Upload
toby-boyd
View
212
Download
0
Embed Size (px)
Citation preview
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.
13th Nov 2006 3
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)
13th Nov 2006 4
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)
13th Nov 2006 5
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.
13th Nov 2006 6
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
13th Nov 2006 7
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
13th Nov 2006 8
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.
13th Nov 2006 9
• 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.
13th Nov 2006 10
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.
13th Nov 2006 12
• 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
13th Nov 2006 13
Some results for embeddings
We will define
• Ldp = (Rd,Lp).
• Cp(X) = minimum distortion with which X may be embedded in Lp.
13th Nov 2006 14
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.
13th Nov 2006 16
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.
13th Nov 2006 18
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.
13th Nov 2006 19
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
13th Nov 2006 20
• 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.
13th Nov 2006 26
Questions???
13th Nov 2006 27
Thank You !!!