Approximation Algorithms for LowDistortion Embeddings into LowDimensional Spaces
Badoiu et al. (SODA 2005)
Presented by:
Ethan Phelps-Goodman
Atri Rudra
1
Moving from Seattle to NY
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Looks like a 2hr drive
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Looks like a 10hr plane ride
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Contraction is bad
Expansion is bad
“Faithful” representation
2
Metric Spaces
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Set of points
Distance function
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Non-negative
Triangle Inequality
Metric  Graph
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1
2
3
5
6
5
weighted graph
shortest path distance
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Embeddings
(Y,d2(¢,¢))
(X,d1(¢,¢))
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Mapping : X! Y
Exp()=maxa,b2 X d2((a),(b))/d1(a,b)
Contr()=Exp(-1)
Dist()=Exp() with Contr()¸ 1
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Edge property
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Mapping from a unwt graph G=(V,E)
Exp()=max(x,y)2 E d2(x’,y’)
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Exp()=maxx2 V,y2 V d2(x’,y’)/d1(x,y)
¸ is obvious
x=v1,v2,,vL=y be the shortest path in G
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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
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Pick an embedding
Non-contractive
Minimum distortion
Embedding into specific
Metric [KRS04],[PS05]
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Previous work
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Start with a problem in some metric space
Embed inputs into another “nice” metric
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Combinatorial in nature
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Problem is easy to solve in the nice metric
For e.g., embed graphs into trees
Try to upper bound the distortion
Survey by Indyk
D
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Algorithmic Task
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Find embedding with min possible distortion
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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 ?
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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
9
Hardness of Approximation
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NP-Hard to a-approximate for some a>1
Reduction from TSP on (1,2)-metric
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All distances are in {1,2}
NP-Hard to a-approximate for any a <5381/5380
(X,D) ) G=(X,E)
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D(u,v)=1  (u,v)2 E
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Hardness of Approximation
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TSP on (1,2)-metric
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Input M:- (V,D(¢,¢))
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Output:- Permutation of V which is a tour
M  G=(V,E)
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G
x
y
D(x,y)=1 (x,y)2 E
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The Construction
G
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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
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Showing that it works
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M has a tour of len t) H embeds into R with distr · t
Let the tour be v1,v2,,vn,v1
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Start with v1
Lay out v2,,vn according to their distances
Put in o
Lay out G’ in the same way as G
Non-contractive
Expansion=t
v1
0
v2
vi
vi+1
D(vi,vi+1)
vn
v’1
o
1
v’n
R
1
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The other direction
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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
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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
¸1
R
f(u2n)
f(u1)
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Getting a nice ordering
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Total ‘span’ still · 2s
“Boundary” nodes are at distance ¸ 2
Swap blue and green boxes
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Total Span still · 2s
Still non-contractive
Keep on swapping till nice ordering is reached
¸2
R
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What is in store for you ?
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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
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Create an Eulerian tour
Embed according to tour
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preserve order
preserve distances
No contraction
Distortion · 2n-1
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Embedding general graphs
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Every graph embeds into R w/ O(n) distortion
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Coming Soon: c-approx algo
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Use any spanning tree
c is the value of the optimal distortion
Combining both gives O(n1/2)-approx
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If c· n1/2, c-approx algo works
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If c> n1/2, spanning tree algo works
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c-approx algorithm
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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
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x closest to vi ) x2 Vi
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c-approx algo (contd.)
V1
V2
VL-1
Vi
VL
R
|Vi|
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|Vi|
Embed Vi (i=1, L) by the spanning tree algo
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2|Vi|
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
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Notations
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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
V1
Vi
V2
x
y
|Vi|
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2|V
i| |
4|V
i
|Vi|
Clm1: D(vi,x)· c/2
Clm2: |Vi|+|Vi+1|++|Vi+c-1|· 2c2
f* has distortion c
Clm3: Embedding is non-contracting
Will now show |f(x)-f(y)|· 16c2
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VL
Vj
R
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VL-1
|i-j|· 2c as D(vi,vj)· D(x,vi)+D(x,y)+D(y,v+j)· c/2+1+c/2
Span= 4¢[ (|Vi|++|Vi+c-1|) + (|Vi+c|++|Vj|) ]
The constructed embedding has distortion O(c2)
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Non-contractive embedding
V1
Vi
V2
x
y
y
VL-1
VL
Vj
R
|Vi|
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2|Vi|
|Vi|
x,y2 Vi
x2 Vi, y2 Vj
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|f(x)-f(y)|¸ |Vi|
Should be |Vj|+1
(root goes first)
+ 2(|Vi+1|+ |Vj-1|) +|Vj|
¸ |Vi|
+ |j-i|
¸ D(x,vi) + D(vi,vj)
¸ D(x,y)
+ |Vj|
+ D(vj,y)
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Proof of Clm1
vj
vi
Vj+1
x
R
f*(vj)
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f*(vi)
f*(vj+1)
2¢ D(x,vi) · ? c
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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
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Embedding Trees into the line
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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:
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If c > n2/3 then use simple spanning tree algorithm
If c < n2/3 then use this algorithm
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Tree Embeddings
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Similar to previous
algorithm
Select endpoints &
compute shortest
path
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Tree Embeddings
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Similar to previous
algorithm
Select endpoints &
compute shortest
path
Group every c
vertices
Embed each
component, then
concatenate
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Local Density
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Define local density by
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Then c > .
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 = maxv, r (|B(v, r)|-1)/2r.
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
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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
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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
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Visit all neighbors of Ci
Visit next g(c) vertices in light-path DFS order
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Bounding the distortion
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Outline:
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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
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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 set
or, where did g(c) come from?
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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:
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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
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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
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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
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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
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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
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For lines:
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Better approximation ratios
Lower bounds
Weighted graphs (with large distortion)
Bigger question: algorithmic embeddings of
graphs into the plane.
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