a light metric spanner lee-ad gottlieb. graph spanners a spanner for graph g is a subgraph h ◦ h...
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A light metric spanner
Lee-Ad Gottlieb
Graph spanners A spanner for graph G is a subgraph
H◦ H contains vertices, subset of edges of
G Some qualities of a spanner
◦ Degree, diameter, stretch, weight◦ Applications: networks, routing, TSP…
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Euclidean spanners
Seminal work in 90’s: Euclidean, planar Das et al. [SoCG ‘93][SODA ‘95], Arya et al. [FOCS ’94]
[STOC ’95], Soares [DCG ‘94], etc.
Remarkable result of Das et al.: ◦ d-dimensional Euclidean spanner◦ Stretch: (1+є) ◦ Weight: WE w(MST)
WE = є–O(d)
◦ Application: faster PTAS for Euclidean TSP Rao-Smith [STOC ‘98] improving Arora [JACM ‘98]
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Metric spanners
Recent focus: Spanners in general metric spaces◦ Problem: Metric spaces can be complex◦ Include high-dimensional Euclidean space
Solution: use doubling dimension to characterize complexity of the space◦ Doubling constant : Every ball can be covered by balls
of half the radius.◦ ddim= log
Analogue to Euclidean:◦ ddim = O(d)
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Metric spanners
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.
Upshot:◦ Many results for Euclidean space carry over to
doubling spaces, ◦ Dependence on Euclidean d replaced with ddim.
Light metric spanners Central open question: Low weight???
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
Light metric spanners Central open question: Low weight???
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
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
Spanners via hierarchies1-net2-net4-net8-net
Spanners via hierarchies1-net2-net4-net8-net
Radius = 1
Covering: all points are covered
Packing
Spanners via hierarchies1-net2-net4-net8-net
Radius = 2
Spanners via hierarchies1-net2-net4-net8-net
Spanners via hierarchies1-net2-net4-net8-net
Spanners via hierarchies1-net2-net4-net8-net
Spanners via hierarchies1-net2-net4-net8-net
Spanners via hierarchies1-net2-net4-net8-net
Spanners via hierarchies1-net2-net4-net8-net
Hierarchy: levels of 2i-nets
A simpler view
1-net2-net4-net8-net
Add parent-child edges
Spanner construction
Tree
Parent-childedge
Add lateral edges◦Between 2i-net points within distance
2i/є
Spanner construction
Graph
Lateraledge
Spanner Paths
Graph
Path
:Analysis
Path
2i/є
2i/2 2i/2
2i 2i
Application: paths spannerTheorem:
◦Pair of paths with no stretch (or low stretch) admits a (1+є)-stretch light spanner
Application: paths spannerProof construction: greedy
◦Create hierarchy for each path◦Add lateral edges in order of length
iff stretch on current graph > (1+є)
Application: paths spannerProof 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 spannerProof 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
SparsityA spanning tree is s-sparse
◦If every ball of radius r>0◦Has edges of total weight sr. r
SparsityA spanning tree is s-sparse
◦If every ball of radius r>0◦Has edges of total weight sr.
Reduce doubling to sparse MST:
r
SparsityA spanning tree is s-sparse
◦If every ball of radius r>0◦Has edges of total weight sr.
Reduce doubling to sparse MST:◦Find dense area
r
SparsityA spanning tree is s-sparse
◦If every ball of radius r>0◦Has edges of total weight sr.
Reduce doubling to sparse MST:◦Remove
r
SparsityA 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
SparsityA 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 treesBasic 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
Tree decomposition:◦Step 1: Decompose tree into arbitrary paths◦Step 2: Replace paths with low-stretch paths
Step 1: Tree decompositionGiven a spanning tree, remove
edges of longest path and repeat
Step 2: Path replacementReplace path with low-stretch
paths ◦Small weight increase – geometric
series
AltogetherGiven a graph
◦ 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