A Random Polynomial-Time Algorithm for Approximating the Volume of Convex Bodies

1. A Random Polynomial-Time Algorithm for Approximatingthe Volume of Convex Bodies By Group 7

2. 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

3. Never seen a n-dimensional body before?

4. 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

5. 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)

6. 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])

7. Motivation • 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.

8. 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.

9. 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.

10. 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

11. The Algorithm Chen Jingyuan

12. 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

13. 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?

14. 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.

15. Basic Idea 1

17. 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

18. 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 !

19. The Algorithm After τ steps... … • Proper trial: if , we call it a proper trial. • Success trial: if , we call it a success trial.

20. 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.

21. The Conclusion of the Algorithm

22. Random Walk Chen Min

24. Natural random walk Some notations x Y/N Oracle 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) K 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 … 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

25. Natural random walk K Steps: 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 space. - if the cube across the chosen facet intersects K, the random walk moves to that cube - else, it stays in the present cube …. …. … j m i n k Prob: i j : ¼ i n : ¼ i k : ¼ i m :0 ii : ¼

26. Technical random walk Why need technical random walk? Only given K by an oracle. How to decide whether Cube ? is smoother K Walk through 1. 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 2.

27. 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 . -modification 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 x Terminates: contains C weakly intersects ) The walk will go to cube C The walk will not go to cube C

28. Technical random walk 2nd modification made on natural random walk …. … New rules: j m i n 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 k Prob: i j : 1/8 i n : 1/8 i k : 1/8 i m :0 i i : 5/8 In sum:

30. 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 Formally: Technical random walk is a Markov Chain

31. Irreducible A state j is said to be accessible from a state i if: j i 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. j i 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 irreducible

32. Periodicity vs. Aperiodic A state i has period k if any return to state i must occur in multiples of k. i j 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. Each cube has a self loop Markov chain for technical random walk is aperiodic

33. 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. 0.4 E.g, i j 0.6 0.6 0.4

34. Proof of Correctness Hoo Chin Hau

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

36. : : Number of sub-cubes : Number of border sub-cubes

41. 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 Previously,

42. Probability of error of k volume estimates

43. Probability of error of k volume estimates

44. Complexity of algorithm

45. Rapidly Mixing Markov Chain Nguyen Duy Anh Tuan

46. 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.

47. Ergodic Markov Chain • A Markov chain is ergodic if it is: • Irreducible, that is: • Aperiodic, that is:

48. 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:

49. 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

50. 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