On metric embeddings, shortest path decompositions and face cover - PowerPoint PPT Presentation

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 → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 with distortion O ( k ) . Graph Family Our results. Previous results O ( k 1 / p ) ( 4 k ) k 3 + 1 (only into ℓ 1 ) Pathwidth k [Lee and Sidiropoulos 13] Exponential Improvement for ℓ 1 . 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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 with distortion O ( k ) . Graph Family Our results. Previous results O ( k 1 / p ) ( 4 k ) k 3 + 1 (only into ℓ 1 ) Pathwidth k [LS13] O ( k 1 − 1 / p ⋅ log 1 / p n ) O (( k log n ) 1 / p ) Treewidth k [Krauthgamer, Lee, Mendel, Naor 04] O (( log ( k log n )) 1 − 1 / p ( log 1 / 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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 with distortion O ( k ) . Graph Family Our results. Previous results O ( k 1 / p ) ( 4 k ) k 3 + 1 (only into ℓ 1 ) Pathwidth k [LS13] O ( k 1 − 1 / p ⋅ log 1 / p n ) O (( k log n ) 1 / p ) Treewidth k [KLMN04] O (( log ( k log n )) 1 − 1 / p ( log 1 / p n )) [KK16] 1 / p n ) 1 / p n ) O ( log O ( log Planar [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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 with distortion O ( k ) . Graph Family Our results. Previous results O ( k 1 / p ) ( 4 k ) k 3 + 1 (only into ℓ 1 ) Pathwidth k [LS13] O ( k 1 − 1 / p ⋅ log 1 / p n ) O (( k log n ) 1 / p ) Treewidth k [KLMN04] O (( log ( k log n )) 1 − 1 / p ( log 1 / p n )) [KK16] 1 / p n ) 1 / p n ) O ( log O ( log Planar [Rao99] 1 / p n ) [Abraham, Gavoille, O (( g ( r ) log n ) 1 / p ) O ( r 1 − 1 / p log K r -minor-free 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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 with distortion O ( k ) . Graph Family Our results. Previous results O ( k min { 1 p , 1 2 } ) ( 4 k ) k 3 + 1 (only into ℓ 1 ) Pathwidth k [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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . √ Embed into both ℓ 1 and ℓ 2 with distortion O ( k ) . Graph Family Our results. Previous results O ( k min { 1 p , 1 2 } ) ( 4 k ) k 3 + 1 (only into ℓ 1 ) Pathwidth k [LS13] Corollary √ O ( k ) approximation algorithm for the sparsest cut problem on pathwidth k graphs. Best previous result: ( 4 k ) k 3 + 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 F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 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 [A F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . Using O ( log n ) dimensions for p ∈ [ 1 , 2 ] , and O ( k log n ) dimensions for p > 2. Corollary Every K r -free graph embeds into ℓ ∞ with O ( 1 ) distortion and O ( g ( r ) ⋅ log 2 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 F GN18]) Let G = ( V , E ) be a weighted graph with SPDdepth k . Then there exists an embedding f ∶ V → ℓ p with distortion O ( k min { 1 p , 1 2 } ) . Using O ( log n ) dimensions for p ∈ [ 1 , 2 ] , and O ( k log n ) dimensions for p > 2. Corollary Every K r -free graph embeds into ℓ ∞ with O ( 1 ) distortion and O ( g ( r ) ⋅ log 2 n ) dimensions. [Krauthgamer, Lee, Mendel, Naor 04]: Every K r -free graph embeds into ℓ ∞ with O ( r 2 ) distortion and O ( 3 r ⋅ 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 ! D k ( R d , �·� p ) Ω( k min { 1 p , 1 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 ℓ 1 with distortion Ω ( 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 √ k embeds into ℓ 1 with distortion Ω ( log k ) . X k ( R d , �·� 1 ) � � k � Ω( log k ) � � 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 ∥ f j ( v ) − f j ( u )∥ = O ( d G ( 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 ∥ f j ( v ) − f j ( u )∥ = O ( d G ( v , u )) (Lipschitz) There is some level s.t. ∥ f j ( v ) − f j ( u )∥ = Ω ( d G ( 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 ∥ f j ( v ) − f j ( u )∥ = O ( d G ( v , u )) (Lipschitz) There is some level s.t. ∥ f j ( v ) − f j ( u )∥ = Ω ( d G ( v , u )) . As each vertex will be non-zero in only k coordinates: The distortion will be O ( k 1 / 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 , v ( d = ∆ 1 = d ( v, P ) ∆ 2 P r Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

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

Initial Attempt v ) r , v ( d = ∆ 1 = d ( v, P ) ∆ 2 P r 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

v Initial Attempt ) r , v ( d = ∆ 1 = d ( v, P ) ∆ 2 P r Use different ∆ 1 coordinate for each component. Use the same ∆ 2 coordinate for all components. r 5 P X 9 u v (0 , 2 , 5) (7 , 0 , 9) 2 7 X 1 X 2 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Initial Attempt v ) r , v ( d = ∆ 1 = d ( v, P ) ∆ 2 P r Use different ∆ 1 coordinate for each component. Use the same ∆ 2 coordinate for all components. P P v X 2 X 2 X 1 X 1 ∆ 2 u ∆ X 2 v 1 u ∆ X 1 ∆ 2 1 r r ∣ ∆ 2 ( u ) − ∆ 2 ( v )∣ = 1 ( u ) + ∆ X 2 1 ( v ) = ∆ X 1 d ( u , P ) + d ( v , P ) = Ω ( d G ( u , v )) ∣ d ( u , r ) − d ( v , r )∣ = Ω ( d G ( u , v )) Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 12 / 34

Problem But the expansion is unbounded . P v X v ∆ 1 ∆ 2 r v v u X u d ( v , P v ) , d ( v , r v ) ≫ 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 ∆ 2 = d ( v, r ) ∆ 1 = d ( v, P ) P r 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 , ) r , v ( d { ∆ 1 = min { d ( v, P ) , d ( v, V \ X ) } n i m = ∆ 2 P r 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 r v u ∣ ∆ 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... X r v u ∣ ∆ 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 y 2 t 2 t − 1 x 0 2 t +1 2 · 2 t +1 3 · 2 t +1 4 · 2 t +1 5 · 2 t +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 y x 2 2 t 2 t − 1 x 1 x 3 x 0 2 t +1 2 · 2 t +1 3 · 2 t +1 4 · 2 t +1 5 · 2 t +1 The graph of the scale t “sawtooth” function g t . Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 16 / 34

Sawtooth Function: a Randomized Truncation y x 2 2 t 2 t − 1 x 1 x 3 x 0 2 t +1 2 · 2 t +1 3 · 2 t +1 4 · 2 t +1 5 · 2 t +1 The graph of the scale t “sawtooth” function g t . h t ( x ) = g t ( 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 x 2 2 t 2 t − 1 x 1 x 3 x 0 2 t +1 2 · 2 t +1 3 · 2 t +1 4 · 2 t +1 5 · 2 t +1 The graph of the scale t “sawtooth” function g t . h t ( x ) = g t ( α + β ⋅ x ) : Sawtooth function after a random shift and stretch . Lemma Let x , y ∈ R + , if ∣ x − y ∣ ≤ 2 t − 1 then E α,β [∣ h t ( x ) − h t ( 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 ∣ ≤ 2 t − 1 then E α,β [∣ h t ( x ) − h t ( y )∣] = Ω (∣ x − y ∣) . X r v u E [∣ h t ( ∆ 2 ( v )) − h t ( ∆ 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 ∣ ≤ 2 t − 1 then E α,β [∣ h t ( x ) − h t ( y )∣] = Ω (∣ x − y ∣) . X r v u E [∣ h t ( ∆ 2 ( v )) − h t ( ∆ 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 ∣ ≤ 2 t − 1 then E α,β [∣ h t ( x ) − h t ( y )∣] = Ω (∣ x − y ∣) . X r v u E [∣ h t ( ∆ 2 ( v )) − h t ( ∆ 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 [∣ f j ( v ) − f j ( u )∣] = Ω ( d G ( 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 [∣ f j ( v ) − f j ( u )∣] = Ω ( d G ( u , v )) . j is the minimal level s.t (1) v and u are in different components of X / P X . OR (2) min { d G ( v , P X ) , d G ( u , P X )} ≤ d G ( 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 [∣ f j ( v ) − f j ( u )∣] = Ω ( d G ( u , v )) . j is the minimal level s.t (1) v and u are in different components of X / P X . OR (2) min { d G ( v , P X ) , d G ( u , P X )} ≤ d G ( u , v )/ 12. P P v X 2 X 2 X 1 X 1 ∆ 2 u ∆ X 2 v 1 u ∆ X 1 ∆ 2 1 r r Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 17 / 34

Planar Graphs into ℓ 1

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

Planar Graphs into ℓ 1 Graph Family Our results. Previous results O (√ log n ) O (√ log n ) Planar [Rao99] GNRS Conjecture Planar graphs embed into ℓ 1 with constant distortion.

Planar Graphs into ℓ 1 Terminal Problem Given a set K of terminals , embed K into ℓ 1 .

Planar Graphs into ℓ 1 Terminal Problem Given a set K of terminals , embed K into ℓ 1 . Definition (Face Cover) Set of faces F , s.t. every terminal lays on some face F ∈ F .

Planar Graphs into ℓ 1 Terminal Problem Given a set K of terminals , embed K into ℓ 1 . 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 ℓ 1 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 ℓ 1 . Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ 1 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 ℓ 1 . Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ 1 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 ℓ 1 with distortion : Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ 1 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 ℓ 1 with distortion : [Lee and Sidiropoulos 09] (implicitly): 2 O ( γ ) . Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 19 / 34

Planar Graphs into ℓ 1 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 ℓ 1 with distortion : [Lee and Sidiropoulos 09] (implicitly): 2 O ( 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 ℓ 1 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 ℓ 1 with distortion : [Lee and Sidiropoulos 09] (implicitly): 2 O ( 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 ℓ 1 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 ℓ 1 with distortion : [Lee and Sidiropoulos 09] (implicitly): 2 O ( 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 ℓ 1 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 ℓ 1 with distortion : [Lee and Sidiropoulos 09] (implicitly): 2 O ( γ ) . [Chekuri, Shepherd, Weibel 13]: 3 ⋅ γ . [Krauthgamer, Lee, Rika 19]: O ( log γ ) . Theorem ([ F 19]) K embeds into ℓ 1 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 [A F GN18]) Suppose G has PSPD of depth k with remainder { C , B } . Then there f ∶ G → ℓ 1 s.t. ∀ u , v:

Theorem (Implicit in [A F GN18]) Suppose G has PSPD of depth k with remainder { C , B } . Then there f ∶ G → ℓ 1 s.t. ∀ u , v: Lemma (Expansion Bound) For every scale j, f j is Lipshitz .

Theorem (Implicit in [A F GN18]) Suppose G has PSPD of depth k with remainder { C , B } . Then there f ∶ G → ℓ 1 s.t. ∀ u , v: √ ∥ f ( v ) − f ( u )∥ 1 ≤ O ( k ) ⋅ d G ( u , v ) . 1 Lemma (Expansion Bound) For every scale j, f j is Lipshitz .

Theorem (Implicit in [A F GN18]) Suppose G has PSPD of depth k with remainder { C , B } . Then there f ∶ G → ℓ 1 s.t. ∀ u , v: √ ∥ f ( v ) − f ( u )∥ 1 ≤ O ( k ) ⋅ d G ( u , v ) . 1 Lemma (Contraction Bound) For any vertices u , v, there exists some coordinate j such that E [∣ f j ( v ) − f j ( u )∣] = Ω ( d G ( u , v )) . j is the minimal level s.t (1) v and u are in different components of X / P X . OR (2) min { d G ( v , P X ) , d G ( u , P X )} ≤ d G ( u , v )/ 12.

Theorem (Implicit in [A F GN18]) Suppose G has PSPD of depth k with remainder { C , B } . Then there f ∶ G → ℓ 1 s.t. ∀ u , v: √ ∥ f ( v ) − f ( u )∥ 1 ≤ O ( k ) ⋅ d G ( u , v ) . 1 If either u , v not belong to the same cluster in C , 2 min { d G ( v , B ) , d G ( u , B )} ≤ d G ( u , v ) or then 12 ∥ f ( u ) − f ( v )∥ 1 = Ω ( d G ( u , v )) . Lemma (Contraction Bound) For any vertices u , v, there exists some coordinate j such that E [∣ f j ( v ) − f j ( u )∣] = Ω ( d G ( u , v )) . j is the minimal level s.t (1) v and u are in different components of X / P X . OR (2) min { d G ( v , P X ) , d G ( u , P X )} ≤ d G ( u , v )/ 12.

Theorem (Path Separator) There are shortest paths P 1 , P 2 , s.t. for every connected component C in G /{ P 1 ∪ P 2 } 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 P 1 , P 2 , s.t. for every connected component C in G /{ P 1 ∪ P 2 } 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 P 1 , P 2 , s.t. for every connected component C in G /{ P 1 ∪ P 2 } it holds γ ( G [ C ] , K ∩ C ) ≤ 2 3 ⋅ γ ( G , K ) + 1 . P 1 P 2 Arnold Filtser On metric embeddings, shortest path decompositions and face cover of planar graphs 22 / 34

Theorem (Path Separator) There are shortest paths P 1 , P 2 , s.t. for every connected component C in G /{ P 1 ∪ P 2 } 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 P 1 , P 2 , s.t. for every connected component C in G /{ P 1 ∪ P 2 } 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