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How to Grow Your Balls (Distance Oracles Beyond the Thorup – Zwick Bound)

Explore various techniques to compress dense graphs into a smaller size while still allowing retrieval of (2k-1)-approximate distances. Includes discussion on spanners, data structures, and upper bounds.

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How to Grow Your Balls (Distance Oracles Beyond the Thorup – Zwick Bound)

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  1. How to Grow Your Balls(Distance Oracles Beyond the Thorup–Zwick Bound) Mihai PătrașcuLiam Roditty FOCS’10

  2. Distance Oracles Distance oracle = replacement of APSP matrix [Thorup, Zwick STOC’01] For k=1,2,3,…:Preprocess undirected, weighted graph G to answer:query(s,t): return đ with dG(s,t) ≤ đ ≤ (2k-1) · dG(s,t) Space O(n1+1/k) with O(1) query time

  3. Two Sides of Distance Oracles Compression question:Encode dense graphs with O(n1+1/k) bits, such that (2k-1)-apx distances can be retrieved [Matoušek’96] (∀) finite metric space on n points ↦ ℓ∞ with dimension O(k·n1+1/k lg n) with distortion 2k-1 Spanners: (∀) unweighted graph G(∃) H ⊆ G with O(n1+1/k) edges: dG ≤ dH≤ (2k-1)dG Data structures question: A data structure of size O(n1+1/k) can answer (2k-1)-apx distance queries in constant time ~ ~

  4. Two Sides of Distance Oracles Compression question:Encode dense graphs with O(n1+1/k) bits, such that (2k-1)-apx distances can be retrieved [Matoušek’96] (∀) finite metric space on n points ↦ ℓ∞ with dimension O(k·n1+1/k lg n) with distortion 2k-1 Spanners: (∀) unweighted graph G(∃) H ⊆ G with O(n1+1/k) edges: dG ≤ dH≤ (2k-1)dG Data structures question: A data structure of size O(n1+1/k) can answer (2k-1)-apx distance queries in constant time ~ Optimal space, assuming: Girth conjecture: [Erdős] et al. (∃) graphs with Ω(n1+1/k) edges and girth 2k+2 ~

  5. Two Sides of Distance Oracles Compression question:Encode dense graphs with O(n1+1/k) bits, such that (2k-1)-apx distances can be retrieved [Matoušek’96] (∀) finite metric space on n points ↦ ℓ∞ with dimension O(k·n1+1/k lg n) with distortion 2k-1 Spanners: (∀) unweighted graph G(∃) H ⊆ G with O(n1+1/k) edges: dG ≤ dH≤ (2k-1)dG Data structures question: A data structure of size O(n1+1/k) can answer (2k-1)-apx distance queries in constant time [Sommer, Verbin, Yu FOCS’09] Assumem=O(n) – sparse graphs.If c-apx distance queries take O(1) time ⇒ the space is ≥ n1+Ω(1/c) Beat [Thorup, Zwick] for graphs with ≪ n1+1/k edges? ~ ~ ~

  6. Two Sides of Distance Oracles Many other spanners: [BKMP’05] dH≤ dG+6 with O(n4/3) edges [EP’01] dH≤ (1+ε)dG + O(1) with O(n1+δ) edges … Use additive approximation in distance oracles? Compression question:Encode dense graphs with O(n1+1/k) bits, such that (2k-1)-apx distances can be retrieved [Matoušek’96] (∀) finite metric space on n points ↦ ℓ∞ with dimension O(k·n1+1/k lg n) with distortion 2k-1 Spanners: (∀) unweighted graph G(∃) H ⊆ G with O(n1+1/k) edges: dG ≤ dH≤ (2k-1)dG Data structures question: A data structure of size O(n1+1/k) can answer (2k-1)-apx distance queries in constant time ~ ~

  7. New Upper Bounds Unweighted graphs:Preprocess any graph G ⇒ distance oracle of size O(n5/3) that finds distance ≤ 2dG + 1 Weighted graphs:Preprocess G with m=n2/α edges ⇒ distance oracle of size O(n2 / α1/3) with approximation 2 Can report path in O(1) / edge.

  8. The Algorithm A = { } = sample n2/3 vertices

  9. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices

  10. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V

  11. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V Query(s,t): t s

  12. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V Query(s,t): • Rs, Rt = distance to NN in C • If Ball(s, Rs) ∩ Ball(t, Rt) = ∅ ⇒ min { Rs, Rt } ≤ ½ d(s,t)

  13. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V Query(s,t): • Rs, Rt = distance to NN in C • If Ball(s, Rs) ∩ Ball(t, Rt) = ∅ ⇒ min { Rs, Rt } ≤ ½ d(s,t) ✔

  14. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V • pairs when Ball(s)∩Ball(t) Query(s,t): • Rs, Rt = distance to NN in C • If Ball(s, Rs) ∩ Ball(t, Rt) = ∅ ⇒ min { Rs, Rt } ≤ ½ d(s,t) ✔ • If Ball(s, Rs) ∩ Ball(t, Rt) ≠ ∅

  15. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V • pairs when Ball(s)∩Ball(t) Query(s,t): • Rs, Rt = distance to NN in C • If Ball(s, Rs) ∩ Ball(t, Rt) = ∅ ⇒ min { Rs, Rt } ≤ ½ d(s,t) ✔ • If Ball(s, Rs) ∩ Ball(t, Rt) ≠ ∅ Key observation: E[#pairs] = O(n5/3)

  16. The Algorithm A = { } = sample n2/3 vertices B = { } = sample n1/3 vertices C = ∪u∈BBall(u→ A) E[|C|]=n2/3 Data structure: • distances C ↔ V • pairs when Ball(s)∩Ball(t) Query(s,t): • Rs, Rt = distance to NN in C • If Ball(s, Rs) ∩ Ball(t, Rt) = ∅ ⇒ min { Rs, Rt } ≤ ½ d(s,t) ✔ • If Ball(s, Rs) ∩ Ball(t, Rt) ≠ ∅ ✔ This is a lie.

  17. Geometric balls ≠ Balls in graphs Weighted graphs: Ball(s,r)={edges adjacent to vertices at distance ≤ r} To bound ball, sample edges ⇒ sparsity matters! s t

  18. Geometric balls ≠ Balls in graphs Weighted graphs: Ball(s,r)={edges adjacent to vertices at distance ≤ r} To bound ball, sample edges ⇒ sparsity matters! Unweighted graphs: d(s,u) = d(t,v) = ⌈ ½ d(s,t) ⌉ Just accept additive 1… s t u v

  19. Upper Bounds Can we get rid of “+1” for not-too-dense graphs? Unweighted graphs:Preprocess any graph G ⇒ distance oracle of size O(n5/3) that finds distance ≤ 2dG + 1 Weighted graphs:Preprocess G with m=n2/α edges ⇒ distance oracle of size O(n2 / α1/3) with approximation 2 Better bounds? Milder dependence on m? E.g. O(m + n5/3)

  20. Set-Intersection Hardness Preprocess S1, …, Sn ⊆ [X]query(i,j): is Si ∩ Sj = ∅ ? Conjecture: Let X = lgO(1)n. If query time = O(1), space = Ω(n2) Even in sparse graphs, approximation < 2 requires Ω(n2) space ~ ~ S1 1 X Sn

  21. Set-Intersection Hardness Preprocess S1, …, Sn ⊆ [X]query(i,j): is Si ∩ Sj = ∅ ? Conjecture: Let X = lgO(1)n. If query time = O(1), space = Ω(n2) Even in sparse graphs, approximation < 2 requires Ω(n2) space Apx. (2-ε)dG+ O(1) in unweighted graphs requires Ω(n2) space ~ ~ ~

  22. Set-Intersection Hardness Preprocess S1, …, Sn ⊆ [X]query(i,j): is Si ∩ Sj = ∅ ? Conjecture: Let X = lgO(1)n. If query time = O(1), space = Ω(n2) Even in sparse graphs, approximation < 2 requires Ω(n2) space Apx. (2-ε)dG+ O(1) in unweighted graphs requires Ω(n2) space ~ ~ ~

  23. New Lower Bounds Preprocess S1, …, Sn ⊆ [X]query(i,j): is Si ∩ Sj = ∅ ? Conjecture: Let X = lgO(1)n. If query time = O(1), space = Ω(n2) In unweighted graphs with m=n2/α edges,constant-time 2-approximation needs space Ω(n2 / √α) Constant-time approximation 2dG+O(1) needs space Ω(n1.5) ~ Actually, randomized conjecture… ~ ~

  24. Lower Bound ~ ~ In graphs with O(n)edges, 2-apx needs space Ω(n1.5) (Sab ∩ Tc ≠ ∅) ∧ (Ta ∩ Scd ≠ ∅) ⇒ distance = 3 3-apx ⇒ distance ≤ 5 (a,x,y) Sab× Ta (c,d) (a,b) Tc× Sab (c,x,y) [√n] × X2 [√n] × X2 [√n] ×[√n] [√n] ×[√n]

  25. Lower Bound ~ ~ In graphs with O(n)edges, 2-apx needs space Ω(n1.5) (Sab ∩ Tc ≠ ∅) ∧ (Ta ∩ Scd ≠ ∅) ⇒ distance = 3 3-apx ⇒ distance ≤ 5 ⇒ either Ta ∩ Scd ≠ ∅ or Sab ∩ Tc ≠ ∅ (a,x,y) Sab× Ta (c,d) (a,b) Tc× Sab (c,x,y) (a,b’) [√n] × X2 [√n] × X2 [√n] ×[√n] [√n] ×[√n]

  26. The End • Recent progress: • space Ω(n5/3) for 2-apx • many result for apx > 2 My flight is at 4:36. Questions @ 857-253-1282

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