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

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

3. Spanners H is a t-spanner if:
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 H is a t-spanner if:
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 H is a t-spanner if:
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 H is a t-spanner if:
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
(X,δ) 2
1-spanner
3-spanner 2 1 1 1 1 1 v3 v3 v3

7. Spanners H is a t-spanner if:
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
(X,δ) 2
1-spanner
3-spanner 2 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. focus “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 dimension d if every ball
can be covered by 2d balls 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 2d balls 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 2d balls 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 2d balls 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 “light spanner” THEOREM (Euclidean 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
“light spanner” THEOREM (Euclidean metrics)
Any low-dim Euclidean metric admits
(1+ε)-spanners with lightness [Das et al., SoCG’93]
A metric is doubling if its doubling dimension is constant

23. Light Spanners “light spanner” THEOREM (Euclidean 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
“light spanner” THEOREM (Euclidean metrics)
Any low-dim Euclidean metric admits
(1+ε)-spanners with lightness [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 with lightness
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 ddim d, 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 En route… 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 En route… COROLLARY: 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 v = sk - s1 vi = v’i + v’’i ,
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’i orthogonal to v & v’’I; v’’i parallel to v

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

51. 2-dim intuition v = sk - s1 vi = v’i + v’’i ,
Parallel contribution in :
s6 = sk
v’’4 s4
v’’2 v5 s2 v4
v’’5
v’4
v’5 v3 v2
v’3 v
v’2 s5
v’’3 v1 s3
v’1
v = sk - s1
v’’1 s1
vi = v’i + v’’i ,
v’i orthogonal to v & v’’I; v’’i parallel to v

52. 2-dim intuition Parallel contribution in : s6 = sk v’’4 s4 v’’2 v5 s2

53. 2-dim intuition For parallel vectors, “switching”
Parallel contribution in :
s6 = sk
v’’4 s4
v’’2 v5 s2 v4
v’’5 v3 v2 v s5
v’’3 v1 s3
For parallel vectors,
“switching”
doesn’t “cost” anything:
v’’1 s1
Parallel contribution in :

54. 2-dim intuition s6 = sk v’’4 s4 v’’2 v5 s2 v4 v’’5 v’4 v’5 v3 v2 v’3 v

55. 2-dim intuition CLAIM: Orthogonal contribution in : s6 = sk v’’4 s4

56. 2-dim intuition WHY? (intuition, 2D)
CLAIM: Orthogonal contribution in :
s6 = sk s4 v5 s2 v4
v’4
v’5 v3 v2
v’3 v
v’2 s5 v1 s3
v’1
WHY? (intuition, 2D)
worst-case scenario (as stretch ≤ 1+ε): s1

57. 2-dim intuition WHY? (intuition, 2D)
CLAIM: Orthogonal contribution in :
s6 = sk s4 v5 s2 v4
v’4
v’5 v3 v2
v’3 v
v’2 s5 v1 s3
v’1
WHY? (intuition, 2D)
worst-case scenario (as stretch ≤ 1+ε): s1

58. 2-dim intuition CLAIM: Orthogonal contribution in : s6 = sk s4 v5 s2

59. 2-dim intuition “switching” “costs” a factor of :
CLAIM: Orthogonal contribution in :
s6 = sk s4 v5 s2 v4
v’4
v’5 v3 v2
v’3 v
v’2 s5 v1 s3
v’1
“switching”
“costs” a factor of :
(that’s a small price to pay) s1
Orthogonal contribution in :

60. 2-dim intuition SUMMARY: Parallel contribution in :
Orthogonal contribution in :

61. 2-dim intuition SUMMARY: Parallel contribution in :
Orthogonal contribution in :
triangle ineq.

62. 2-dim intuition = -spanner path under SUMMARY:
Parallel contribution in :
Orthogonal contribution in :
triangle ineq.
= spanner path under

63. 2-dim intuition = -spanner path under SUMMARY:
Parallel contribution in :
Orthogonal contribution in :
triangle ineq.
= spanner path under

64. Snowflake Metrics PROOF II – direct argument
The standard “net-tree spanner” [Gao et al. SoCG’04,
Chan et al. SODA’05] is light!

65. Snowflake Metrics PROOF II – direct argument
The standard “net-tree spanner” [Gao et al. SoCG’04,
Chan et al. SODA’05] is light!
More complicated, but bypasses heavy machinery

66. Snowflake Metrics PROOF II – direct argument
The standard “net-tree spanner” [Gao et al. SoCG’04,
Chan et al. SODA’05] is light!
More complicated, but bypasses heavy machinery
Yields smaller lightness (singly vs. doubly exponential)
 Important for metric TSP

67. Snowflake Metrics PROOF II – direct argument
The standard “net-tree spanner” [Gao et al. SoCG’04,
Chan et al. SODA’05] is light!
More complicated, but bypasses heavy machinery
Yields smaller lightness (singly vs. doubly exponential)
 Important for metric TSP
more advantages (runtime, …)

68. Net-Tree Spanner
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

69. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

70. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

71. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) 2 v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

72. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) 4 2 v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

73. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) 8 4 2 v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

74. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) 8 4 2 v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i

75. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) 8 4 2 v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels
In level i, add ~ (n / 2i) edges of weight ~ 2i
 weight

76. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels sqrt-distances (α = 1/2):
In level i, add ~ (n / 2i) edges of weight ~ 2i 2i/2
 weight

77. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels sqrt-distances (α = 1/2):
In level i, add ~ (n / 2i) edges of weight ~ 2i 2i/2
 weight

78. Net-Tree Spanner
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n) v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17
Based on hierarchical tree of the metric (quadtree-like):
L = log (aspect-ratio) levels sqrt-distances (α = 1/2):
In level i, add ~ (n / 2i) edges of weight ~ 2i 2i/2
 weight

79. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17

80. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance v1 v2 v3 v4 v5 v6 v7 v8 v9
v10
v11
v12
v13
v14
v15
v16
v17

81. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance

82. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance
STRATEGY: from global weight to local “covering”
Spanner edge (vi,vj) covers j-i tour edges (vi,vi+1), …, (vj-1,vj)

83. Net-Tree Spanner INTUITION: Evenly spaced points in 1D
sqrt-distances (α = 1/2):
In level i, add n / 2i edges of weight ~ 2i/2
 weight

84. Net-Tree Spanner INTUITION: Evenly spaced points in 1D
sqrt-distances (α = 1/2):
In level i, add n / 2i edges of weight ~ 2i/2
 weight

85. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance
STRATEGY: from global weight to local “covering”
Spanner edge (vi,vj) covers j-i tour edges (vi,vi+1), …, (vj-1,vj)

86. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance
STRATEGY: from global weight to local “covering”
Spanner edge (vi,vj) covers j-i tour edges (vi,vi+1), …, (vj-1,vj)
Covering of (vi,vi+1) by (vi,vj) :=
snowflake-weight of (vi,vj) ∙
relative weight of (vi,vi+1)

87. Net-Tree Spanner Extension to general case: work on O(1)-approx tour
Two issues:
points on tour are NOT evenly spaced
metric distance may be smaller than tour distance
STRATEGY: from global weight to local “covering”
Spanner edge (vi,vj) covers j-i tour edges (vi,vi+1), …, (vj-1,vj)
Covering of (vi,vi+1) by (vi,vj) :=
snowflake-weight of (vi,vj) ∙
relative weight of (vi,vi+1)
lightness ≤ max covering over tour edges (by spanner edges)

88. Net-Tree Spanner We show: covering of any tour edge is O(1) v1 v2 v3

89. Conclusions and Open Questions
Snowflake doubling metrics admit light spanners
All spaces admit light spanners
Faster PTAS for metric TSP

90. Conclusions and Open Questions
Snowflake doubling metrics admit light spanners
All spaces admit light spanners
Faster PTAS for metric TSP
First step towards general conjecture?

91. THANK YOU!