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On metric embeddings, shortest path decompositions and face cover of planar graphs

Arnold Filtser

Ben-Gurion University

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 1 / 34

This talk is based on the following papers:

Metric Embedding via Shortest Path Decompositions Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman. A face cover perspective to ℓ1 embeddings of planar graphs Arnold Filtser

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 2 / 34

Metric Embeddings

(Rd, ·p)

f : X → Rd

(X, dX)

Embedding f ∶ X → Rd has distortion t if for all x,y ∈ X dX(x,y) ≤ ∥f (x) − f (y)∥p ≤ t ⋅ dX(x,y)

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 3 / 34

Metric Embeddings

(Rd, ·p)

f : X → Rd

(X, dX)

Embedding f ∶ X → Rd has distortion t if for all x,y ∈ X dX(x,y) ≤ ∥f (x) − f (y)∥p ≤ t ⋅ dX(x,y)

Theorem (Bourgain 85)

Every n-point metric (X,dX) is embeddable into Euclidean space with distortion O(log n).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 3 / 34

Metric Embeddings

(Rd, ·p)

f : X → Rd

(X, dX)

Embedding f ∶ X → Rd has distortion t if for all x,y ∈ X dX(x,y) ≤ ∥f (x) − f (y)∥p ≤ t ⋅ dX(x,y)

Theorem (Bourgain 85)

Every n-point metric (X,dX) is embeddable into Euclidean space with distortion O(log n). Tight.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 3 / 34

Sparsest Cut Problem

G = (V ,E,w) is a weighted graph. The sparsity S ⊆ V : φ(S) = w(E(S, ¯ S)) min{∣S∣,∣ ¯ S∣}

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34

Sparsest Cut Problem

G = (V ,E,w) is a weighted graph. The sparsity S ⊆ V : φ(S) = w(E(S, ¯ S)) min{∣S∣,∣ ¯ S∣}

S φ(S) = 15

8

¯ S

2 1 2 3 1 2 1 3 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34

Sparsest Cut Problem

G = (V ,E,w) is a weighted graph. The sparsity S ⊆ V : φ(S) = w(E(S, ¯ S)) min{∣S∣,∣ ¯ S∣}

S φ(S) = 15

8

¯ S

2 1 2 3 1 2 1 3

Sparsest Cut Problem

Find the cut with minimum sparsity minS⊊V φ(S).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34

Sparsest Cut Problem

G = (V ,E,w) is a weighted graph. The sparsity S ⊆ V : φ(S) = w(E(S, ¯ S)) min{∣S∣,∣ ¯ S∣}

S φ(S) = 15

8

¯ S

2 1 2 3 1 2 1 3

Sparsest Cut Problem

Find the cut with minimum sparsity minS⊊V φ(S). NP-hard.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34

Sparsest Cut Problem

G = (V ,E,w) is a weighted graph. The sparsity S ⊆ V : φ(S) = w(E(S, ¯ S)) min{∣S∣,∣ ¯ S∣}

Sparsest Cut Problem

Find the cut with minimum sparsity minS⊊V φ(S). NP-hard. G embeds into ℓ1 = (Rd,∥ ⋅ ∥1) with distortion t.

⇒

t-approximation for sparsest cut.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34

Special Graph Families

Bourgain’s log n distortion: best possible for general metric spaces.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34

Special Graph Families

Bourgain’s log n distortion: best possible for general metric spaces. Try special graph families:

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34

Special Graph Families

Bourgain’s log n distortion: best possible for general metric spaces. Try special graph families: Planar.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34

Special Graph Families

Bourgain’s log n distortion: best possible for general metric spaces. Try special graph families: Planar. Excluding a fixed minor.

Excluded as a minor

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34

Special Graph Families

Bourgain’s log n distortion: best possible for general metric spaces. Try special graph families: Planar. Excluding a fixed minor. Bounded treewidth.

1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34

Special Graph Families

Bourgain’s log n distortion: best possible for general metric spaces. Try special graph families: Planar. Excluding a fixed minor. Bounded treewidth. Bounded Pathwidth.

1 2 3 4 5 6 7 8 9 10 11 12

1 3 5 1 3 6 1 3 7 2 1 3 8 2 1 11 2 4 1 12 2 4 1 9 4 1 9 4

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in G/P has an SPD of depth k − 1.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in G/P has an SPD of depth k − 1. The diamond graph D4 and its SPD . The SPD depth is 4.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in G/P has an SPD of depth k − 1.

• Planar. G Planar ⇒ SPDdepth O(log n). (Cycle separator).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in G/P has an SPD of depth k − 1.

• Planar. G Planar ⇒ SPDdepth O(log n). (Cycle separator).

Minor-free. G excludes H as a minor ⇒ SPDdepth O(g(H) ⋅ log n).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in G/P has an SPD of depth k − 1.

• Planar. G Planar ⇒ SPDdepth O(log n). (Cycle separator).

Minor-free. G excludes H as a minor ⇒ SPDdepth O(g(H) ⋅ log n).

• Treewidth. G has treewidth-k ⇒ SPDdepth O(k log n).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

Definition (Depth of a Shortest Path Decompositions)

Every (weighted) path graph has an SPD of depth 1. G has an SPD of depth k if after removing some shortest path P, every connected component in G/P has an SPD of depth k − 1.

• Planar. G Planar ⇒ SPDdepth O(log n). (Cycle separator).

Minor-free. G excludes H as a minor ⇒ SPDdepth O(g(H) ⋅ log n).

• Treewidth. G has treewidth-k ⇒ SPDdepth O(k log n).
• Pathwidth. G has pathwidth-k ⇒ SPDdepth k + 1.

1 3 5 1 3 6 1 3 7 2 1 3 8 2 1 11 2 4 1 12 2 4 1 9 4 1 4 10

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 6 / 34

Shortest Path Decompositions

There is a graph with SPDdepth 2, which contain Kn as a minor.

1 1 1 1 1 1 1 1 n n n

Shortest Path Decompositions

There is a graph with SPDdepth 2, which contain Kn as a minor.

1 1 1 1 1 1 1 1 n n n

Shortest Path Decompositions

There is a unweighted graph with SPDdepth 3, containing Kn.

1 1 1 1 1 1 1 1 n

Main Result

Theorem (Embeddings by SPDdepth [Abraham, F, Gupta, Neiman 18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(k

1 p ). Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [Abraham, F, Gupta, Neiman 18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Graph Family Our results. Previous results Pathwidth k O(k

1/p)

(4k)k3+1 (only into ℓ1) [Lee and Sidiropoulos 13]

Exponential Improvement for ℓ1. First result for any p > 1.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Graph Family Our results. Previous results Pathwidth k O(k

1/p)

(4k)k3+1 (only into ℓ1) [LS13] Treewidth k O((k log n)

1/p)

O(k1−1/p ⋅ log1/p n) [Krauthgamer, Lee, Mendel, Naor 04] O((log(k log n))1−1/p(log1/p n)) [Kamma and Krauthgamer 16]

Improvement in the regime where p > 2 and n ≫ k.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Graph Family Our results. Previous results Pathwidth k O(k

1/p)

(4k)k3+1 (only into ℓ1) [LS13] Treewidth k O((k log n)

1/p)

O(k1−1/p ⋅ log1/p n) [KLMN04] O((log(k log n))1−1/p(log1/p n)) [KK16] Planar O(log

1/p n)

O(log

1/p n)

[Rao99]

New&completely different proof of important result.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Graph Family Our results. Previous results Pathwidth k O(k

1/p)

(4k)k3+1 (only into ℓ1) [LS13] Treewidth k O((k log n)

1/p)

O(k1−1/p ⋅ log1/p n) [KLMN04] O((log(k log n))1−1/p(log1/p n)) [KK16] Planar O(log

1/p n)

O(log

1/p n)

[Rao99] Kr-minor-free O((g(r)log n)

1/p)

O(r 1−1/p log

1/p n) [Abraham, Gavoille,

Gupta, Neiman, Talwar 14] + [Krauthgamer, Lee, Mendel, Naor 04]

Improvement for large enough p.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Graph Family Our results. Previous results Pathwidth k O(kmin{ 1

p , 1 2 })

(4k)k3+1 (only into ℓ1) [LS13]

Corollary

O( √ k) approximation algorithm for the sparsest cut problem on pathwidth k graphs.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Embed into both ℓ1 and ℓ2 with distortion O( √ k).

Graph Family Our results. Previous results Pathwidth k O(kmin{ 1

p , 1 2 })

(4k)k3+1 (only into ℓ1) [LS13]

Corollary

O( √ k) approximation algorithm for the sparsest cut problem on pathwidth k graphs. Best previous result: (4k)k3+1 [LS13].

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Using O(log n) dimensions for p ∈ [1,2], and O(k log n) dimensions for p > 2.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 9 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Using O(log n) dimensions for p ∈ [1,2], and O(k log n) dimensions for p > 2.

Corollary

Every Kr-free graph embeds into ℓ∞ with O(1) distortion and O(g(r) ⋅ log2 n) dimensions.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 9 / 34

Main Result

Theorem (Embeddings by SPDdepth [AFGN18])

Let G = (V ,E) be a weighted graph with SPDdepth k. Then there exists an embedding f ∶ V → ℓp with distortion O(kmin{ 1

p , 1 2}).

Using O(log n) dimensions for p ∈ [1,2], and O(k log n) dimensions for p > 2.

Corollary

Every Kr-free graph embeds into ℓ∞ with O(1) distortion and O(g(r) ⋅ log2 n) dimensions. [Krauthgamer, Lee, Mendel, Naor 04]: Every Kr-free graph embeds into ℓ∞ with O(r 2) distortion and O(3r ⋅ log r ⋅ log n) dimensions.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 9 / 34

Lower Bound

Theorem ([Newman and Rabinovich 03] [Lee and Naor 04] [Mendel and Naor 13] )

For any fixed p > 1 and every k ≥ 1, the main theorem is tight!

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 10 / 34

Lower Bound

Theorem ([Newman and Rabinovich 03] [Lee and Naor 04] [Mendel and Naor 13] )

For any fixed p > 1 and every k ≥ 1, the main theorem is tight!

(Rd, ·p) Ω(kmin{1

p,1 2})

Dk

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 10 / 34

Lower Bound

Theorem (Based on [Lee and Sidiropoulos 11])

For every k ≥ 1, there is a graph G with SPDdepth O(k) that embeds into ℓ1 with distortion Ω( √

k log k ).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 10 / 34

Lower Bound

Theorem (Based on [Lee and Sidiropoulos 11])

For every k ≥ 1, there is a graph G with SPDdepth O(k) that embeds into ℓ1 with distortion Ω( √

k log k ).

(Rd, ·1) Ω(

• k

log k)

Xk

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 10 / 34

Technical Ideas

Removing the shortest path induces a hierarchical partition. The embedding will be defined recursively using different coordinates for each cluster.

P

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 11 / 34

Technical Ideas

Removing the shortest path induces a hierarchical partition. The embedding will be defined recursively using different coordinates for each cluster.

P

Consider u,v ∈ V : For every level ∥fj(v) − fj(u)∥ = O(dG(v,u)) (Lipschitz)

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 11 / 34

Technical Ideas

Removing the shortest path induces a hierarchical partition. The embedding will be defined recursively using different coordinates for each cluster.

P

Consider u,v ∈ V : For every level ∥fj(v) − fj(u)∥ = O(dG(v,u)) (Lipschitz) There is some level s.t. ∥fj(v) − fj(u)∥ = Ω(dG(v,u)).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 11 / 34

Technical Ideas

Removing the shortest path induces a hierarchical partition. The embedding will be defined recursively using different coordinates for each cluster.

P

Consider u,v ∈ V : For every level ∥fj(v) − fj(u)∥ = O(dG(v,u)) (Lipschitz) There is some level s.t. ∥fj(v) − fj(u)∥ = Ω(dG(v,u)). As each vertex will be non-zero in only k coordinates:

The distortion will be O(k1/p).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 11 / 34

Initial Attempt

Embed vertex v relative to geodesic path P using two dim’s: First coordinate ∆1: distance to path d(v,P). Second coordinate ∆2: distance d(v,r) to endpoint of path, called its “root”.

v r

∆1 = d(v, P) ∆

2

= d ( v , r )

P

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Initial Attempt

v r

∆1 = d(v, P) ∆

2

= d ( v , r )

P

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Initial Attempt

v r

∆1 = d(v, P) ∆

2

= d ( v , r )

P

Use different ∆1 coordinate for each component. Use the same ∆2 coordinate for all components.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Initial Attempt

v r

∆1 = d(v, P) ∆

2

= d ( v , r )

P

Use different ∆1 coordinate for each component. Use the same ∆2 coordinate for all components.

X1 X2

PX (7, 0, 9) (0, 2, 5)

r

5 2 7 9

v u

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Initial Attempt

v r

∆1 = d(v, P) ∆

2

= d ( v , r )

P

Use different ∆1 coordinate for each component. Use the same ∆2 coordinate for all components.

X1 X2

P

v u r ∆X1

1

∆X2

1

∆X1

1 (u) + ∆X2 1 (v) =

d(u,P) + d(v,P) = Ω(dG(u,v))

P

v u r X1 X2 ∆2 ∆2

∣∆2(u) − ∆2(v)∣ = ∣d(u,r) − d(v,r)∣ = Ω(dG(u,v))

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Problem

But the expansion is unbounded.

Xv Xu Pv

v u rv

∆1 ∆2

d(v,Pv),d(v,rv) ≫ d(v,u)

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 13 / 34

Truncation

To avoid unbounded distortion in future levels, “truncate”!

v r

∆1 = d(v, P) ∆2 = d(v, r)

P

X

d(v, V \ X)

For each v in the cluster X, both ∆1 and ∆2 will be truncated by d(v,V /X).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 14 / 34

Truncation

To avoid unbounded distortion in future levels, “truncate”!

v r P

X

d(v, V \ X)

∆2 = m i n { d ( v , r ) , d ( v , V \ X ) } ∆1 = min{d(v, P), d(v, V \ X)}

For each v in the cluster X, both ∆1 and ∆2 will be truncated by d(v,V /X).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 14 / 34

Yet another problem...

v u

r X

∣∆2(v) − ∆2(u)∣ = ∣d(v,V /X) − d(u,V /X)∣ = 0

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 15 / 34

Yet another problem...

v u

r X

∣∆2(v) − ∆2(u)∣ = ∣d(v,V /X) − d(u,V /X)∣ = 0

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 15 / 34

Sawtooth Function: a Randomized Truncation

2t 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 2t+1

y x

2t−1

The graph of the truncation function at scale t.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation

2t 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 2t+1 x1 x2 x3

y x

2t−1

The graph of the scale t “sawtooth” function gt.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation

2t 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 2t+1 x1 x2 x3

y x

2t−1

The graph of the scale t “sawtooth” function gt. ht(x) = gt(α + β ⋅ x): Sawtooth function after a random shift and stretch.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation

2t 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 2t+1 x1 x2 x3

y x

2t−1

The graph of the scale t “sawtooth” function gt. ht(x) = gt(α + β ⋅ x): Sawtooth function after a random shift and stretch.

Lemma

Let x,y ∈ R+, if ∣x − y∣ ≤ 2t−1 then Eα,β [∣ht(x) − ht(y)∣] = Ω(∣x − y∣) .

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation

Lemma

Let x,y ∈ R+, if ∣x − y∣ ≤ 2t−1 then Eα,β [∣ht(x) − ht(y)∣] = Ω(∣x − y∣) .

v u

r X

E[∣ht(∆2(v)) − ht(∆2(u))∣] = Ω(∣∆2(v) − ∆2(u)∣) = Ω(d(u,v))

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation

Lemma

Let x,y ∈ R+, if ∣x − y∣ ≤ 2t−1 then Eα,β [∣ht(x) − ht(y)∣] = Ω(∣x − y∣) .

v u

r X

E[∣ht(∆2(v)) − ht(∆2(u))∣] = Ω(∣∆2(v) − ∆2(u)∣) = Ω(d(u,v))

This all nice, but which scale should we use?

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation

Lemma

Let x,y ∈ R+, if ∣x − y∣ ≤ 2t−1 then Eα,β [∣ht(x) − ht(y)∣] = Ω(∣x − y∣) .

v u

r X

E[∣ht(∆2(v)) − ht(∆2(u))∣] = Ω(∣∆2(v) − ∆2(u)∣) = Ω(d(u,v))

This all nice, but which scale should we use? A smooth combination of the scales around d(v,V /X).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Lemma (Contraction Bound)

For any vertices u,v, there exists some coordinate j such that E[∣fj(v) − fj(u)∣] = Ω(dG(u,v)) .

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 17 / 34

Lemma (Contraction Bound)

For any vertices u,v, there exists some coordinate j such that E[∣fj(v) − fj(u)∣] = Ω(dG(u,v)) . j is the minimal level s.t (1) v and u are in different components of X/PX. OR (2) min{dG(v,PX),dG(u,PX)} ≤ dG(u,v)/12.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 17 / 34

Lemma (Contraction Bound)

For any vertices u,v, there exists some coordinate j such that E[∣fj(v) − fj(u)∣] = Ω(dG(u,v)) . j is the minimal level s.t (1) v and u are in different components of X/PX. OR (2) min{dG(v,PX),dG(u,PX)} ≤ dG(u,v)/12.

P

v u r X1 X2 ∆2 ∆2 X1 X2

P

v u r ∆X1

1

∆X2

1 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 17 / 34

Planar Graphs into ℓ1

Planar Graphs into ℓ1

Graph Family Our results. Previous results Planar O(√log n) O(√log n) [Rao99]

Planar Graphs into ℓ1

Graph Family Our results. Previous results Planar O(√log n) O(√log n) [Rao99]

GNRS Conjecture

Planar graphs embed into ℓ1 with constant distortion.

Planar Graphs into ℓ1

Terminal Problem

Given a set K of terminals, embed K into ℓ1.

Planar Graphs into ℓ1

Terminal Problem

Given a set K of terminals, embed K into ℓ1.

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F.

Planar Graphs into ℓ1

Terminal Problem

Given a set K of terminals, embed K into ℓ1.

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover.

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover.

Theorem ([Okamura and Seymour 81])

If γ(G,K) = 1, then K embeds isometrically into ℓ1.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover.

Theorem ([Okamura and Seymour 81])

If γ(G,K) = 1, then K embeds isometrically into ℓ1.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover. Suppose γ(G,K) = γ, then K embeds into ℓ1 with distortion:

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover. Suppose γ(G,K) = γ, then K embeds into ℓ1 with distortion: [Lee and Sidiropoulos 09] (implicitly): 2O(γ).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover. Suppose γ(G,K) = γ, then K embeds into ℓ1 with distortion: [Lee and Sidiropoulos 09] (implicitly): 2O(γ). [Chekuri, Shepherd, Weibel 13]: 3 ⋅ γ.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover. Suppose γ(G,K) = γ, then K embeds into ℓ1 with distortion: [Lee and Sidiropoulos 09] (implicitly): 2O(γ). [Chekuri, Shepherd, Weibel 13]: 3 ⋅ γ. [Krauthgamer, Lee, Rika 19]: O(log γ).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover. Suppose γ(G,K) = γ, then K embeds into ℓ1 with distortion: [Lee and Sidiropoulos 09] (implicitly): 2O(γ). [Chekuri, Shepherd, Weibel 13]: 3 ⋅ γ. [Krauthgamer, Lee, Rika 19]: O(log γ). (Actually this is a stochastic embedding into trees).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ1

Definition (Face Cover)

Set of faces F, s.t. every terminal lays on some face F ∈ F. γ(G,K) ∶ minimal size of a face cover. Suppose γ(G,K) = γ, then K embeds into ℓ1 with distortion: [Lee and Sidiropoulos 09] (implicitly): 2O(γ). [Chekuri, Shepherd, Weibel 13]: 3 ⋅ γ. [Krauthgamer, Lee, Rika 19]: O(log γ).

Theorem ([F19])

K embeds into ℓ1 with distortion O(√log γ).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Partial Shortest Path Decompositions

Definition (PSPD )

Similarly to SPD , PSPD is hierarchical decomposition of a graph. However, we allow to leave non-empty subgraphs in the final hierarchal stage.

Partial Shortest Path Decompositions

Definition (PSPD )

Similarly to SPD , PSPD is hierarchical decomposition of a graph. However, we allow to leave non-empty subgraphs in the final hierarchal stage. The remainder of the PSPD is a pair {C,B}. C: is the set of final level clusters. B: all the removed paths, also called boundary.

Partial Shortest Path Decompositions

Definition (PSPD )

Similarly to SPD , PSPD is hierarchical decomposition of a graph. However, we allow to leave non-empty subgraphs in the final hierarchal stage. The remainder of the PSPD is a pair {C,B}. C: is the set of final level clusters. B: all the removed paths, also called boundary. In SPD , C = ∅, B = V .

Partial Shortest Path Decompositions

Partial Shortest Path Decompositions

Partial Shortest Path Decompositions

Partial Shortest Path Decompositions

Partial Shortest Path Decompositions

Partial Shortest Path Decompositions

PSPD depth=2

Partial Shortest Path Decompositions

PSPD depth=2 B: Boundary

Partial Shortest Path Decompositions

PSPD depth=2 B: Boundary C: Remaining clusters

Theorem (Implicit in [AFGN18])

Suppose G has PSPD of depth k with remainder {C,B}. Then there f ∶ G → ℓ1 s.t. ∀u,v:

Theorem (Implicit in [AFGN18])

Suppose G has PSPD of depth k with remainder {C,B}. Then there f ∶ G → ℓ1 s.t. ∀u,v:

Lemma (Expansion Bound)

For every scale j, fj is Lipshitz.

Theorem (Implicit in [AFGN18])

Suppose G has PSPD of depth k with remainder {C,B}. Then there f ∶ G → ℓ1 s.t. ∀u,v:

1

∥f (v) − f (u)∥1 ≤ O( √ k) ⋅ dG(u,v).

Lemma (Expansion Bound)

For every scale j, fj is Lipshitz.

Theorem (Implicit in [AFGN18])

Suppose G has PSPD of depth k with remainder {C,B}. Then there f ∶ G → ℓ1 s.t. ∀u,v:

1

∥f (v) − f (u)∥1 ≤ O( √ k) ⋅ dG(u,v).

Lemma (Contraction Bound)

For any vertices u,v, there exists some coordinate j such that E[∣fj(v) − fj(u)∣] = Ω(dG(u,v)) . j is the minimal level s.t (1) v and u are in different components of X/PX. OR (2) min{dG(v,PX),dG(u,PX)} ≤ dG(u,v)/12.

Theorem (Implicit in [AFGN18])

Suppose G has PSPD of depth k with remainder {C,B}. Then there f ∶ G → ℓ1 s.t. ∀u,v:

1

∥f (v) − f (u)∥1 ≤ O( √ k) ⋅ dG(u,v).

2

If either u,v not belong to the same cluster in C,

• r

min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) .

Lemma (Contraction Bound)

For any vertices u,v, there exists some coordinate j such that E[∣fj(v) − fj(u)∣] = Ω(dG(u,v)) . j is the minimal level s.t (1) v and u are in different components of X/PX. OR (2) min{dG(v,PX),dG(u,PX)} ≤ dG(u,v)/12.

Theorem (Path Separator)

There are shortest paths P1,P2, s.t. for every connected component C in G/{P1 ∪ P2} it holds γ(G[C],K ∩ C) ≤ 2 3 ⋅ γ(G,K) + 1 .

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 22 / 34

Theorem (Path Separator)

There are shortest paths P1,P2, s.t. for every connected component C in G/{P1 ∪ P2} it holds γ(G[C],K ∩ C) ≤ 2 3 ⋅ γ(G,K) + 1 .

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 22 / 34

Theorem (Path Separator)

There are shortest paths P1,P2, s.t. for every connected component C in G/{P1 ∪ P2} it holds γ(G[C],K ∩ C) ≤ 2 3 ⋅ γ(G,K) + 1 .

P1 P2

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 22 / 34

Theorem (Path Separator)

There are shortest paths P1,P2, s.t. for every connected component C in G/{P1 ∪ P2} it holds γ(G[C],K ∩ C) ≤ 2 3 ⋅ γ(G,K) + 1 .

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 22 / 34

Theorem (Path Separator)

There are shortest paths P1,P2, s.t. for every connected component C in G/{P1 ∪ P2} it holds γ(G[C],K ∩ C) ≤ 2 3 ⋅ γ(G,K) + 1 .

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 22 / 34

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1. There is an embedding f ∶ G → ℓ1 s.t. for every u,v:

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1. There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v).

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1. There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v). If either u,v not belong to the same cluster,

• r min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) .

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1. There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v). If either u,v not belong to the same cluster,

• r min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) . Each C ∈ C is OS-graph.

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1. There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v). If either u,v not belong to the same cluster,

• r min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) . Each C ∈ C is OS-graph. OS-graphs embed isometrically into ℓ1.

Corollary

There is a PSPD of depth O(log(γ)) with remainder (C,B), s.t. for every cluster C ∈ C, γ(C,K ∩ C) ≤ 1. There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v). If either u,v not belong to the same cluster,

• r min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) . Each C ∈ C is OS-graph. OS-graphs embed isometrically into ℓ1. Embed each C ∈ C using different coordinates!

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

ε

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

ε

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 24 / 34

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I].

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 25 / 34

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I].

I B

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 25 / 34

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I].

F

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 25 / 34

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I]. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = dG(v,B).

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , if min{dG(v,B),dG(u,B)} ≥ dG (u,v)

12

then ∥f (v) − f (u)∥1 = Ω(dG(v,u)).

B F

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 25 / 34

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I]. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = dG(v,B).

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , if min{dG(v,B),dG(u,B)} ≥ dG (u,v)

12

then ∥f (v) − f (u)∥1 = Ω(dG(v,u)). There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v). If either u,v not belong to the same cluster,

• r min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) .

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I]. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = dG(v,B).

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , if min{dG(v,B),dG(u,B)} ≥ dG (u,v)

12

then ∥f (v) − f (u)∥1 = Ω(dG(v,u)). There is an embedding f ∶ G → ℓ1 s.t. for every u,v: ∥f (u) − f (v)∥1 ≤ O( √ log(γ)) ⋅ dG(u,v). If either u,v not belong to the same cluster,

• r min{dG(v,B),dG(u,B)} ≤ dG (u,v)

12

then ∥f (u) − f (v)∥1 = Ω(dG(u,v)) .

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Theorem ([F19])

K embeds into ℓ1 with distortion O( √ log γ(G,K)).

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I]. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = dG(v,B).

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , if min{dG(v,B),dG(u,B)} ≥ dG (u,v)

12

then ∥f (v) − f (u)∥1 = Ω(dG(v,u)).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 27 / 34

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I]. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = dG(v,B).

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , if min{dG(v,B),dG(u,B)} ≥ dG (u,v)

12

then ∥f (v) − f (u)∥1 = Ω(dG(v,u)).

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F face, t > 0 parameter. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 27 / 34

Theorem (OS into Outer-Planar [Englert, Gupta, Krauthgamer, R¨ acke, Talgam-Cohen, Talwar 14])

G planar, F a face. Then ∃ stochastic embedding of F into

• uter-planar graphs with O(1) distortion.

f1 f2 fs

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 28 / 34

Theorem (OS into Outer-Planar [EGKRTT14])

G planar, F a face. Then ∃ stochastic embedding of F into

• uter-planar graphs with O(1) distortion.

f1 f2 fs

∀u,v ∈ F, Ei[dGi(fi(u),fi(v))] = O(dG(u,v))

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 28 / 34

Theorem (Outer-Planar into Trees [Gupta, Newman, Rabinovich, Sinclair 04])

G outer-planar, then ∃ stochastic embedding of G into trees with O(1) distortion.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 29 / 34

Theorem (Outer-Planar into Trees [GNRS04])

G outer-planar, then ∃ stochastic embedding of G into trees with O(1) distortion.

f1 f2 fs

∀u,v ∈ G, Ei[dTi(fi(u),fi(v))] = O(dG(u,v))

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 29 / 34

Theorem (OS into Outer-Planar [EGKRTT14])

G planar, F a face. Then ∃ stochastic embedding of F into

• uter-planar graphs with O(1) distortion.

Theorem (Outer-Planar into Trees [GNRS04])

G outer-planar, then ∃ stochastic embedding of G into trees with O(1) distortion.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 30 / 34

Theorem (OS into Outer-Planar [EGKRTT14])

G planar, F a face. Then ∃ stochastic embedding of F into

• uter-planar graphs with O(1) distortion.

Theorem (Outer-Planar into Trees [GNRS04])

G outer-planar, then ∃ stochastic embedding of G into trees with O(1) distortion.

Corollary (OS into Trees)

G planar, F a face. Then ∃ stochastic embedding of F into trees with O(1) distortion.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 30 / 34

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

t 2

x

Add new vertex x.

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

t 2

x

Add new vertex x. T ∪ {x} has treewidth 2.

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

t 2

x

Add new vertex x. T ∪ {x} has treewidth 2.

Theorem ([CJLV08])

Treewidth 2 graphs embed into ℓ1 with distortion 2.

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

t 2

x

Add new vertex x. T ∪ {x} has treewidth 2.

Theorem ([CJLV08])

Treewidth 2 graphs embed into ℓ1 with distortion 2. W.l.o.g. f (x) = ⃗ 0.

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

Corollary (OS into Trees)

G planar, F a face. Then ∃ stochastic embedding of F into trees with O(1) distortion.

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → ℓ1 s.t.:

1

∀v ∈ T, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

Corollary (OS into Trees)

G planar, F a face. Then ∃ stochastic embedding of F into trees with O(1) distortion.

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F face, t > 0 parameter. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}.

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F a face, t > 0 parameter. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}. Smooth combination of all scales.

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F a face, t > 0 parameter. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = t.

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , ∥f (v) − f (u)∥1 ≥ min{dG(v,u),t}. Smooth combination of all scales.

Lemma (Truncated Okamura Seymour)

I ⊍B = V , F face in G[I]. Then ∃f ∶ F → ℓ1 s.t.:

1

∀v ∈ F, ∥f (v)∥1 = dG(v,B).

2

Lipschitz: ∀u,v ∈ F, ∥f (v) − f (u)∥1 ≤ O(dG(v,u)).

3

Contraction: ∀u,v ∈ V , if min{dG(v,B),dG(u,B)} ≥ dG (u,v)

12

then ∥f (v) − f (u)∥1 = Ω(dG(v,u)).

A Sample of Open Questions

Embed planar graphs into ℓ1 (or show a L.B.). Embed graphs with treewidth k into ℓ1 with distortion g(k) (or show a L.B.).

▸ At least graphs with treewidth 3?

Embed graphs with pathwidth k into distribution over tress with expected distortion O(k) (or show a L.B.). Embed graphs with pathwidth k into distribution over spanning tress with expected distortion g(k) (or show a L.B.).

GNRS Conjecture

For every graph family F F embeds to ℓ1 with distortion O(1) ⇐ ⇒ F excludes a fixed minor

Thank you for listening!

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 34 / 34