Lecturer : Alistair Sinclair Scribes : Yanlei Diao and Xiaofeng Ren
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چکیده
The Metropolis process is an extremely general recipe for constructing a Markov chain which has any desired stationary distribution π on a finite set Ω. Moreover, this distribution can be specified just by a weight function w : Ω → R, so that π(x) = w(x) Z where Z is an unknown normalizing factor. The Metropolis process is named after one of its inventors [MR+53]. To specify the Metropolis process, we need to provide two things:
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Lecturer : Alistair Sinclair Scribes : Francis Bach ,
We can do so by treating the vertices one by one. Consider the intermediate situation where V1 is the set of vertices that have already got their final color (i.e., the color in X2) and V2 = V \V1. Now, pick some vertex v ∈ V2. This vertex still has the color c1 it has in state X1 and needs to be recolored to the color c2 it is supposed to have in state X2. This is only possible if the neighbor...
متن کاملLecturer: Alistair Sinclair Based on Scribe Notes By: 14.1 Finding Hamilton Cycles in Random Graphs
Moreover, there is a polynomial time algorithm to find a Hamilton cycle in a random graph G ∈ Gn,p w.h.p. for all p > lnn+ln lnn+c(n) n , c(n) → ∞ [BFF85]. In this section, we look at a simpler algorithm [AV77] that works for all p ≥ c lnn n−1 for a sufficiently large constant c. This is a slightly weaker result as the value of c, determined by Chernoff bound arguments used in the algorithm’s a...
متن کاملLecturer : Alistair Sinclair Based on scribe notes by :
Recall that a (proper) coloring maps each vertex of a given graph G to a color such that no two adjacent vertices share the same color. The chromatic number χ(G) is the minimum number of colors in a proper coloring of G, and is NP-complete to compute for general G. However, we shall now see that for a random graph G, χ(G) has a fairly well-determined value. In the last lecture we proved the fol...
متن کاملRandomness & Computation Fall 2011 Lecture 20 : November 1 Lecturer : Alistair Sinclair Based on scribe notes by :
In this lecture we use Azuma’s inequality to analyze the randomized Quicksort algorithm. Quicksort takes as input a set S of numbers, which can be assumed to be distinct without loss of generality, and sorts the set S as follows: it picks a pivot x ∈ S uniformly at random, then it partitions the set S as Sx = {y ∈ S | y > x}, and recursively sorts Sx . The fo...
متن کاملCS 271 Randomness & Computation Fall 2011 Lecture 3 : September 1 Lecturer : Alistair Sinclair Based on scribe notes by :
Observe that if a = b then Bob will always be correct. However, if b 6= a then there may be an error: this happens iff the fingerprints of a and b happen to coincide. We now show that, even for a modest value of T (exponentially smaller than a and b), if a 6= b then Pr[Fp(a) = Fp(b)] is small. First observe that, if Fp(a) = Fp(b), then a = b mod p, so p must divide |a − b|. But |a − b| is an n-...
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