Doubling Dimension: a short survey Anupam Gupta Carnegie Mellon University Barriers in Computational Complexity II, CCI, Princeton

Metric space M = (V, d) (finite) set V of points symmetric non-negative distances d(x,y) triangle inequality d(x,y) ≤ d(x,z) + d(z,y) x y z

Dimension dim D (M) is the smallest k such that every set S with diameter D S can be covered by 2 k sets of diameter ½D S D doubling dimension ¸ = 2 dim_D = doubling constant

doubling generalizes geometric dimension Take k-dim Euclidean space R k Claim: dim D (R k ) ≈ Θ(k) Easy to see for boxes Argument for spheres a bit more involved. 2 3 boxes to cover larger box in R 3

facts about doubling The notion of doubling dimension behaves smoothly under metric distortion definition closed under taking submetrics jargon: “doubling” = family of metrics with doubling dimension bounded by some absolute constant c independent of n.

Suppose a metric (X,d) has doubling dimension k. If any subset S µ X of points has all inter-point distances lying between ± and ¢ then |S| ≤ (2 ¢ / ± ) k useful property of doubling Proof: recursively apply the definition…

Suppose a metric (X,d) has doubling dimension k. If any subset S µ X of points has all inter-point distances lying between ± and ¢ then |S| ≤ (2 ¢ / ± ) k useful property of doubling   this 2-dim set has O(  /  ) 2 points

Uniform metric: All non-zero distances equal to R 2-uniform metric: All non-zero distances in [R,2R] Doubling Dimension k iff largest 2-uniform submetric has ¼ 2 O(k) points alternate characterization

what is not a doubling metric? The equidistant metric U n on n points has dimension  (log n) Hence low doubling dimension captures the fact that the metric does not have large (near)-equidistant metrics.

the picture thus far… Doubling dimension k Euclidean dimension £ (k) Metrics with >> 2 k nearly-equidistant points

btw, just to check Natural Q: Do all doubling metrics embed into ℓ2 with distortion O(1)? No. The Laakso fractals require  ( √ log n) distortion to embed into ℓ2 with any number of dimensions. [GKL’03] In fact, the right behavior is £ ( √ dim D log n) [KLMN’04, ABN’05, JLM’09] The Laakso fractals require  ( √ log n) distortion to embed into ℓ2 with any number of dimensions. [GKL’03] In fact, the right behavior is £ ( √ dim D log n) [KLMN’04, ABN’05, JLM’09]

Many geometric algorithms can be extended to doubling spaces… Near neighbor search Compact routing Distance labeling Network triangulation Sensor placements Small-world networks Traveling Salesman Sparse Spanners Approx. inference Network Design Clustering problems Well-separated pair decomposition Data structures Learnability a substantial(?) generalization Doubling dimension k Euclidean dimension £ (k)

example application Assign labels L(x) to each host x in a metric space Looking just at L(x) and L(y), can infer distance d(x,y) Results labels with (O( 1 )/ε) dim × log n bits estimates within ( 1 + ε) factor Contrast with lower bound of n bit labels in general for any factor < 2 x y f(, ) ≈ d(x,y)

[Arora 95] showed that TSP on R k was (1+ ² )-approximable in time [Talwar 04] extended the first result to metrics with doubling dimension k another example Can we get the PTAS as well?

example in action: sparse spanners for doubling metrics

spanners Given a metric M = (V, d), a graph G = (V, E) is an (m, ² )-spanner if 1) number of edges in G is m 2) d(x,y) ≤ d G (x,y) ≤ (1 + ² ) d(x,y) A reasonable goal: ² = 0.1, m = O(n) Fact: For the equidistant metric U n, if ² < 1 then G = K n

spanners for doubling metrics Theorem: Given any metric M, and any ² < ½, we can efficiently find an spanner G with stretch ² and number of edges m = n (1 + 1/ ² ) dim D (M) Hence, for doubling metrics, linear-sized spanners! Generalizes a similar theorem for Euclidean metrics.

standard tool: nets Nets: A set of points N is an r-net of a set S if – d(u,v) ≥ r for any u,v 2 N – For every w 2 S \ N, there is a u 2 N with d(u,w) < r r

standard tool: nets Nets: A set of points N is an r-net of a set S if – d(u,v) ≥ r for any u,v 2 N – For every w 2 S \ N, there is a u 2 N with d(u,w) < r Fact: If a metric has doubling dim k and N is an r-net ) B(x,2r) \ N has O(1) k points.

recursive nets so you take a 2-net N 1 of these points Now you can take a 4-net N 2 of this net And so on… Suppose all the points were at least unit distance apart

recursive nets N 0 = V N t is a 2 t -net of the set N t-1 N1N1 N2N2 N3N3 N4N4  N t is a 2 t+1 -net of the set V (almost)

the spanner construction N 0 = V N t is a 2 t -net of the set N t-1 N1N1 N2N2 N3N3 N4N4  N t is a 2 t+1 -net of the set V (almost) Connect each net point in N t to other net points at distance at most O(1/ ² ) 2 t

the number of edges Number of points in N t within O(1/ ² ) 2 t of some net point at most O(1/ ² ) k Number of levels = O(log diameter) Number of nodes in net at each level ≤ n Hence, number of edges ≤ n × log diameter × O(1/ ² ) k Can be improved to n × O(1/ ² ) k

the stretch factor

spanners for doubling metrics Theorem: Given any metric M, and any ² < ½, we can efficiently find an (m, ² )-spanner G with number of edges m = n (1 + 1/ ² ) dim D (M) Hence, for doubling metrics, linear-sized spanners!

example in action: TSP for doubling metrics

plan of attack We have PTASs for TSP for points in constant-dimensional ℓ2. If we could embed doubling metrics into constant-dimensional ℓ2 that maintains distances to within (1+ ² ) (in expectation) we’d be done. completely ridiculous strategy, but maybe we’ll get somewhere.

embedding doubling trees into ℓ2 Recall: embedding doubling metrics into ℓ2 requires  ( √ log n) distortion, regardless of dim’n. however… Theorem: if a doubling metric is also a tree metric, embeds into ℓ2 with distortion O(1) and dimension O(1) poly( ¸ )

embedding doubling metrics into doubling trees Bad news: 2-d grids require  (log n) distortion to embed into distributions over trees Good news: All doubling metrics embed into distributions over doubling trees with distortion O(log n).

plan of attack We have PTASs for TSP for points in constant-dimensional ℓ2. If we could embed doubling metrics into constant-dimensional ℓ2 that maintains distances to within (1+ ² ) (in expectation) we’d be done. revised

Arora’s simpler TSP idea Given any TSP tour of length L in d-dim space find B = (log n/ ± ) d portals in each cluster and show there exists a portal-respecting tour which increases length by ≤ ± L Now dynamic program to find best portal-resp tour  runtime ~ (n log n) B B

Arora’s simpler TSP idea Given any TSP tour of length L in d-dim space find B = (log n/ ± ) d portals in each cluster and show there exists a portal-respecting tour which increases length by ≤ ± L define portals, choosing ± = ² /O(log n) OPT tour of length L* in original doubling metric embeds into O(1)-dim space with length L = O(log n)L* increase in length = ± L = ² L* and now find the best portal-respecting tour in original doubling metric!

recap for TSP embedded doubling metric randomly into doubling trees embedded those into constant-dimensional ℓ2 use that to find clusters/portals and claim existence of (1+ ² ) OPT tour find best tour in original metric using dynamic programming. Talwar’s algorithm does it better, dependence on dim D, not on ¸

open problem Is there a PTAS for TSP on doubling metrics? Can we embed doubling trees into ℓ2 of O(dim D ) dimensions with O(dim D ) distortion? (suffices to consider unweighted doubling trees)

low dimensional embeddings (and dimensionality reduction)

dimensionality reduction If a Euclidean metric embeds into R k for some dimension k with distortion O(1) the Euclidean metric has doubling dimension O(k) we want to efficiently find an Euclidean embedding into R O(k) with distortion O(1)   We just saw: embed any metric with doubling dimension k into distribution over 2 O(k) -dimensional ℓ1 spaces with distortion O(log n)2 O(k). We just saw: embed any metric with doubling dimension k into distribution over 2 O(k) -dimensional ℓ1 spaces with distortion O(log n)2 O(k).

dimensionality reduction   We just saw: embed any metric with doubling dimension k into distribution over 2 O(k) -dimensional ℓ1 spaces with distortion O(log n)2 O(k). We just saw: embed any metric with doubling dimension k into distribution over 2 O(k) -dimensional ℓ1 spaces with distortion O(log n)2 O(k). If a Euclidean metric embeds into R k for some dimension k with distortion O(1) the Euclidean metric has doubling dimension O(k) we want to efficiently find an Euclidean embedding into R O(k) with distortion O(1)

dimensionality reduction   We just saw: embed any metric with doubling dimension k into distribution over 2 O(k) -dimensional ℓ1 spaces with distortion O(log n)2 O(k). We just saw: embed any metric with doubling dimension k into distribution over 2 O(k) -dimensional ℓ1 spaces with distortion O(log n)2 O(k). O(k) ℓ2 space O*(log n) Better: If a Euclidean metric embeds into R k for some dimension k with distortion O(1) the Euclidean metric has doubling dimension O(k) we want to efficiently find an Euclidean embedding into R O(k) with distortion O(1)

a more general bound Example Theorem: Any metric with doubling dimension dim D embeds into Euclidean space with T dimensions with distortion (where T 2 [ dim D log log n, log n]) All these techniques are ultimately limited by fact that they embed all doubling metrics, and not just Euclidean ones. log n dim D T

special cases of interest Distortion on using O(dim D (M)) Euclidean dimensions Distortion on using O(log n) Euclidean dimensions General metrics Euclidean This generalizes result we talked about in Lecture #2: any metric embeds into Euclidean space with O(log n) distortion This is just the Johnson-Lindenstrauss lemma. If the metric is doubling, this quantity is sqrt{log n}. In general, this is never more than O(log n). Again generalizes the previous result.

weaken requirements? Low-dimensional projection preserving near-neighbors O(log dim D poly ² -1 ) dimension random projection [IN05?] (random projections also work for points on smooth manifolds) Give low-dim set of points approximating d(x,y) 0.99 Again, can get similar dimensionality… [GK10, BRS10]

one more useful tool.. Given a metric M, want to partition it randomly into pieces of “small” diameter such that “nearby” vertices lie in different pieces only with “small” probability. “random metric decompositions”

“padded” decompositions A metric (V,d) admits ¯ -padded decompositions, if for every ¢, we can output a random partition V = V 1 ] V 2 ] … ] V k 1.each V j has diameter ≤ ¢ 2.Pr[ B(x, ½ ) split ] ≤

the facts Thm: Doubling metrics admit O(dim D )-padded decompositions Useful wherever padded decompositions are useful E.g.: can prove that all doubling metrics embed into ℓ2 with distortion

last slide: some questions For specific metric space problems, can we match the performance for their geometric counterparts? Which problems admit algorithms whose performance can be parameterized using such a notion of dimension? Other notions of dimension that are algorithmically significant?

thank you!