On Metric Embeddings, Shortest Path Decompositions and Face Cover of Planar Graphs

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 `1 embeddings of planar graphs Arnold Filtser

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 2 / 34 Theorem (Bourgain 85)

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

Metric Embeddings

(X, dX)

(Rd, k·k ) f : X → Rd p

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

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

Metric Embeddings

(X, dX)

(Rd, k·k ) f : X → Rd p

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

(X, dX)

(Rd, k·k ) f : X → Rd p

Embedding f ∶ X → Rd has distortion t if for all x, y ∈ X dX (x, y) ≤ Yf (x) − f (y)Yp ≤ 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 NP-hard.

Sparsest Cut Problem

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

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

Sparsest Cut Problem

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

1 1 3 15 2 φ(S) = 8 2 3 2 1 S¯

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

Sparsest Cut Problem

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

1 1 3 15 2 φ(S) = 8 2 3 2 1 S¯

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 : ¯ wE(S, S) S φ( ) = ¯ min SSS, SSS S

1 1 3 15 2 φ(S) = 8 2 3 2 1 S¯

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 : ¯ wE(S, S) S φ( ) = ¯ min SSS, SSS

Sparsest Cut Problem

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

NP-hard.

d G embeds into `1 = (R , Y ⋅ Y1) t-approximation for with distortion t. ⇒ sparsest cut.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 4 / 34 Planar. Excluding a ﬁxed minor. Bounded treewidth. Bounded Pathwidth.

Try special graph families:

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 Planar. Excluding a ﬁxed minor. Bounded treewidth. Bounded Pathwidth.

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 Excluding a ﬁxed minor. Bounded treewidth. Bounded Pathwidth.

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 Bounded treewidth. Bounded Pathwidth.

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 Bounded Pathwidth.

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

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. 3 Excluding a ﬁxed minor. 5 7 Bounded treewidth. Bounded Pathwidth. 6 8 1 2 9 11

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

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 5 / 34 G has an SPD of depth k if after removing some shortest path P, every connected component in GP has an SPD of depth k − 1.

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 GP 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 GP 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

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

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

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

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

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. n n 1 1

1 1

1 1 n Shortest Path Decompositions

There is a graph with SPDdepth 2, which contain Kn as a minor. n n 1 1

1 1

1 1 n Shortest Path Decompositions

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

1 1

1 1 Main Result

Theorem (Embeddings by SPDdepth [Abraham, F, Gupta, Neiman 18]) Let G = (V , E) be a weighted graph with SPDdepth k. Then there 1 exists an embedding f ∶ V → `p with distortion O(k 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 min{ 1 , 1 } exists an embedding f ∶ V → `p with distortion O(k p 2 ). √ Embed into both `1 and `2 with distortion O( k).

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34 [LS13]

Main Result

Theorem (Embeddings by SPDdepth[A FGN18]) Let G = (V , E) be a weighted graph with SPDdepth k. Then there min{ 1 , 1 } exists an embedding f ∶ V → `p with distortion O(k p 2 ). √ Embed into both `1 and `2 with distortion O( k).

Graph Family Our results. Previous results

1~p k3+1 Pathwidth k O(k ) (4k) (only into `1) [Lee and Sidiropoulos 13]

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

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 8 / 34 [KLMN04]

[KK16]

Main Result

Graph Family Our results. Previous results

1~p k3+1 Pathwidth k O(k ) (4k) (only into `1) [LS13] 1~ − ~ 1~p Treewidth k O((k log n) p) O(k1 1 p ⋅ log n) [Krauthgamer, Lee, Mendel, Naor 04] − ~ 1~p O((log(k log n))1 1 p(log 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

Graph Family Our results. Previous results

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

Graph Family Our results. Previous results

1~p k3+1 Pathwidth k O(k ) (4k) (only into `1) [LS13] 1~ − ~ 1~p Treewidth k O((k log n) p) O(k1 1 p ⋅ log n) [KLMN04] − ~ 1~p O((log(k log n))1 1 p(log n)) [KK16] 1~p 1~p Planar O(log n) O(log n) [Rao99] 1 1~p 1−1~p ~p Kr -minor-free O((g(r) log n) ) O(r log 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

Graph Family Our results. Previous results 1 1 min{ , } k3+1 Pathwidth k O(k p 2 ) (4k) (only into `1) [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

3+ Best previous result: (4k)k 1 [LS13].

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

Theorem (Embeddings by SPDdepth[A FGN18]) Let G = (V , E) be a weighted graph with SPDdepth k. Then there min{ 1 , 1 } exists an embedding f ∶ V → `p with distortion O(k p 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[A FGN18]) Let G = (V , E) be a weighted graph with SPDdepth k. Then there min{ 1 , 1 } exists an embedding f ∶ V → `p with distortion O(k p 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 2 into `∞ with O 1 distortion and O g r log n dimensions.

( ) ( ( ) ⋅ )

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

Theorem (Embeddings by SPDdepth[A FGN18]) Let G = (V , E) be a weighted graph with SPDdepth k. Then there min{ 1 , 1 } exists an embedding f ∶ V → `p with distortion O(k p 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 2 into `∞ with O 1 distortion and O g r log n dimensions.

( ) ( ( ) ⋅ ) [Krauthgamer, Lee, Mendel, Naor 04]: Every Kr -free graph embeds 2 r into `∞ with O r distortion and O 3 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!

Dk d 1 1 (R , k·kp) Ω(kmin{p,2})

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 k embeds into `1 with distortion Ω log k . ¼ ( )

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 k embeds into `1 with distortion Ω log k . ¼ ( )

Xk d v (R , k·k ) u 1 u u k Ω(tlog k)

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 10 / 34 Consider u, v ∈ V : For every level Yfj (v) − fj (u)Y = O(dG (v, u)) (Lipschitz) There is some level s.t. Yfj (v) − fj (u)Y = Ω(dG (v, u)). As each vertex will be non-zero in only k coordinates: 1 The distortion will be O(k ~p).

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

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 11 / 34 There is some level s.t. Yfj (v) − fj (u)Y = Ω(dG (v, u)). As each vertex will be non-zero in only k coordinates: 1 The distortion will be O(k ~p).

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

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

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 11 / 34 As each vertex will be non-zero in only k coordinates: 1 The distortion will be O(k ~p).

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

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

Consider u, v ∈ V : For every level Yfj (v) − fj (u)Y = O(dG (v, u)) (Lipschitz) There is some level s.t. Yfj (v) − fj (u)Y = Ω(dG (v, u)). As each vertex will be non-zero in only k coordinates: 1 The distortion will be O(k ~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 ) (v, r = d ∆ = d(v, P ) ∆2 1 r P

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34 Initial Attempt v ) (v, r = d ∆ = d(v, P ) ∆2 1 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 ) (v, r = d ∆ = d(v, P ) ∆2 1 r P

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

PX 5 9 u v (7, 0, 9) 2 (0, 2, 5) 7 X1 X2

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34 Initial Attempt v ) (v, r = d ∆ = d(v, P ) ∆2 1 r P ∆∆2 2

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

P P v

X2 X2 X1 X1 X2 u ∆2 ∆1 v u X1 ∆1 ∆2

r r X1 X2 ∆1 (u) + ∆1 (v) = S∆2(u) − ∆2(v)S = d(u, P) + d(v, P) = Ω(dG (u, v)) Sd(u, r) − d(v, r)S = 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.

∆1 Xv rv ∆2 v u Xu

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”!

X d(v, V \ X) v ) (v, r ∆1 = d(v, P ) = d ∆2 r P

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”!

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

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... X v u r

S∆2(v) − ∆2(u)S = Sd(v, V X ) − d(u, V X )S = 0

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 15 / 34 Yet another problem... X v u r

S∆2(v) − ∆2(u)S = Sd(v, V X ) − d(u, V X )S = 0

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 15 / 34 Sawtooth Function: a Randomized Truncation

y 2t

2t−1 x 0 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 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 ht (x) = gt (α + β ⋅ x): Sawtooth function after a random shift and stretch.

Sawtooth Function: a Randomized Truncation

y 2t x2

t−1 x1 2 x3 x 0 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 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

y 2t x2

t−1 x1 2 x3 x 0 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 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

y 2t x2

t−1 x1 2 x3 x 0 2t+1 2 · 2t+1 3 · 2t+1 4 · 2t+1 5 · 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 t−1 Let x, y ∈ R+, if Sx − yS ≤ 2 then

Eα,β [Sht (x) − ht (y)S] = Ω(Sx − yS) .

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34 This all nice, but which scale should we use? A smooth combination of the scales around d(v, V X ).

Sawtooth Function: a Randomized Truncation Lemma t−1 Let x, y ∈ R+, if Sx − yS ≤ 2 then

Eα,β [Sht (x) − ht (y)S] = Ω(Sx − yS) . X v u r

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

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34 A smooth combination of the scales around d(v, V X ).

Sawtooth Function: a Randomized Truncation Lemma t−1 Let x, y ∈ R+, if Sx − yS ≤ 2 then

Eα,β [Sht (x) − ht (y)S] = Ω(Sx − yS) . X v u r

E [Sht(∆2(v)) − ht(∆2(u))S] = Ω (S∆2(v) − ∆2(u)S) = 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 t−1 Let x, y ∈ R+, if Sx − yS ≤ 2 then

Eα,β [Sht (x) − ht (y)S] = Ω(Sx − yS) . X v u r

E [Sht(∆2(v)) − ht(∆2(u))S] = Ω (S∆2(v) − ∆2(u)S) = 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 [Sfj (v) − fj (u)S] = 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 [Sfj (v) − fj (u)S] = 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. ≤

(1) v and u are in diﬀerent components of X PX . ∆2 OR (2) min {dG (v, PX ), dG (u, PX )} dG (u, v)~12. ≤ P P v

X2 X2 X1 X1 X2 u ∆2 ∆1 v u X1 ∆1 ∆2

r r

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

Graph Family Our results. Previous results √ √ Planar O( log n) O( log n) [Rao99] Planar Graphs into `1

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

Planar graphs embed into `1 with constant distortion. γ G, K ∶ minimal size of a face cover. ( )

Deﬁnition (Face Cover) Set of faces , s.t. every terminal lays on some face F ∈ F. F

Planar Graphs into `1

Terminal Problem

Given a set K of terminals, embed K into `1. γ G, K ∶ minimal size of a face cover. ( )

Planar Graphs into `1

Terminal Problem

Given a set K of terminals, embed K into `1.

Deﬁnition (Face Cover) Set of faces , s.t. every terminal lays on some face F ∈ F. F Planar Graphs into `1

Terminal Problem

Given a set K of terminals, embed K into `1.

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

Planar Graphs into `1 Deﬁnition (Face Cover) Set of faces , s.t. every terminal lays on some face F ∈ F. γ G, K ∶ minimal size of a face cover. F Theorem( ) ([Okamura and Seymour 81])

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

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34 [Lee and Sidiropoulos 09] (implicitly): 2O(γ). [Chekuri, Shepherd, Weibel 13]: 3 ⋅ γ. [Krauthgamer, Lee, Rika 19]: O(log γ).

Planar Graphs into `1 Deﬁnition (Face Cover) Set of faces , s.t. every terminal lays on some face F ∈ F. γ G, K ∶ minimal size of a face cover. F Theorem( ) ([Okamura and Seymour 81])

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

Planar Graphs into `1 Deﬁnition (Face Cover) Set of faces , s.t. every terminal lays on some face F ∈ F. γ G, K ∶ minimal size of a face cover. F Suppose( )γ(G, K) = γ, then K embeds into `1 with distortion:

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34 [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 [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 `1 Deﬁnition (Face Cover) Set of faces , s.t. every terminal lays on some face F ∈ F. γ G, K ∶ minimal size of a face cover. F Suppose( )γ(G, K) = γ, then K embeds into `1 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 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 . PSPD depth=2 B: Boundary C: Remaining clusters

Partial Shortest Path Decompositions PSPD depth=2 B: Boundary C: Remaining clusters

Partial Shortest Path Decompositions B: Boundary C: Remaining clusters

Partial Shortest Path Decompositions

PSPD depth=2 C: Remaining clusters

Partial Shortest Path Decompositions

PSPD depth=2 B: Boundary Partial Shortest Path Decompositions

PSPD depth=2 B: Boundary C: Remaining clusters if her

√ 1 Yf (v) − f (u)Y1 ≤ O( k) ⋅ dG (u, v). 2 If either u, v not belong to the same cluster in C, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

Theorem (Implicit in[A FGN18]) Suppose G has PSPD of depth k with remainder {C, B}. Then there f ∶ G → `1 s.t. u, v: ∀ if her

√ 1 Yf (v) − f (u)Y1 ≤ O( k) ⋅ dG (u, v). 2 If either u, v not belong to the same cluster in C, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

Theorem (Implicit in[A FGN18]) Suppose G has PSPD of depth k with remainder {C, B}. Then there f ∶ G → `1 s.t. u, v: ∀

Lemma (Expansion Bound)

For every scale j, fj is Lipshitz. if her

2 If either u, v not belong to the same cluster in C, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

Theorem (Implicit in[A FGN18]) Suppose G has PSPD of depth k with remainder {C, B}. Then there f ∶ G → `1 s.t. u, v: √ 1 Yf (v) − f (u)Y1 ≤ O( k) ⋅ dG (u, v). ∀

Lemma (Expansion Bound)

For every scale j, fj is Lipshitz. if her

2 If either u, v not belong to the same cluster in C, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

Theorem (Implicit in[A FGN18]) Suppose G has PSPD of depth k with remainder {C, B}. Then there f ∶ G → `1 s.t. u, v: √ 1 Yf (v) − f (u)Y1 ≤ O( k) ⋅ dG (u, v). ∀

Lemma (Contraction Bound) For any vertices u, v, there exists some coordinate j such that E [Sfj (v) − fj (u)S] = 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. ≤ if her

Theorem (Implicit in[A FGN18]) Suppose G has PSPD of depth k with remainder {C, B}. Then there f ∶ G → `1 s.t. u, v: √ 1 Yf (v) − f (u)Y1 ≤ O( k) ⋅ dG (u, v). ∀ 2 If either u, v not belong to the same cluster in C, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

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

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 2 γ G C , K C γ G, K 1 . ( [ ] ∩ ) ≤ 3 ⋅ ( ) +

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 2 γ G C , K C γ G, K 1 . ( [ ] ∩ ) ≤ 3 ⋅ ( ) +

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 2 γ G C , K C γ G, K 1 . ( [ ] ∩ ) ≤ 3 ⋅ ( ) +

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 2 γ G C , K C γ G, K 1 . ( [ ] ∩ ) ≤ 3 ⋅ ( ) +

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 » Yf (u) − f (v)Y1 ≤ O( log(γ)) ⋅ dG (u, v). If either u, v not belong to the same cluster, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = 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 → `1 s.t. for every u, v: If either u, v not belong to the same cluster, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = 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 → `1 s.t. for every u, v: » Yf (u) − f (v)Y1 ≤ 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 → `1 s.t. for every u, v: » Yf (u) − f (v)Y1 ≤ O( log(γ)) ⋅ dG (u, v). If either u, v not belong to the same cluster, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then

Yf (u) − f (v)Y1 = 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.

Yf (u) − f (v)Y1 = 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.

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

Each C ∈ C is OS-graph.

OS-graphs embed isometrically into `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.

Yf (u) − f (v)Y1 = Ω(dG (u, v)) .

Each C ∈ C is OS-graph.

OS-graphs embed isometrically into `1.

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 Then ∃f ∶ F → `1 s.t.: 1 ∀v ∈ F , Yf (v)Y1 = dG (v, B). 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). d (u,v) 3 G Contraction: ∀u, v ∈ V , if min {dG (v, B), dG (u, B)} ≥ 12 then Yf (v) − f (u)Y1 = Ω(dG (v, u)).

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 Then ∃f ∶ F → `1 s.t.: 1 ∀v ∈ F , Yf (v)Y1 = dG (v, B). 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). d (u,v) 3 G Contraction: ∀u, v ∈ V , if min {dG (v, B), dG (u, B)} ≥ 12 then Yf (v) − f (u)Y1 = Ω(dG (v, u)).

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)

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 25 / 34 Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)).

Lemma (Truncated Okamura Seymour)

I " B = V , F face in G[I]. Then ∃f ∶ F → `1 s.t.: 1 ∀v ∈ F , Yf (v)Y1 = dG (v, B). 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). d (u,v) 3 G Contraction: ∀u, v ∈ V , if min {dG (v, B), dG (u, B)} ≥ 12 then Yf (v) − f (u)Y1 = Ω(dG (v, u)). There is an embedding f G ` s.t. for every u, v: »∶ → 1 Yf (u) − f (v)Y1 ≤ O( log(γ)) ⋅ dG (u, v). If either u, v not belong to the same cluster, dG (u,v) or min {dG (v, B), dG (u, B)} ≤ 12 then Yf (u) − f (v)Y1 = Ω(dG (u, v)) . Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Theorem ([F19]) » K embeds into `1 with distortion O( log γ(G, K)). Lemma (Truncated Okamura Seymour)

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 27 / 34 Lemma (Truncated Okamura Seymour)

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F face, t > 0 parameter. Then ∃f ∶ F → `1 s.t.: 1 ∀v ∈ F , Yf (v)Y1 = t. 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). 3 Contraction: ∀u, v ∈ V , Yf (v) − f (u)Y1 ≥ min{dG (v, u), t}.

Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 27 / 34 ∀u, v ∈ F , Ei [dGi (fi (u), fi (v))] = O(dG (u, v))

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

Theorem (OS into Outer-Planar [EGKRTT14]) G planar, F a face. Then ∃ stochastic embedding of F into outer-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 outer-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 → `1 s.t.: 1 ∀v ∈ T , Yf (v)Y1 = t. 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). 3 Contraction: ∀u, v ∈ V , Yf (v) − f (u)Y1 ≥ min{dG (v, u), t}. Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → `1 s.t.: 1 ∀v ∈ T , Yf (v)Y1 = t. 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). 3 Contraction: ∀u, v ∈ V , Yf (v) − f (u)Y1 ≥ min{dG (v, u), t}.

Add new vertex x.

x t 2 Lemma (Uniformly Truncated Trees)

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

x t 2 Lemma (Uniformly Truncated Trees)

Add new vertex x. T ∪ {x} has treewidth 2. Theorem ([CJLV08]) Treewidth 2 graphs embed into `1 with distortion 2.

x t 2 Lemma (Uniformly Truncated Trees)

Add new vertex x. T ∪ {x} has treewidth 2. Theorem ([CJLV08]) Treewidth 2 graphs embed into `1 with distortion 2.

W.l.o.g. f (x) = ⃗0.

x t 2 Lemma (Uniformly Truncated Okamura Seymour)

Lemma (Uniformly Truncated Trees)

T a tree, t > 0 parameter. Then ∃f ∶ T → `1 s.t.: 1 ∀v ∈ T , Yf (v)Y1 = t. 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). 3 Contraction: ∀u, v ∈ V , Yf (v) − f (u)Y1 ≥ 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)

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F face, t > 0 parameter. Then ∃f ∶ F → `1 s.t.: 1 ∀v ∈ F , Yf (v)Y1 = t. 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). 3 Contraction: ∀u, v ∈ V , Yf (v) − f (u)Y1 ≥ min{dG (v, u), t}. Lemma (Truncated Okamura Seymour)

Lemma (Uniformly Truncated Okamura Seymour)

G planar, F a face, t > 0 parameter. Then ∃f ∶ F → `1 s.t.: 1 ∀v ∈ F , Yf (v)Y1 = t. 2 Lipschitz: ∀u, v ∈ F , Yf (v) − f (u)Y1 ≤ O(dG (v, u)). 3 Contraction: ∀u, v ∈ V , Yf (v) − f (u)Y1 ≥ min{dG (v, u), t}.

Smooth combination of all scales. Lemma (Uniformly Truncated Okamura Seymour)

Smooth combination of all scales.

Lemma (Truncated Okamura Seymour)

Embed planar graphs into `1 (or show a L.B.).

Embed graphs with treewidth k into `1 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 `1 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