A Random Polynomial-Time Algorithm for Approximatingthe Volume of Convex Bodies

By Group 7

The Problem DefinitionThe main result of the paper is a randomized algorithm for finding an approximation to the volume of a convex body ĸ in n-dimensional Euclidean spaceThe paper is a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan presented in 1991.This is done by assuming the existence of a membership oracle which returns yes if a query point lies inside the convex body or not.n is definitely ≥3

Never seen a n-dimensional body before?

What is a convex body?

• In Euclidean space, an object is defined as convex – if for every pair of points within the object, – every point on the straight line segment that joins the pair of

points also lies within the object.

Convex Body Non- Convex body

Well Roundedness?

The algorithm mentions well rounded convex body which means the dimensions of the convex body are fixed and finite.Well roundedness is defined as a property of a convex body which lies between two spheres having the radii:-

1 & √ (n)x(n+1)(where n= no. of dimensions)

The running time of the algorithm

This algorithm takes time bounded by a polynomial in n, the dimension of the body ĸ and 1/ε where ε is relative bound error.The expression for the running time is:-

O(n23(log n)5 ε-2 log[1/ε])

Motivation

• There is no deterministic approach of finding the volume of an n-dimensional convex body in polynomial time, therefore it was a major challenge for the authors.

• The authors worked on a probabilistic approach to find the volume of the n-dimensional convex body using the concept of rapidly mixing markov chains.

• They reduced the probability of error by repeating the same technique multiple number of times.

• It was also the FIRST polynomial time bound algorithm of its kind.

Deterministic approach and why it doesn’t work?

Membership oracle answers in the following way: It says yes, if a point lies inside the unit sphere and says no otherwise.

After polynomial no of. queries, we have a set of points, which we call P, from which must form the hull of the actual figure.

But possible candidates for the figure can range from the convex hull of P to the unit sphere.

Deterministic approach and why it doesn’t work contd.

• The ratio of convex hull (P) and unit sphere is at least

poly(n)/2^n. • So, there is no deterministic approximation

algorithm that runs in polynomial time.

Overview of today’s presentation

The algorithm itself will be covered by Chen JingyuanChen Min will introduce the concept of Random walk.Proof of correctness and the complexity of algorithm is covered by Chin HauTuan Nguyen will elaborate on the concept of Rapidly Mixing Markov’s Chains(RMMC).Zheng Leong will elaborate on the proof of why the markov’s chain in rapidly mixing.Anurag will conclude by providing the applications and improvements to the current algorithm

The Algorithm

Chen Jingyuan

The Dilation of a Convex Body

}:{ KxxK

For any convex body K and a nonnegative real number ɑ,

The dilation of K by a factor of ɑ is denoted as

The Problem Definition

nRK Input: A convex body

Goal: Compute the volume of , .

• Here, n is the dimension of the body K.

K )(Kvoln

How to describe K?

Well-guaranteed Membership Oracle&Well-rounded

A sphere contained in the body: B.• B is the unit ball with the origin as center.

A sphere containing the body: rB.• Here , n is the dimension of the body.

A black box• which presented with any point x in space, either replies

that x is in the convex body or that it is not.

)1( nnr

Basic Idea

rB

K

B

rBKB

kK

rKrB 1K 2K rKKk

)( rBKvol )()()(

)()( 1

1rBKvol

rBKvolrBKvol

rBKvolrBKvol

kk

k

)(Kvol

)()()(

)()()()( 1

1rBKvol

rBKvolrBKvol

rBKvolrBKvolrBKvolKvol k

k

k

)(rBvol)()( 1

rBKvolrBKvol

l

l

rBKl 1

rBK l

The Algorithm

How to generate a group dilations of K?Let , and .For i=1, 2, …, k, the algorithm will generate a

group dilations of K, and the ratios equals to

)1(1 n rk 1log }1,max{ rii

rBKrBK

i

i

1

rKK 0 KKk

Ki

The Algorithm

How to find an approximation to the ratio

The ratio will be found by a sequence of "trials" using random walk.

In the following discussion, let rBKK ii

)()(

1 rBKvolrBKvol

in

in

Sample uniformly at random from Ki !

The Algorithm

iK 1iK

After τ steps...

…

})1(:{ iii qxqxC

rqqqx nn,,, 2

21

10

• Proper trial: if , we call it a proper trial.

• Success trial: if , we call it a success trial.

10 iKx

iKx 0

12},,2,1,0{,, 21 rn

The Algorithm

m̂mRepeat until we have made proper trials.

And of them are success trials.

The ratio, , will be a good approximation to the ratio of volumes that we want to compute.

mm̂

mm

rBKvolrBKvol

in

in ˆ)(

)(

1

)()()(

)()()()( 0

0

1

1

rBKvolrBKvolrBKvol

rBKvolrBKvolrBKvolKvol n

n

n

kn

knknn

)(rBvoln

The Conclusion of the Algorithm

Random Walk

Chen Min

Natural random walk

Technical random walk

Natural random walkSome notations

2. For any set in and a nonnegative real number , we denote by the set of points at distance at most from K.

K

is smoother than K

3.cubes:We assume that space () is divided into cubes of side . Formally, a cube is defined as:

Where are integers

…

…

Any convex body can be filled with cubes

1.Oracle: A black box tells you whether a point x belongs to K or not (e.g, a convex body is given by an oracle)

OracleY/Nx

Natural random walk

….

Steps:

1. Starts at any cube intersecting

2. It chooses a facet of the present cube each with probability 1/(2n), where n is the dimension of the space.

- if the cube across the chosen facet intersects K, the random walk moves to that cube

- else, it stays in the present cube

Prob:i j : ¼i n : ¼i k : ¼i m :0i i : ¼

K

….i nmk

j

…

Technical random walk

Walk through

Only given K by an oracle.

How to decide whetherCube ?

Why need technical

random walk?

1.

2. Apply the theorem of Sinclair and Jerrum

Prove rapidly mixing Satisfy the constraint:Random walk has ½

probability stay in the same cube.

is smoother

K

is smoother than K

Technical random walkQ: We want to walk through . But we are only given K by an oracle, and this will not let us decide precisely whether a particular cube .

-modificationrandom walk is executed includes all of those cubes that intersect plus some other cubes each of which intersects , where .

Ellipsoid algorithm

contains

C weakly intersects )The walk will go to cube C

Terminates:

offers a terminate condition

The walk will not go to cube C

𝐾 (𝛼+𝛼 ′ )

x

Technical random walk2nd modification made on natural random walk

….i nmk

j

…

Prob:i j : 1/8i n : 1/8i k : 1/8i m :0i i : 5/8

New rules:

1. The walk has ½ probability stays in the present cube

2. With probability 1/(4n) each, it picks one of the facets to move across to an adjacent cube

• 1/2nnatural

• • 1/4n, 1/2stay

technical

In sum:

Background on Markov chain

Technical random walk will converge to uniform distribution

Discrete-time Markov Chain

A simple two-state Markov Chain

A Markov Chain is a sequence of random variables With Markov Property.

Markov Property:The future states only depend on current state.

Formally:

Technical random walk is a Markov Chain

1 1 1 2 2 1Pr( | , ,..., ) Pr( | )n n n n n nX x X x X x X x X x X x

IrreducibleA state j is said to be accessible from a state i if: i j

j is accessible from i i is not accessible from j

A state i is said to communicate with state j if they are mutually accessible. i j

A Markov chain is said to be irreducible if its state space is a single communicating class.

Markov chain for technical random walk is irreducible

The graph of random walk is

connected

( )0Pr( | ) 0ijn

n ijX j X i p

Periodicity vs. Aperiodic

A state i has period k if any return to state i must occur in multiples of k. i j

0gcd{ : Pr( | ) 0}nk n X i X i

If k=1, then the state is said to be aperiodic, which means that returns to state i can occur at irregular times.

i j

A Markov chain is aperiodic if very state is aperiodic.

Markov chain for technical random walk is aperiodic

Each cube has a self loop

Stationary distributionThe stationary distribution π is a vector, whose entries are non-negative and add up to 1. π is unchanged by the operation of transition matrix P on it, and is defined by:

P Property of Markov chain:If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π .

Markov chain for technical random walk has a stationary distribution

Uniformly random

generator

Since P is symmetric for technical random walk, it is easy to see that all ’s are equal.

i jE.g, 0.4

0.40.6 0.6

Proof of CorrectnessHoo Chin Hau

Overview

1. Relate to

2. Show that approximates within a certain bound with a probability of at least ¾

Pr (𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )

: : Number of sub-cubes: Number of border sub-cubes

𝐶𝐾 𝑖−1

𝛿

Pr (𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )

Pr (𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )

Pr (𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )≥𝑉𝑜𝑙𝑛 (𝐾 𝑖− 1 )|W|𝛿𝑛 (1− 𝜖

100𝑘 )≥0.33

∑𝐶∈𝑊

𝑁𝐶𝐵

𝑁𝐶≤3𝑛

32 𝜂𝑉𝑜𝑙 (𝐾 𝑖 −1 )𝛿𝑛

Pr (𝑠𝑢𝑐𝑐𝑒𝑠𝑠∩𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )

𝑉𝑜𝑙𝑛 (𝐾 𝑖 )|W|𝛿𝑛 (1− 𝜖

100𝑘 )≤ Pr (𝑠𝑢𝑐𝑐𝑒𝑠𝑠∩𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )≤𝑉𝑜𝑙𝑛 (𝐾 𝑖 )|W|𝛿𝑛

(1+ 𝜖100𝑘 )

Pr (𝑠𝑢𝑐𝑐𝑒𝑠𝑠∨𝑝𝑟𝑜𝑝𝑒𝑟 𝑡𝑟𝑖𝑎𝑙 )

𝑣 (1− 𝜖100𝑘 )(1+ 𝜖

100𝑘 )− 1

≤𝑝 ≤𝑣 (1+ 𝜖100𝑘 )(1− 𝜖

100𝑘 )−1

,𝑣=𝑉𝑜𝑙𝑛 (𝐾 𝑖 ) /𝑉𝑜𝑙𝑛 (𝐾 𝑖−1 )

Probability of error of a single volume estimateBased on Hoeffding’s inequality , we can relate the result of the algorithm () and p as follows:

: Number of successes: Number of proper trials

Previously,

𝑝 ≥ 15

Probability of error of k volume estimates

(1−𝑥 )𝑛≥1−𝑛𝑥 ,𝑥 ≤1

𝐴𝑠𝑠𝑢𝑚𝑖𝑛𝑔𝑉𝑜𝑙𝑛 (𝐾 0 )𝑐𝑎𝑛𝑏𝑒𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑𝑡𝑜 h𝑤𝑖𝑡 𝑖𝑛1± 𝜖2 , h𝑡 𝑒 h𝑎𝑙𝑔𝑜𝑟𝑖𝑡 𝑚𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑠 𝑎𝑛𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑉 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔

Probability of error of k volume estimates

(1−𝜖 )≤ 𝑉𝑉𝑜𝑙𝑛 (𝐾 )

≤ (1+𝜖 )with a probability of 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 34

Complexity of algorithm

𝑂 (𝑘𝑚𝜏 )=𝑂 (𝑛23 (𝑙𝑜𝑔𝑛 )5𝜖− 2 log( 1𝜖 ))

Rapidly Mixing Markov ChainNguyen Duy Anh Tuan

Recap Random walk – Markov chain

A random walk is a process in which at every step we are at a node in an undirected graph and follow an outgoing edge chosen uniformly at random.

A Markov chain is similar, except the outgoing edge is chosen according to an arbitrary distribution.

Ergodic Markov Chain

A Markov chain is ergodic if it is:1. Irreducible, that is:

2. Aperiodic, that is:jipNs s

ji ,,0: )(,

jips sji ,,1}0:gcd{ )(

,

Markov Chain Steady-state

Lemma:

Any finite, ergodic Markov chain converges to a unique stationary distribution π after an infinite number of steps, that is:

jip jsji

s,)(

,lim

j

j 1

Markov Chain Mixing time

Mixing time is the time a Markov chain takes to converge to its stationary distribution

It is measured in terms of the total variation distance between the distribution at time s and the stationary distribution

Total variation distance

Letting denotes the probability of going from i to j after s steps, the total variation distance at time s is:

)(,sjip

j

jsjiitv

s pp ,21max,

Ω is the set of all states

Bounded Mixing Time

Since it is not possible to obtain the stationary distribution by running infinite number of steps, a small value ε > 0 is introduced to relax the convergent condition.

Hence, the mixing time τ(ε) is defined as:

}',,:min{)( ' sspstv

s

Rapidly Mixing

A Markov chain is rapidly mixing if the mixing time τ(ε) is O(poly(log(N/ε))) with N is the number of states.

If N is exponential in problem size n, τ(ε) would be only O(poly(n)).

Rapidly Mixing

In our case:• n is the dimension of the convex body • and the number of states would be (3r/δ)n (δ is the size of the cube, r is the radius of the

bound ball).

krns

n 3003log10 1917

jin

pt

jtji ,,

1011 1917

)(,

Rapidly MixingIf the value of τ is substituted to the inequality in Theorem 1 of the paper

krp

ep

np

np

n

jji

kr

jji

krn

jji

t

jtji

n

n

3003

1011

1011

)(,

3003log1)(,

3003log10

1917)(

,

1917)(

,

1917

Rapidly MixingThen, we take the summation of all the states to calculate the total variation distance:

jikr

pn

jtji ,,

3003)(

,

kp

krrp

pp

s

nns

jj

sjiitv

s

30021,

30033

21,

21max, ,

Proof of Rapidly Mixing Markov Chain

Chua Zheng Leong

Anurag Anshu

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

Proof of Rapidly Mixing Markov Chain

ApplicationsShows that P ≠ BPP relative to this oracle. This means the implementation of oracle cannot be in polynomial time. Further, its surprising since P=BPP is believed to be true.

Technique can be used to integrate well behaved and bounded functions over a convex body.

Improvements in running time of algorithm would require improvement in mixing time of random walk. This is useful because the random walk introduced in paper is frequently studied in literature.

ConclusionLets revisit the algorithm, briefly.Given a well rounded figure K, we consider a series of rescaled figures, such that the ratio of volume for consecutive ones is a constant fraction.We perform a technical random walk on each figure, and look for the ‘success’, which gives us the ratio of volumes between consecutive figures to good approximation. We use it to obtain the volume of K, given that we know the volume of bounding sphere.Technical challenge is to prove convergence of markov process.

Improvements in Algorithm

A novel technique of using Markov process to approximate the volume of a convex body.In current analysis, the diameter of random walk was O(n^4). So algorithm could not have been improved beyond O(n^8), without improving the diameter. Algorithm improved to O(n^7) by Lovasz and Simonovitz in “Random walks in a convex body and an improved volume algorithm”. Current algorithms reach up to O(n^4), as noted here.

Thank you!