A light metric spanner Lee-Ad Gottlieb. Graph spanners A spanner for graph G is a subgraph H ◦ H...

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…

G

2

11

H

2

11

11 1

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]

2

11

1

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)

1

2

3

4

6

5

7

8

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