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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

  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 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  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

  35. 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

  36. 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

  37. 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

  38. 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

  39. Planar Graphs into ℓ 1

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

  41. 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.

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

  43. 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 .

  44. 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.

  45. 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

  46. 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

  47. 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

  48. 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

  49. 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

  50. 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

  51. 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

  52. 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

  53. 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.

  54. 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 .

  55. 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 .

  56. Partial Shortest Path Decompositions

  57. Partial Shortest Path Decompositions

  58. Partial Shortest Path Decompositions

  59. Partial Shortest Path Decompositions

  60. Partial Shortest Path Decompositions

  61. Partial Shortest Path Decompositions PSPD depth=2

  62. Partial Shortest Path Decompositions PSPD depth=2 B : Boundary

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

  64. 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:

  65. 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 .

  66. 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 .

  67. 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.

  68. 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.

  69. 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

  70. 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

  71. 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

  72. 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

  73. 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

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