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Light Spanners for Snowflake Metrics

Light Spanners for Snowflake Metrics. Lee-Ad Gottlieb Shay Solomon Ariel University Weizmann Institute . SoCG 2014. Spanners. metric (complete graph + triangle inequality) spanning subgraph of the metric. Spanners.

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Light Spanners for Snowflake Metrics

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  1. Light Spanners for Snowflake Metrics Lee-Ad GottliebShay Solomon Ariel University Weizmann Institute SoCG 2014

  2. Spanners • metric (complete graph + triangle inequality) • spanning subgraph of the metric

  3. Spanners • metric (complete graph + triangle inequality) • spanning subgraph of the metric H is a t-spanner if: it preserves all pairwise distances up to a factor of t

  4. Spanners • metric (complete graph + triangle inequality) • spanning subgraph of the metric H is a t-spanner if: it preserves all pairwise distances up to a factor of t there is a path in H between p and q with weight t = stretch of H

  5. Spanners • metric (complete graph + triangle inequality) • spanning subgraph of the metric H is a t-spanner if: it preserves all pairwise distances up to a factor of t there is a path in H between p and q with weight t = stretch of H -spanner path

  6. Spanners • metric (complete graph + triangle inequality) • spanning subgraph of the metric H is a t-spanner if: it preserves all pairwise distances up to a factor of t there is a path in H between p and q with weight t = stretch of H -spanner path v1 v2 v1 v2 v1 v2 2 1-spanner 3-spanner 2 (X,δ) 1 1 1 1 1 v3 v3 v3

  7. Spanners • metric (complete graph + triangle inequality) • spanning subgraph of the metric H is a t-spanner if: it preserves all pairwise distances up to a factor of t there is a path in H between p and q with weight t = stretch of H, t = 1+ε -spanner path v1 v2 v1 v2 v1 v2 2 1-spanner 3-spanner 2 (X,δ) 1 1 1 1 1 v3 v3 v3

  8. “Good” Spanners stretch 1+ε • Small number of edges, ideally O(n) Applications: distributed computing, TSP, …

  9. “Good” Spanners stretch 1+ε • Small number of edges, ideally O(n) • small weight, ideally O(w(MST)) Applications: distributed computing, TSP, …

  10. “Good” Spanners stretch 1+ε • Small number of edges, ideally O(n) • small weight, ideally O(w(MST)) • lightness = normalized weight • Lt(H) = w(H) / w(MST) Applications: distributed computing, TSP, …

  11. “Good” Spanners stretch 1+ε • Small number of edges, ideally O(n) • small weight, ideally O(w(MST)) • lightness = normalized weight • Lt(H) = w(H) / w(MST) focus Applications: distributed computing, TSP, …

  12. Doubling Metrics “Good” spanners for arbitrary metrics?

  13. Doubling Metrics “Good” spanners for arbitrary metrics? NO!

  14. Doubling Metrics “Good” spanners for arbitrary metrics? NO! For the uniform metric: (1+ε)-spanner (ε < 1)  complete graph 1 1 1

  15. Doubling Metrics “Good” spanners for arbitrary metrics? NO! For the uniform metric: (1+ε)-spanner (ε < 1)  complete graph What about “simpler” metrics? 1 1 1

  16. Doubling Metrics Definition (doubling dimension) • Metric (X,δ) has doubling dimensiond if every ball • can be covered by 2dballs of half the radius. • A metric is doubling if its doubling dimension is constant

  17. Doubling Metrics Definition (doubling dimension) • Metric (X,δ) has doubling dimension d if every ball • can be covered by 2dballs of half the radius. • FACT: Euclidean space ℝd has doubling dimension Ѳ(d)

  18. Doubling Metrics Definition (doubling dimension) • Metric (X,δ) has doubling dimension d if every ball • can be covered by 2dballs of half the radius. • FACT: Euclidean space ℝd has doubling dimension Ѳ(d) • Doubling metric = constant doubling dimension • Extensively studied [Assouad83, Clarkson97, GKL03, …]

  19. Doubling Metrics Definition (doubling dimension) • Metric (X,δ) has doubling dimension d if every ball • can be covered by 2dballs of half the radius. • FACT: Euclidean space ℝd has doubling dimension Ѳ(d) • Doubling metric = constant doubling dimension • constant-dim Euclidean metrics • Extensively studied [Assouad83, Clarkson97, GKL03, …]

  20. Doubling Metrics “Good” spanners for arbitrary metrics? NO! For the uniform metric: (1+ε)-spanner (ε < 1)  complete graph 1 1 1

  21. Doubling Metrics “Good” spanners for arbitrary metrics? NO! For the uniform metric: doubling dimension Ω(log n)) (1+ε)-spanner (ε < 1)  complete graph 1 1 1

  22. Light Spanners “Good” spanners for arbitrary metrics? NO! For the uniform metric: doubling dimension Ω(log n)) (1+ε)-spanner (ε < 1)  complete graph 1 1 “light spanner” THEOREM (Euclidean metrics) • Any low-dim Euclidean metric admits • (1+ε)-spanners withlightness [Das et al., SoCG’93] • A metric is doubling if its doubling dimension is constant

  23. Light Spanners “Good” spanners for arbitrary metrics? NO! For the uniform metric: doubling dimension Ω(log n)) (1+ε)-spanner (ε < 1)  complete graph 1 1 “light spanner” THEOREM (Euclidean metrics) • Any low-dim Euclidean metric admits • (1+ε)-spanners withlightness [Das et al., SoCG’93] • A metric is doubling if its doubling dimension is constant “light spanner” CONJECTURE (doubling metrics) • Doubling metrics admit (1+ε)-spanners withlightness • naïve bound = lightness

  24. Light Spanners APPLICATION: Euclidean traveling salesman problem (TSP) • PTAS, (1+ε)-approx tour, runtime [Arora JACM’98, Mitchell SICOMP’99] • Using light spanners, runtime • [Rao-Smith, STOC’98]

  25. Light Spanners APPLICATION: Euclidean traveling salesman problem (TSP) • PTAS, (1+ε)-approx tour, runtime [Arora JACM’98, Mitchell SICOMP’99] • Using light spanners, runtime • [Rao-Smith, STOC’98] APPLICATION: metric TSP • PTAS, (1+ε)-approx tour, runtime [Bartal et al., STOC’12] • Using conjecture, runtime

  26. Snowflake Metrics α-snowflake • Given metric (X,δ) with ddimd, snowflake param’0 < α < 1 • α-snowflake of (X,δ) = metric (X,δα) with ddim ≤ d/α • snowflake doubling metrics [Assouad 1983, Gupta et al. • FOCS’03, Abraham et al. SODA’08, …]

  27. Snowflake Metrics MAIN RESULT Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness

  28. Snowflake Metrics MAIN RESULT Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness En route… • All spaces admit light (1+ε)-spanners

  29. Snowflake Metrics MAIN RESULT Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness En route… • All spaces admit light (1+ε)-spanners COROLLARY: • Faster PTAS for TSP (via Rao-Smith): • snowflake doubling metrics: • all spaces:

  30. PROOFS

  31. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma

  32. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05]

  33. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05] new goal: light spanners under

  34. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05]

  35. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05] • WE SAW: light spanners in low-dim Euclidean metrics • (under ) [Das et al., SoCG’93]

  36. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05] • WE SAW: light spanners in low-dim Euclidean metrics • (under ) [Das et al., SoCG’93] missing:

  37. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05] • WE SAW: light spanners in low-dim Euclidean metrics • (under ) [Das et al., SoCG’93]

  38. Snowflake Metrics MAIN RESULT • Any snowflake doubling metric (X,δα) admits • (1+ε)-spanners with constant lightness PROOF I – combine known results with new lemma • α-snowflake metric embed into , distortion 1+ε • target dim[Har-Peled & Mendel SoCG’05] • WE SAW: light spanners in low-dim Euclidean metrics • (under ) [Das et al., SoCG’93] • NEW LEMMA: light (1+ε)-spanner under  • light -spanner under

  39. Snowflake Metrics NEW LEMMA • S = set of points in ℝd • H = (1+ε)-spanner for , of lightness c

  40. Snowflake Metrics NEW LEMMA • S = set of points in ℝd • H = (1+ε)-spanner for , of lightness c • Then for : • H = -spanner • lightness

  41. Snowflake Metrics NEW LEMMA • S = set of points in ℝd • H = (1+ε)-spanner for , of lightness c • Then for : • H = -spanner • lightness Distances change by a factor of <

  42. Snowflake Metrics NEW LEMMA • S = set of points in ℝd • H = (1+ε)-spanner for , of lightness c • Then for : • H = -spanner • lightness NAÏVE ? Distances change by a factor of <

  43. Snowflake Metrics NEW LEMMA • S = set of points in ℝd • H = (1+ε)-spanner for , of lightness c • Then for : • H = -spanner NAÏVE -spanner • lightness NAÏVE ? Distances change by a factor of <

  44. Snowflake Metrics CLAIM S = set of points in ℝd = (s1, s2, …, sk) = (1+ε)-spanner path under Then = -spanner path under PROOF.

  45. Snowflake Metrics CLAIM S = set of points in ℝd = (s1, s2, …, sk) = (1+ε)-spanner path under Then = -spanner path under s6 = sk s4 s2 PROOF. (2D) s5 s3 s1

  46. 2-dim intuition s6 = sk s4 s2 s5 s3 s1

  47. 2-dim intuition s6 = sk s4 v5 s2 v4 v3 v2 s5 v1 s3 s1

  48. 2-dim intuition s6 = sk s4 v5 s2 v4 v3 v2 v s5 v1 s3 v = sk- s1 s1

  49. 2-dim intuition s6 = sk s4 v5 s2 v4 v3 v2 v s5 v1 s3 v’1 v = sk- s1 v’’1 s1 vi = v’i+ v’’i, v’iorthogonal to v & v’’I; v’’iparallel to v

  50. 2-dim intuition s6 = sk v’’4 s4 v’’2 v5 s2 v4 v’’5 v’4 v’5 v3 v2 v’3 v s5 v’2 v’’3 v1 s3 v’1 v = sk- s1 v’’1 s1 vi = v’i+ v’’i, v’iorthogonal to v & v’’I; v’’iparallel to v

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