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Light Spanners for Snowflake Metrics
SoCG 2014
Lee-Ad Gottlieb Shay Solomon
Ariel University Weizmann Institute
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• metric (complete graph + triangle inequality)• spanning subgraph of the metric
),( XH
Spanners
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• metric (complete graph + triangle inequality)• spanning subgraph of the metric
),( X
H is a t-spanner if: it preserves all pairwise distances up to a factor of t
H
Spanners
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• metric (complete graph + triangle inequality)• spanning subgraph of the metric
),( X
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
H
Spanners
Xqp ,),( qpt
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• metric (complete graph + triangle inequality)• spanning subgraph of the metric
),( X
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
H
Spanners
Xqp ,),( qpt - spanner
patht
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• metric (complete graph + triangle inequality)• spanning subgraph of the metric
),( X
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
H
Spanners
Xqp ,),( qpt - spanner
patht
111-spanner 3-spanner(X,δ)
v3
v1 v2211
v3
v1 v2 21
v3
v1 v2
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• metric (complete graph + triangle inequality)• spanning subgraph of the metric
),( X
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
H
Spanners
Xqp ,),( qpt - spanner
patht
111-spanner 3-spanner(X,δ)
v3
v1 v2211
v3
v1 v2 21
v3
v1 v2
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• Small number of edges, ideally O(n)
“Good” Spanners stretch 1+ε
Applications: distributed computing, TSP, …
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• Small number of edges, ideally O(n)
• small weight, ideally O(w(MST))
stretch 1+ε
“Good” Spanners
Applications: distributed computing, TSP, …
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• Small number of edges, ideally O(n)
• small weight, ideally O(w(MST))
lightness = normalized weightLt(H) = w(H) / w(MST)
stretch 1+ε
“Good” Spanners
Applications: distributed computing, TSP, …
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• Small number of edges, ideally O(n)
• small weight, ideally O(w(MST))
lightness = normalized weightLt(H) = w(H) / w(MST)
stretch 1+ε
“Good” Spanners
Applications: distributed computing, TSP, …
focus
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“Good” spanners for arbitrary metrics?
Doubling Metrics
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“Good” spanners for arbitrary metrics? NO!
Doubling Metrics
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“Good” spanners for arbitrary metrics? NO!
For the uniform metric:(1+ε)-spanner (ε < 1) complete graph
1
11
Doubling Metrics
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“Good” spanners for arbitrary metrics? NO!
For the uniform metric:(1+ε)-spanner (ε < 1) complete graph
What about “simpler” metrics?
1
11
Doubling Metrics
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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
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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)
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• FACT: Euclidean space ℝd has doubling dimension Ѳ(d)
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.
Doubling metric = constant doubling dimension
Extensively studied [Assouad83, Clarkson97, GKL03, …]
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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.
Doubling metric = constant doubling dimension
constant-dim Euclidean metrics
Extensively studied [Assouad83, Clarkson97, GKL03, …]
• FACT: Euclidean space ℝd has doubling dimension Ѳ(d)
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“Good” spanners for arbitrary metrics? NO!
For the uniform metric: (1+ε)-spanner (ε < 1) complete graph 1
11
Doubling Metrics
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“Good” spanners for arbitrary metrics? NO!
For the uniform metric: doubling dimension Ω(log n))(1+ε)-spanner (ε < 1) complete graph 1
11
Doubling Metrics
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“Good” spanners for arbitrary metrics? NO!
For the uniform metric: doubling dimension Ω(log n))(1+ε)-spanner (ε < 1) complete graph
11
Light Spanners
A metric is doubling if its doubling dimension is constant
• Any low-dim Euclidean metric admits (1+ε)-spanners with lightness [Das et al., SoCG’93]
“light spanner” THEOREM (Euclidean metrics)
)(dO
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“Good” spanners for arbitrary metrics? NO!
For the uniform metric: doubling dimension Ω(log n))(1+ε)-spanner (ε < 1) complete graph
11
Light Spanners
A metric is doubling if its doubling dimension is constant
• Any low-dim Euclidean metric admits (1+ε)-spanners with lightness [Das et al., SoCG’93]
“light spanner” THEOREM (Euclidean metrics)
)(dO
• Doubling metrics admit (1+ε)-spanners with lightness
• naïve bound = lightness
“light spanner” CONJECTURE (doubling metrics))(dO
ndO log)(
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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]
)(
)(logdO
nn
nnn dOdO
log22 )()(
Light Spanners
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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]
)(
)(logdO
nn
nnn dOdO
log22 )()(
APPLICATION: metric TSP
• PTAS, (1+ε)-approx tour, runtime [Bartal et al., STOC’12]
• Using conjecture, runtime
)(dO
n
nnn dOdO
log22 )(~)(~
Light Spanners
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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, …]
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Snowflake Metrics MAIN RESULT
Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness
)1/(/ dO
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Snowflake Metrics MAIN RESULT
Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness
)1/(/ dO
En route…
All spaces admit light (1+ε)-spanners
p
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Snowflake Metrics MAIN RESULT
Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness
)1/(/ dO
En route…
All spaces admit light (1+ε)-spanners
p
nnn dOdO
log22 )(~)(~
COROLLARY:
Faster PTAS for TSP (via Rao-Smith):• snowflake doubling metrics:
• all spaces:
pnnn dOdO
log22 )/(~)1/()/(~
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PROOFS
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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
•
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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]
•
)1/(/ dO
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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]
•
)1/(/ dO
new goal :light spanners under
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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]
•
)1/(/ dO
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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]
2
)1/(/ dO
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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]
2
)1/(/ dO
missing:
2
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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]
2
)1/(/ dO
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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
2
)1/(/ dO
2 pp 1,)1( d
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Snowflake Metrics NEW LEMMA
S = set of points in ℝd
H = (1+ε)-spanner for , of lightness c
),( 2S
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Snowflake Metrics NEW LEMMA
S = set of points in ℝd
H = (1+ε)-spanner for , of lightness c
Then for :
• H = -spanner • lightness
),( 2S
pS p 1),,(
))(1( dO
dc
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Snowflake Metrics NEW LEMMA
S = set of points in ℝd
H = (1+ε)-spanner for , of lightness c
Then for :
• H = -spanner • lightness
),( 2S
pS p 1),,(
))(1( dO
dc
Distances change by a factor of < d:1,2 pp
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Snowflake Metrics NEW LEMMA
S = set of points in ℝd
H = (1+ε)-spanner for , of lightness c
Then for :
• H = -spanner • lightness NAÏVE
),( 2S
pS p 1),,(
))(1( dO
dc
Distances change by a factor of < d:1,2 pp
?
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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
),( 2S
pS p 1),,(
))(1( dO
dc
Distances change by a factor of < d:1,2 pp
)( dO?
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Snowflake Metrics CLAIM
S = set of points in ℝd
= (s1, s2, …, sk) = (1+ε)-spanner path under
Then = -spanner path under
2
)](1[ dO
PROOF.
pp 1,
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Snowflake Metrics CLAIM
S = set of points in ℝd
= (s1, s2, …, sk) = (1+ε)-spanner path under
Then = -spanner path under
s1
s2
s3
s4
s5
s6 = sk
PROOF. (2D)
)](1[ dO
2
pp 1,
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2-dim intuition
s1
s2
s3
s4
s5
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v = sk - s1
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’’1
v = sk - s1
vi = v’i + v’’i , v’i orthogonal to v & v’’I; v’’i parallel to v
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’’1
v’’2
v’2 v’’3
v’3
v’4
v’’4
v’5v’’5
v = sk - s1
vi = v’i + v’’i , v’i orthogonal to v & v’’I; v’’i parallel to v
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’’1
v’’2
v’2 v’’3
v’3
v’4
v’’4
v’5v’’5
v = sk - s1
Parallel contribution in : 222)1('' vvv ii 2
vi = v’i + v’’i , v’i orthogonal to v & v’’I; v’’i parallel to v
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’’1
v’’2
v’’3
v’’4
v’’5
Parallel contribution in : 222)1('' vvv ii 2
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’’1
v’’2
v’’3
v’’4
v’’5
Parallel contribution in : 222)1('' vvv ii
For parallel vectors, “switching”doesn’t “cost” anything:
p 2
ppivv )1(''
2
Parallel contribution in : p
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’’1
v’’2
v’2 v’’3
v’3
v’4
v’’4
v’5v’’5
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’’1
v’’2
v’2 v’’3
v’3
v’4
v’’4
v’5v’’5
22)(' vOv i CLAIM: Orthogonal contribution in : 2
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’2
v’3
v’4 v’5
22)(' vOv i CLAIM: Orthogonal contribution in : 2
WHY? (intuition, 2D)worst-case scenario (as stretch ≤ 1+ε):
s6 = sk
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v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’2
v’3
v’4 v’5
22)(' vOv i CLAIM: Orthogonal contribution in : 2
WHY? (intuition, 2D)worst-case scenario (as stretch ≤ 1+ε):
2)( vO
s6 = sk
![Page 58: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/58.jpg)
v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’2
v’3
v’4 v’5
22)(' vOv i CLAIM: Orthogonal contribution in : 2
s6 = sk
![Page 59: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/59.jpg)
v1
v2v3
v4v5
2-dim intuition
s1
s2
s3
s4
s5v
v’1
v’2
v’3
v’4 v’5
22)(' vOv i CLAIM: Orthogonal contribution in : 2
“switching”“costs” a factor of : (that’s a small price to pay)
p 2d
ppivdOv )(' Orthogonal contribution in : p
s6 = sk
![Page 60: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/60.jpg)
2-dim intuition
ppivdOv )(' Orthogonal contribution in : p
ppipivvv )1('' Parallel contribution in : p
SUMMARY:
pipipip vvvw ''')(
pvdO )](1[
![Page 61: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/61.jpg)
pipipip vvvw ''')(
2-dim intuition
ppivdOv )(' Orthogonal contribution in : p
ppipivvv )1('' Parallel contribution in : p
SUMMARY:
pvdO )](1[
triangle ineq.
![Page 62: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/62.jpg)
2-dim intuition
ppivdOv )(' Orthogonal contribution in : p
ppipivvv )1('' Parallel contribution in : p
SUMMARY:
= -spanner path under pp 1,)](1[ dO
pipipip vvvw ''')(
pvdO )](1[
triangle ineq.
![Page 63: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/63.jpg)
2-dim intuition
ppivdOv )(' Orthogonal contribution in : p
ppipivvv )1('' Parallel contribution in : p
SUMMARY:
= -spanner path under pp 1,)](1[ dO
pipipip vvvw ''')(
pvdO )](1[
triangle ineq.
![Page 64: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/64.jpg)
Snowflake Metrics PROOF II – direct argument
• The standard “net-tree spanner” [Gao et al. SoCG’04, Chan et al. SODA’05] is light!
![Page 65: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/65.jpg)
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
![Page 66: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/66.jpg)
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
![Page 67: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/67.jpg)
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, …)
![Page 68: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/68.jpg)
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
Net-Tree Spanner
![Page 69: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/69.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
![Page 70: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/70.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
![Page 71: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/71.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
2
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
![Page 72: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/72.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
2
4
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
![Page 73: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/73.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
2
4
8INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
![Page 74: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/74.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
Based on hierarchical tree of the metric (quadtree-like):
2
4
8INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
![Page 75: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/75.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels In level i, add ~ (n / 2i) edges of weight ~ 2i
weight
Based on hierarchical tree of the metric (quadtree-like):
2
4
8INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
)log()1(...4)4/(2)2/(1)( nnOnnnn
![Page 76: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/76.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels sqrt-distances (α = 1/2): In level i, add ~ (n / 2i) edges of weight ~ 2i 2i/2
weight
Based on hierarchical tree of the metric (quadtree-like):
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
)log()1(...4)4/(2)2/(1)( nnOnnnn
22
44
88
![Page 77: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/77.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels sqrt-distances (α = 1/2): In level i, add ~ (n / 2i) edges of weight ~ 2i 2i/2
weight
Based on hierarchical tree of the metric (quadtree-like):
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
22
44
88
![Page 78: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/78.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
L = log (aspect-ratio) levels sqrt-distances (α = 1/2): In level i, add ~ (n / 2i) edges of weight ~ 2i 2i/2
weight
Based on hierarchical tree of the metric (quadtree-like):
INTUITION: Evenly spaced points in 1D (coordinates 1,…, n)
)()1(...4)4/(2)2/(1)( nOnnnn
22
44
88
![Page 79: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/79.jpg)
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11 v15v14 v17v16
![Page 80: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/80.jpg)
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11 v15v14 v17v16
![Page 81: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/81.jpg)
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
![Page 82: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/82.jpg)
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
STRATEGY: from global weight to local “covering”
Spanner edge (vi,vj) covers j-i tour edges (vi,vi+1), …, (vj-1,vj)
![Page 83: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/83.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
In level i, add n / 2i edges of weight ~ 2i/2
weight
sqrt-distances (α = 1/2):
Net-Tree Spanner
v15v14
INTUITION: Evenly spaced points in 1D
v17v16
22
44
88
)()1(...4)4/(2)2/(1)( nOnnnn
![Page 84: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/84.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
In level i, add n / 2i edges of weight ~ 2i/2
weight
sqrt-distances (α = 1/2):
Net-Tree Spanner
v15v14
INTUITION: Evenly spaced points in 1D
v17v16
22
44
88
)()1(...4)4/(2)2/(1)( nOnnnn
![Page 85: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/85.jpg)
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
STRATEGY: from global weight to local “covering”
Spanner edge (vi,vj) covers j-i tour edges (vi,vi+1), …, (vj-1,vj)
![Page 86: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/86.jpg)
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) ∙
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
relative weight of (vi,vi+1)
![Page 87: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/87.jpg)
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) ∙
points on tour are NOT evenly spaced metric distance may be smaller than tour distance
Extension to general case: work on O(1)-approx tour
Two issues:
Net-Tree Spanner
relative weight of (vi,vi+1)
lightness ≤ max covering over tour edges (by spanner edges)
![Page 88: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/88.jpg)
v1 v2 v3 v4 v5 v6 v8 v10 v12 v13v7 v9 v11
Net-Tree Spanner
v15v14 v17v16
We show: covering of any tour edge is O(1)
![Page 89: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/89.jpg)
• Snowflake doubling metrics admit light spanners All spaces admit light spanners
• Faster PTAS for metric TSP
Conclusions and Open Questions
p
![Page 90: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/90.jpg)
• Snowflake doubling metrics admit light spanners All spaces admit light spanners
• Faster PTAS for metric TSP
• First step towards general conjecture?
Conclusions and Open Questions
p
![Page 91: Light Spanners for Snowflake Metrics](https://reader035.vdocument.in/reader035/viewer/2022062501/56816243550346895dd27c92/html5/thumbnails/91.jpg)
THANK YOU!