A light metric spanner
Lee-Ad Gottlieb
Graph spanners
A spanner for graph G is a subgraph H
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H contains vertices, subset of edges of G
Some qualities of a spanner
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Degree, diameter, stretch, weight
Applications: networks, routing, TSP…
G
1
H
1
1
2
2
1
1
1
1
Euclidean spanners
Seminal work in 90’s: Euclidean, planar
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Das et al. [SoCG ‘93][SODA ‘95], Arya et al. [FOCS ’94][STOC
’95], Soares [DCG ‘94], etc.
Remarkable result of Das et al.:
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◦
d-dimensional Euclidean spanner
Stretch: (1+є)
Weight: WE w(MST)
WE = є–O(d)
1
2
1
Application: faster PTAS for Euclidean TSP
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Rao-Smith [STOC ‘98] improving Arora [JACM ‘98]
1
Metric spanners
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Recent focus: Spanners in general metric spaces
◦ Problem: Metric spaces can be complex
◦ Include high-dimensional Euclidean space
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Solution: use doubling dimension to characterize
complexity of the space
◦ Doubling constant : Every ball can be covered by  balls of
half the radius.
4
5
◦ ddim= log 
3
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Analogue to Euclidean:
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8
◦ ddim = O(d)
2
7
1
Metric spanners
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Recent focus: doubling metric spaces
◦ Gao et al. [CGTA ‘06]: low-stretch metric spanners
◦ Related to WSPD [Callahan-Kosaraju STOC ‘92]
◦ Spawned a line of work
 Low degree, hop-diameter, efficient construction…
 Gottlieb-Roditty [SODA‘08][ESA‘08], Smid [EA‘09], Chan et al.
[SICOMP‘15], Solomon [SODA‘11][STOC‘14], etc.
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Upshot:
◦ Many results for Euclidean space carry over to doubling
spaces,
◦ Dependence on Euclidean d replaced with ddim.
Light metric spanners
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Central open question: Low weight???
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Do metrics admit (1+є)-stretch spanners of weight: WDw(MST)
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for WD independent of n?
for WD = є-O(ddim)?
Best known bounds: WD = O(log n)
Smid [EA ‘09], Elkin-Solomon [STOC ‘13]
Euclidean proof doesn’t carry over
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Very Euclidean-oriented
Uses “leapfrog” property, dumbbell trees
Light metric spanners
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Central open question: Low weight???
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Do metrics admit (1+є)-stretch spanners of weight: WDw(MST)
◦
◦

◦
for WD independent of n?
for WD = є-O(ddim)?
Best known bounds: WD = O(log n)
Smid [EA ‘09], Elkin-Solomon [STOC ‘13]
Euclidean proof doesn’t carry over

◦
◦
Very Euclidean-oriented
Uses “leapfrog” property, dumbbell trees
This paper: Yes!
WD = (ddim/є)O(ddim)
Outline
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Review spanner construction via hierarchies

Gao et al. [CGTA ‘06]
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Reduce doubling spaces to spaces with
sparse spanning trees
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Build light spanner for sparse spaces
Spanners via hierarchies
1-net
2-net
4-net
8-net
Spanners via hierarchies
1-net
2-net
4-net
8-net
Packing
Radius = 1
Covering: all points
are covered
Spanners via hierarchies
1-net
2-net
4-net
8-net
Radius = 2
Spanners via hierarchies
1-net
2-net
4-net
8-net
Spanners via hierarchies
1-net
2-net
4-net
8-net
Spanners via hierarchies
1-net
2-net
4-net
8-net
Spanners via hierarchies
1-net
2-net
4-net
8-net
Spanners via hierarchies
1-net
2-net
4-net
8-net
Spanners via hierarchies
1-net
2-net
4-net
8-net
A simpler view
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Hierarchy: levels of 2i-nets
1-net
2-net
4-net
8-net
Spanner construction
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Add parent-child edges
Tree
Parent-child
edge
Spanner construction
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Add lateral edges
◦ Between 2i-net points within distance 2i/є
Graph
Lateral
edge
Spanner Paths
Graph
Path
Analysis:
2i/є
Path
2i
2i/2
2i
2i/2
Application: paths spanner
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Theorem:
◦ Pair of paths with no stretch (or low stretch)
admits a (1+є)-stretch light spanner
Application: paths spanner
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Proof construction: greedy
◦ Create hierarchy for each path
◦ Add lateral edges in order of length iff stretch
on current graph > (1+є)
Application: paths spanner
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Proof construction: greedy
◦ Create hierarchy for each path
◦ Add lateral edges in order of length iff stretch
on current graph > (1+є)
◦ Claim I: low-stretch (immediate)
Application: paths spanner
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Proof construction: greedy
◦ Create hierarchy for each path
◦ Add lateral edges in order of length iff stretch
on current graph > (1+є)
◦ Claim I: low-stretch
◦ Claim II: light (charging argument)
Outline

Review spanner construction via hierarchies

Gao et al. [CGTA ‘06]

Reduce doubling spaces to spaces with
sparse spanning trees

Build light spanner for sparse spaces
Sparsity
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A spanning tree is s-sparse
◦ If every ball of radius r>0
◦ Has edges of total weight sr.
r
Sparsity
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A spanning tree is s-sparse
◦ If every ball of radius r>0
◦ Has edges of total weight sr.
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Reduce doubling to sparse MST:
r
Sparsity
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A spanning tree is s-sparse
◦ If every ball of radius r>0
◦ Has edges of total weight sr.
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Reduce doubling to sparse MST:
◦ Find dense area
r
Sparsity
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A spanning tree is s-sparse
◦ If every ball of radius r>0
◦ Has edges of total weight sr.
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Reduce doubling to sparse MST:
◦ Remove
r
Sparsity
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A spanning tree is s-sparse
◦ If every ball of radius r>0
◦ Has edges of total weight sr.

Reduce doubling to sparse MST:
◦ Repeat
r
Sparsity

A spanning tree is s-sparse
◦ If every ball of radius r>0
◦ Has edges of total weight sr.

Reduce doubling to sparse MST:
◦ Sparsity s = (ddim/є)O(ddim)
r
Outline

Review spanner construction via hierarchies

Gao et al. [CGTA ‘06]

Reduce doubling spaces to spaces with
sparse spanning trees

Build light spanner for sparse spaces
Spanner for sparse trees
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Basic idea:
◦ Pairs of low-stretch paths admit light spanner
◦ Decompose tree into many low-stretch paths
◦ Build light spanner for every close pair
 Tree sparsity guarantees only a small number of close
pairs
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Tree decomposition:
◦ Step 1: Decompose tree into arbitrary paths
◦ Step 2: Replace paths with low-stretch paths
Step 1: Tree decomposition
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Given a spanning tree, remove edges of
longest path and repeat
Step 2: Path replacement
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Replace path with low-stretch paths
◦ Small weight increase – geometric series
Altogether
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Given a graph
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Decompose into sparse trees
Decompose sparse tree into paths
Replace paths with low-stretch paths
Build path spanners
Outline

Review spanner construction via hierarchies

Gao et al. [CGTA ‘06]

Reduce doubling spaces to spaces with
sparse spanning trees admit

Build light spanner for sparse spaces