Light Spanners for Snowflake Metrics

SoCG 2014

Lee-Ad Gottlieb Shay Solomon

Ariel University Weizmann Institute

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

),( XH

Spanners

• 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

• 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

• 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

• 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

• 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

• Small number of edges, ideally O(n)

“Good” Spanners stretch 1+ε

Applications: distributed computing, TSP, …

• Small number of edges, ideally O(n)

• small weight, ideally O(w(MST))

stretch 1+ε

“Good” Spanners

Applications: distributed computing, TSP, …

• 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, …

• 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

“Good” spanners for arbitrary metrics?

Doubling Metrics

“Good” spanners for arbitrary metrics? NO!

Doubling Metrics

“Good” spanners for arbitrary metrics? NO!

For the uniform metric:(1+ε)-spanner (ε < 1) complete graph

1

11

Doubling Metrics

“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

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

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)

• 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, …]

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)

“Good” spanners for arbitrary metrics? NO!

For the uniform metric: (1+ε)-spanner (ε < 1) complete graph 1

11

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

11

Doubling Metrics

“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

“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)(

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

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

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, …]

Snowflake Metrics MAIN RESULT

Any snowflake doubling metric (X,δα) admits (1+ε)-spanners with lightness

)1/(/ dO

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

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/()/(~

PROOFS

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

•

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

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

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

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

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

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

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

Snowflake Metrics NEW LEMMA

S = set of points in ℝd

H = (1+ε)-spanner for , of lightness c

),( 2S

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

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

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

?

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?

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,

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,

2-dim intuition

s1

s2

s3

s4

s5

s6 = sk

v1

v2v3

v4v5

2-dim intuition

s1

s2

s3

s4

s5

s6 = sk

v1

v2v3

v4v5

2-dim intuition

s1

s2

s3

s4

s5v

v = sk - s1

s6 = sk

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

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

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

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

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

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

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

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

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

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

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

2-dim intuition

ppivdOv )(' Orthogonal contribution in : p

ppipivvv )1('' Parallel contribution in : p

SUMMARY:

pipipip vvvw ''')(

pvdO )](1[

pipipip vvvw ''')(

2-dim intuition

ppivdOv )(' Orthogonal contribution in : p

ppipivvv )1('' Parallel contribution in : p

SUMMARY:

pvdO )](1[

triangle ineq.

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.

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.

Snowflake Metrics PROOF II – direct argument

• The standard “net-tree spanner” [Gao et al. SoCG’04, Chan et al. SODA’05] is light!

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

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

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, …)

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

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):

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)

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)

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)

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)

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)

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

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

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

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

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

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

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

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)

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

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

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)

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)

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)

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)

• Snowflake doubling metrics admit light spanners All spaces admit light spanners

• Faster PTAS for metric TSP

Conclusions and Open Questions

p

• 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

THANK YOU!