılka, Buchin, Fulek, Kiyomi, Tanigawa, T´

• th

Swanepoel, Valtr Tiwary Fukuda, Weibel Halman, Onn, Rothblum linear expected ⇔ ∃c > 1 : Kc,c,c,c ∈ 3D − Gs

a

· ≤ mn