a random polynomial time algorithm for approximating the volume of convex bodies n.
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A Random Polynomial-Time Algorithm for Approximating the Volume of Convex Bodies. By Group 7. The Problem Definition. The 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 space

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the problem definition
The Problem Definition
  • The 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 space
  • The 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 membershiporacle which returns yes if a query point lies inside the convex body or not.
  • n is definitely ≥3
what is a convex body
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
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
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)5ε-2 log[1/ε])

  • There is no deterministic approach of finding the volume of an n-dimensional convex body in polynomial time, therefore itwas 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
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
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
Overview of today’s presentation
  • The algorithm itself will be covered by Chen Jingyuan
  • Chen Min will introduce the concept of Random walk.
  • Proof of correctness and the complexity of algorithm is covered by Chin Hau
  • Tuan 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

The Algorithm

Chen Jingyuan

the dilation of a convex body
The Dilation of a Convex Body
  • 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 definition1
The Problem Definition
  • Input: A convex body
  • Goal: Compute the volume of , .
    • Here, n is the dimension of the body K.

How to describe K?

well guaranteed membership oracle well rounded
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.
the algorithm1
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
the algorithm2
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

Sample uniformly at random from Ki !

the algorithm3
The Algorithm

After τ steps...

  • Proper trial: if , we call it a proper trial.
  • Success trial: if , we call it a success trial.
the algorithm4
The Algorithm
  • Repeat 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.
random walk

Random Walk

Chen Min

natural random walk
Natural random walk

Some notations





A black box tells you whether a point x belongs

to K or not (e.g, a convex body is given by an oracle)


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

is smoother than K


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

natural random walk1
Natural random walk



Starts at any cube intersecting

It chooses a facet of the present cube each with

probability 1/(2n), where n is the dimension of the


- if the cube across the chosen facet intersects K,

the random walk moves to that cube

- else, it stays in the present cube









i j : ¼

i n : ¼

i k : ¼

i m :0

ii : ¼

technical random walk
Technical random walk

Why need technical random walk?

Only given K by an oracle.

How to decide whether

Cube ?

is smoother


Walk through


is smoother than K

Prove rapidly mixing

Satisfy the constraint:

Random walk has ½ probability stay in the same cube.

Apply the theorem of Sinclair and Jerrum


technical random walk1
Technical random walk
  • Q: 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 .


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

Ellipsoid algorithm

offers a terminate condition




C weakly intersects )

The walk will go to cube C

The walk will not go to cube C

technical random walk2
Technical random walk

2nd modification made on natural random walk


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



i j : 1/8

i n : 1/8

i k : 1/8

i m :0

i i : 5/8

In sum:

discrete time markov chain
Discrete-time Markov Chain

A Markov Chain is a sequence of random variables

With Markov Property.

Markov Property:

The future states only depend on current state.

A simple two-state Markov Chain


Technical random walk is a Markov Chain


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



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.



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

communicating class.

The graph of random walk is connected

Markov chain for technical random walk is


periodicity vs aperiodic
Periodicity vs. Aperiodic

A state i has period k if any return to state i

must occur in multiples of k.



If k=1, then the state is said to be aperiodic,

which means that returns to state i can occur

at irregular times.



A Markov chain is aperiodic if very state is aperiodic.

Each cube has a self loop

Markov chain for technical random walk is


stationary distribution
Stationary distribution

The 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:

Property of Markov chain:

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

Uniformly random generator

Markov chain for technical random walk has a

stationary distribution

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








  • Relate to
  • Show that approximates within a certain bound with a probability of at least ¾


: Number of sub-cubes

: Number of border sub-cubes

probability of error of a single volume estimate
Probability of error of a single volume estimate

Based on Hoeffding’s inequality , we can relate the result of the algorithm () and p as follows:

: Number of successes

: Number of proper trials


rapidly mixing markov chain

Rapidly Mixing Markov Chain

Nguyen Duy Anh Tuan

recap random walk markov chain
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
Ergodic Markov Chain
  • A Markov chain is ergodic if it is:
    • Irreducible, that is:
    • Aperiodic, that is:
markov chain steady state
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:

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

Ω is the set of all states

bounded mixing time
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:
rapidly mixing
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 mixing1
Rapidly Mixing
  • In our case:
    • n is the dimension of the convex body
    • and the number of states would be (3r/δ)n (δis the size of the cube, r is the radius of the bound ball).
rapidly mixing2
Rapidly Mixing
  • If the value of τ is substituted to the inequality in Theorem 1 of the paper
rapidly mixing3
Rapidly Mixing
  • Then, we take the summation of all the states to calculate the total variation distance:
proof of rapidly mixing markov chain

Proof of Rapidly Mixing Markov Chain

Chua Zheng Leong

Anurag Anshu

  • Shows 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.
  • Lets 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
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.