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 H

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: W E w( MST)  W E = є –O(d) ◦ Application: faster PTAS for Euclidean TSP  Rao-Smith [STOC ‘98] improving Arora [JACM ‘98]

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)

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: W D w(MST) ◦ for W D independent of n? ◦ for W D = є -O(ddim) ? Best known bounds: W D = 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: W D w(MST) ◦ for W D independent of n? ◦ for W D = є -O(ddim) ? Best known bounds: W D = 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! W D = (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 hierarchies 1-net 2-net 4-net 8-net

Spanners via hierarchies 1-net 2-net 4-net 8-net Radius = 1 Covering: all points are covered Packing

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

Hierarchy: levels of 2 i -nets A simpler view 1-net 2-net 4-net 8-net

Add parent-child edges Spanner construction Tree Parent-child edge

Add lateral edges ◦ Between 2 i -net points within distance 2 i / є Spanner construction Graph Lateral edge

Spanner Paths Graph Path

:Analysis Path 2i/є2i/є 2 i /2 2i2i 2i2i

Application: paths spanner Theorem: ◦ Pair of paths with no stretch (or low stretch) admits a (1+ є )-stretch light spanner

Application: paths spanner 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 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 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 A spanning tree is s-sparse ◦ If every ball of radius r>0 ◦ Has edges of total weight sr. 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: 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: ◦ Find dense area 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: ◦ Remove 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: ◦ 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 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 Tree decomposition: ◦ Step 1: Decompose tree into arbitrary paths ◦ Step 2: Replace paths with low-stretch paths

Step 1: Tree decomposition Given a spanning tree, remove edges of longest path and repeat

Step 2: Path replacement Replace path with low-stretch paths ◦ Small weight increase – geometric series

Altogether Given 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