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Approximation Algorithms for Low-Distortion Embeddings into Low-Dimensional Spaces

Badoiu et al. (SODA 2005)Presented by: Ethan Phelps-Goodman Atri Rudra

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Moving from Seattle to NY

Looks like a 2hr drive Contraction is bad

Looks like a 10hr plane ride Expansion is bad

“Faithful” representation

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Metric Spaces Set of points Distance function

Non-negative Triangle Inequality

Metric Graph weighted graph shortest path distance

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Embeddings

Mapping : X! Y Exp()=maxa,b2 X d2((a),(b))/d1(a,b) Contr()=Exp(-1) Dist()=Exp() with Contr()¸ 1

(X,d1(¢,¢))

(Y,d2(¢,¢))

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Edge property

Mapping from a unwt graph G=(V,E) Exp()=max(x,y)2 E d2(x’,y’)

Exp()=maxx2 V,y2 V d2(x’,y’)/d1(x,y) ¸ is obvious x=v1,v2,,vL=y be the shortest path in G

maxi d2(v’i,v’i+1)¸ 1/L¢ i d2(v’i,v’i+1) The sum ¸ d2(x’,y’) by triangle inequality

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Embedding into a class of metrics

Pick an embedding Non-contractive Minimum distortion

Embedding into specific Metric [KRS04],[PS05]

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Previous work Start with a problem in some metric space Embed inputs into another “nice” metric

Problem is easy to solve in the nice metric For e.g., embed graphs into trees

Combinatorial in nature Try to upper bound the distortion

Survey by Indyk

D

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Algorithmic Task

Find embedding with min possible distortion Don’t care about the value of distortion Recall the map example

Embed unweighted graph into R (line) as well as possible

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What is in store for you ?

Embed unwt graph ! Line Hardness of the algorithmic Q Approx Algo for general unwt graph Approx Algo for unwt tree What else is there in the paper ? Open problems

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Hardness of Approximation

NP-Hard to a-approximate for some a>1 Reduction from TSP on (1,2)-metric

All distances are in {1,2} NP-Hard to a-approximate for any a <5381/5380

(X,D) ) G=(X,E) D(u,v)=1 (u,v)2 E

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Hardness of Approximation TSP on (1,2)-metric

Input M:- (V,D(¢,¢)) Output:- Permutation of V which is a tour M G=(V,E)

G

x y D(x,y)=1 (x,y)2 E

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The Construction

Given metric M G Make a copy of G called G’=(V’,E’) Add a special node o connected to V[ V’ The constrcuted graph H has diameter=2

G

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Showing that it works

M has a tour of len t) H embeds into R with distr · t Let the tour be v1,v2,,vn,v1

Start with v1

Lay out v2,,vn according to their distances Put in o Lay out G’ in the same way as G

Rv1

0

v2vi vi+1 vn

D(vi,vi+1)

o

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v’1 v’n

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Non-contractive

Expansion=t

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The other direction

f embeds H into R w/ distr s ) 9 tour in M of len ·s+1 Let u1,,u2n be the ordering of V[ V’ by f

Assume “nice” ordering |f(u2n)-f(u1)|· 2s (Wlog) blue box · (2s-2)/2=s-1 Blue box gives a tour of length · (s-1)+2=s+1

R

f(u1)f(u2n)

¸ 1

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Getting a nice ordering

Total ‘span’ still · 2s “Boundary” nodes are at distance ¸ 2 Swap blue and green boxes

Total Span still · 2s Still non-contractive

Keep on swapping till nice ordering is reached

R¸ 2

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What is in store for you ?

Embed unwt graph ! Line Hardness of the algorithmic Q Approx Algo for general unwt graph Approx Algo for unwt tree What else is there in the paper ? Open problems

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Embedding a tree into R Create an Eulerian tour Embed according to tour

preserve order preserve distances

No contraction Distortion · 2n-1

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Embedding general graphs

Every graph embeds into R w/ O(n) distortion Use any spanning tree

Coming Soon: c-approx algo c is the value of the optimal distortion

Combining both gives O(n1/2)-approx If c· n1/2, c-approx algo works If c> n1/2, spanning tree algo works

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c-approx algorithm

Embed G=(V,E) into R Let f* be the optimal embedding Let t1,t2 be the ‘end-points’ of f*

t1=v1,v2,,vL=t2 shortest path V=V1[ V2 [ VL

x closest to vi ) x2 Vi

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c-approx algo (contd.)

Embed Vi (i=1, L) by the spanning tree algo Layout vi first Recall that the max span is · 2|Vi|

Leave a gap of |Vi| on each side Run for all possible values of t1 and t2

V1 VLV2VL-1Vi

R

|Vi| |Vi|2|Vi|

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Notations

f* is the optimal embedding c is the optimal distortion f is the computed embedding D(¢,¢) shortest path in G

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Analysis

Clm1: D(vi,x)· c/2 Clm2: |Vi|+|Vi+1|++|Vi+c-1|· 2c2

Clm3: Embedding is non-contracting Will now show |f(x)-f(y)|· 16c2

|i-j|· 2c Span= 4¢[ (|Vi|++|Vi+c-1|) + (|Vi+c|++|Vj|) ]

The constructed embedding has distortion O(c2)

V1 VLV2VL-1

Vi

R|Vi| |Vi|2|Vi|

x

Vj

y

4|Vi|

as D(vi,vj)· D(x,vi)+D(x,y)+D(y,v+j)· c/2+1+c/2

f* has distortion c

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Non-contractive embedding

x,y2 Vi

x2 Vi, y2 Vj |f(x)-f(y)|¸ |Vi| + 2(|Vi+1|+ |Vj-1|) +|Vj|

¸ |Vi| + |j-i| + |Vj| ¸ D(x,vi) + D(vi,vj) + D(vj,y) ¸ D(x,y)

V1 VLV2VL-1

Vi

R|Vi| |Vi|2|Vi|

x

Vj

yy

Should be |Vj|+1(root goes first)

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Proof of Clm1

2¢ D(x,vi) · ? c D(x,vi) + D(x,vi)

· D(x,vj)+D(x,vj+1)

· ( f*(vj) – f*(x) ) + ( f*(x) – f*(vj+1)

= f*(vj) – f*(vj+1)

· c

R

x

vivj Vj+1

f*(vi)f*(vj) f*(vj+1)

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Embedding Trees into the line Problem: Given an unweighted tree that

embeds into the line with distortion c, find the smallest distortion line embedding.

They give (c logc)1/2-approximation, Can also be stated as O((n logn)1/3)-

approximation: If c > n2/3 then use simple spanning tree algorithm If c < n2/3 then use this algorithm

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Tree Embeddings

Similar to previous algorithm

Select endpoints & compute shortest path

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Tree Embeddings

Similar to previous algorithm

Select endpoints & compute shortest path

Group every c vertices

Embed each component, then concatenate

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Local Density

Define local density by = maxv, r (|B(v, r)|-1)/2r.

Then c > . In max density ball, there are 2r vertices, so end

points of embedding have distance at least 2r. But max distance is 2r, so endpoints have

distortion at least . Also, any component with diameter d has at

most d vertices.

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Component Embedding

Note: only care about order of vertices. Distances computed from shortest path

Want to embed in roughly depth-first order But don’t want neighbors too far away Algorithm alternates between laying out

neighbors of previously visited vertices (BFS) and DFS.

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Component Embedding in action

Ci

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Component embedding algorithm Magic number g(c)= 2(clogc)1/2 + c Pick a leftmost vertex r Let Ci be vertices visited up to round i While there are unvisited vertices

Visit all neighbors of Ci

Visit next g(c) vertices in light-path DFS order

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Bounding the distortion

Outline: Bound number of iterations Bound span of neighbor step Bound total distortion in component Bound distortion from concatenation

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Number of iterations

Diameter of tree is at most 2c:

So total # vertices is 2c

At least g(c) added at each iteration

Number of iterations is (2c)/g(c) (clog-1c)1/2

c

c/2 c/2

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Distortion of neighbor setor, where did g(c) come from? Claim: neighbor set is spanned by tree of size

g(c). Idea: Vertices in neighbor set can’t be too far

from “active” DFS vertices: at most (i+1) < (clog-1c)1/2 away.

So spanned by tree of size (clog-1c)1/2

2c2 vertices in component, so 2logc can be active

2logc * (clog-1c)1/2 + c = g(c).

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Distortion for full component

Vertices added in neighbor step are spanned by tree of size g(c)

g(c) connected vertices added in DFS step So distortion at most 2g(c) for each iteration 2 adjacent vertices could be on opposite

ends of iteration i and i+1, so total distortion 4g(c) over all iterations

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Concatenating the embeddings There is only one edge (vi, vi+1) connecting

components Xi and Xi+1 Modify the DFS ordering of Xi so that vi is last

visited Doesn’t affect distortion of Xi, and distortion of

edge (vi, vi+1) is at most 2g(c) Total distortion is at most

4g(c) = 8(clogc)1/2 + 4c 8 c3/2log1/2c + 4c Or O((c logc)1/2) times optimal

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Other algorithms in paper

An exact algorithm for embedding a general graph into the line, with runtime O(nc).

A geometric 3-approximation for embedding the sphere into the plane.

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Open questions

For lines: Better approximation ratios Lower bounds Weighted graphs (with large distortion)

Bigger question: algorithmic embeddings of graphs into the plane.