13th Nov 2006 1

Geometry of Graphsand It’s Applications

Suijt P Gujar.

Topics in Approximation Algorithms

Instructor : T Kavitha

13th Nov 2006 2

Agenda• Introduction

• Definitions

• Some Important Results

• Embedding Finite metric space into (Rd, Lp)

• Multi Commodity Flow via Low Distortion Embeddings

• Applications.

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Geometry Graphs

• Geometry of Graphs simply viewing Graphs from Geometric perspective

• Topological Models• Adjacency Models• Metric Models

In this talk we will be discussing Paper

“The Geometry of Grpahs and some of its algorithmic applications” by London, Linial, Rabinovich (LLR’94)

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What is Metric Space?

( , ) ( , ) 0

( , ) 0

( , ) ( , ) ( , ) (Triangle Inequality)

d x y d y x

d x y x y

d x y d y z d x z

Metric Space: A pair (X,d ) where X is a set and d is a distance function such that for x,y in X :

Banach Space: A vector space and a norm |v |, which defines a metric d (u,v)=|v-u|.

Hilbert Space :A vector space with inner product along with induced norm |v |, which defines a metric d (u,v)=|v-u|. E.g. (Rd, Lp)

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Examples of MetricsMinkowski Lp Metric: Let X = Rd.

1/

( , )p

p

i ii

d x y x y

Linf (Chessboard): ( , ) maxi i id x y x y

Hamming Distance: Let X = {0,1}k. Number of 1-bits in the exclusive-or

.x y

L1 : Manhattan Distance , L2 : Euclidian Distance

Cut Metric : X = A U B where A,B is partition of X d (x,y) = 0 iff x,y both Є A or both Є B = 1 otherwise.

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Embedding

• We will be considering Embedding of Metric Spaces to Banach Spaces esp. (Rd,Lp)

• Metric embedding is a function

• f : (X,dx) (Y,dy)

• Distortion : The embedding is said to have distortion C if for any x1,x2 in X

),(1

))(),((),( 212121 xxdc

xfxfdxxd xyx

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Example• Consider Graph G with 4 vertices with unit distance

between any pair of Vertices.

• Embed this in (R2,L2) with 4 vertices as vertices as square with diagonal length ‘1’.

),(2

1))(),((),( 2121221 xxdxfxfLxxd GG

A B

C D

d (A,B)

G

1

R2

0.7071

d (A,C) 1 0.7071

d (A,D) 1 1

d (B,C) 1 1

d (B,D) 1 0.7071

d (C,D) 1 0.7071

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Isometrics

• The isometry is mapping f from Metric space (X,dx) to metric space (Y,dy) which preserves distance. i.e. Distortion C = 1.

• Isometric Dimension of Metric space (X,dx) is the least dimension for which there exists embedding of X into any real normed space.

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• dim (X) ≤ n for ‘n’ point metric space.

• Let X = {x1, x2, …, xn} with dij = d(xi,xj).

• Map each point xi to zi Є Rn whose kth coordinate is zi

k = dik.

• || zi – zj ||inf = maxk | zik - zj

k | ≥ | zij - zj

j |

= |dij - djj | = dij

• On other hand, | zik - zj

k | = |dik - djk| ≤ dij

(Triangular inequality)

so, || zi – zj ||inf = dij.

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Johnson – Lindenstrauss Theorem (84)

• Any set of n points in a n - dimesional Euclidian space can be mapped to Rd where d = O(ε-2log n) with distortion ≤

1 + ε. Such mapping may be found in random polynomial time.

• Idea is to project n dimensional space orthogonally to d dimensional subspace

13th Nov 2006 11

JL Theorem contd…

• Take A1, A2, …, Ad set of orthonormal Vectors randomly chosen in Rn.

• A = [A1 A2 … Ad]t • For any x in X, x’ = Ax.

Consider x Є Rn st || x ||2 = 1.

So, E[xi2] = 1/n. E[x’.x’] = d/n.

E[||x’||] = √(d/n) = m.

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• Let x,y be two vectors in Rn. And x’, y’ be corresponding embedding in Rd.

• X’ = Ax, y’ = Ay. • ||x’-y’|| = A(x-y).

• Pr ( | ||x’-y’|| - m||x-y|| | > εm||x-y|| )

≤ e Ω(-d/ ε* ε)

When d O(ln n / ε* ε ), this Probability of failure < 1/n2.

Best known bound is d = 16*ln n / ε2

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Some results for embeddings

We will define

• Ldp = (Rd,Lp).

• Cp(X) = minimum distortion with which X may be embedded in Lp.

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X Y C

L2n, n points L2

O(log n) 1 + ε i.e. O(1)

R JL(84)

n point metric space

L22^n O (log n) D Bourgain(85)

L2n C2(X) + ε D

L2O(logn) O (log n) R

LpO(logn)

1 ≤ p ≤ 2

O (log n) R LLR(94)

LpO(n^2)

1 ≤ p ≤ 2

O ( C2(X) ) D LLR(94)

LpO(logn^2)

1 ≤ p

O (log n) R LLR(94)

13th Nov 2006 15

LLR Algorithm for Embedding

• Let q = O(log n). [Constant affects the constant in distortion.]

• For i = 1,2,…,log n doFor j = 1 to q do

Ai,j = random subset of X of size 2i.

• Map x to the vector {di,j}/Q1/p • di,j is the distance from x to the closest

point in Ai,j and Q is the total number of subsets.

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Theorem 1

• Let (X,d) be a finite metric space and

{(si,ti) | i = 1,2,…,k} Є X x X. There exists a deterministic algorithm that finds an embedding f : X → l1O(n^2), so that

d (x,y) ≥ ||f(x) – f(y)||1 for every x,y in X and

||f(si) – f(ti)||1 ≥ Ω(1/log k)*d (si,ti)

for every i = 1,2,…,k.

13th Nov 2006 17

Multi commodity flows via low distortion embeddings

• Problem :

Given an undirected Graph G(V,E) with n vertices, Capacity Ce associated with every edge in E. There are k source-sink pairs (s i,ti) and Demand Di associated with it. Flow conservation law should hold true. Total flow through each edge should not exceed the capacity. Find the maxflow, largest f such that, it is possible to simultaneously flow f*Di, between (si,ti) for all i.

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Max flow – Min Cut gap• f* be the maxflow. • Trivial upper bound )(

)(min*

SDemSCap

aS

f* ≤ a*Are these two equal? No.

*f*a

gapCut Min-Max Flow

Leigthon-Rao (‘87) showed in some cases this gap ≤ O (log n) Garg, Vazerani (‘93) showed in case of unit demand among all source-sink pairs, this gap ≤ O (log k)LLR (‘94) : This gap is always ≤ O (log k) using, least distortion embedding of graph in L1.

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LP for Max flow multi commodity

• Garg, Vazerani :

k

i i

jiji ji

dDdC

ts ii

f1 ,

,,

.

.min*

Where minimum is over all metrics over G

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• Let d be optimizing metric.• Apply theorem 1 to embed (V,d) into L1

m.

say {x1,x2,…,xn}.• ||xi-xj||1 ≤ di,j for all i,j. And,

||x_si – x_ti)||1 ≥ Ω(1/log k)*d (si,ti) for every i = 1,2,…,k.

Lets denote, xi,j = ||xi - xj||1

)(log *f .

,1

,, kOxDxC

ts ii

k

i i

jiji ji

13th Nov 2006 21

Lemma

yb

xa

ybxa

ii

i

ii

i

i

ii

i

i

)(min

||

||.

||

||..

,,

,,

1

,,,

1

11

,,,1

,1

,,

minxxDxxC

xxD

xxC

xDxC

rtrs

rtrsts

ii

iiii

k

i i

rjriji ji

mr

k

i i

m

r

rjriji ji

m

rk

i i

jiji ji

13th Nov 2006 22

Max Flow Min Cut gap

• Suppose the for minimizing r,

all xi,r in {0,1}.

Then, for that r,

}1|{ where

)(

)(

||

||.

,

1

,,,

,,

x

xxDxxC

ri

k

i i

rjriji ji

iS

SDem

SCap

rtrs ii

a* ≤ Cap(S)/Dem(S) ≤ f* O (log k)So, Max-flow min cut gap is bounded by O (log k)

13th Nov 2006 23

Variational ArgumentConsider the expression

||.

||.

,

,

xxbxxajiji ji

jiji ji

1. If all x’s take only two values, the valuation can be replaced by 0,1

2. Suppose x’s take three values, s > t > u. Then Consider the x’s which take value t. Fixing all other values let t varies over [u,s],

3. The expression is linear function in t. So changing t to u or s, the value of expression won’t increase.

4. Repeat this procedure till all variables take only two values.

13th Nov 2006 24

Algorithm• Solve LP to find f*.• Embed Graph with optimizing metric,

into L1m.

• Find r which minimizes,

||

||.

,,1

,,,

xxDxxC

rtrs ii

k

i i

rjriji ji

• Using Variational Argument, get near Optimal Cut

13th Nov 2006 25

Limitations

Limitations of the LLR embedding:O(log2 n) dimension: This is a real problem.

O(n 2) distance computations must be performed in the process of embedding and embedding a query point requires O(n) distance computations: Too high if distance function is complex.

O(log n) distortion: Experiments show that the actual distortions may be much smaller.

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Questions???

13th Nov 2006 27

Thank You !!!