Randomness & Computation Fall 2011 Lecture 20 : November 1 Lecturer : Alistair Sinclair Based on scribe notes by :
نویسنده
چکیده
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 formal description of the algorithm is as follows:
منابع مشابه
CS 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-...
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 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 کاملSinclair Based on scribe notes by Rafael Frongillo
Recall from Lecture 4 that the probabilistic method provides a useful non-constructive strategy for proving the existence (or non-existence) of an object that satisfies some prescribed property. Generally, the argument involves selecting an object randomly from a specific set and demonstrating that it has the desired property with strictly positive probability. This in turn proves the existence...
متن کاملCS 49 : Data Stream Algorithms Lecture Notes , Fall 2011 Amit
Acknowledgements These lecture notes began as rough scribe notes for a Fall 2009 offering of the course " Data Stream Algorithms " at Dartmouth College. The initial scribe notes were prepared mostly by students enrolled in the course in 2009. Subsequently, during a Fall 2011 offering of the course, I edited the notes heavily, bringing them into presentable form, with the aim being to create a r...
متن کامل