6.856 Randomized Algorithms
David Karger
Handout #24, December 5th, 2002 Homework 11 Solutions
Problem 1
(a) Let D be the disjoint union, and N := |D|. We will denote by (a, x) D a particular
assignment/clause pair in the disjoint union. That is, assignme
6.856 Randomized Algorithms
David Karger
Handout #12, October 14, 2002 Homework 5 Solutions
M. R. refers to this text:
Motwani, Rajeez, and Prabhakar Raghavan. Randomized Algorithms. Cambridge: Cambridge
University Press, 1995.
Problem 1 Randomized select
6.856 Randomized Algorithms
David Karger
Handout #14, October 20, 2002 Homework 6 Solutions
M. R. refers to this text:
Motwani, Rajeez, and Prabhakar Raghavan. Randomized Algorithms. Cambridge: Cambridge
University Press, 1995.
Problem 1 Consider a perfec
November 17, 2002
6.856
Homework 10 Solutions
1. (a) Let the graph be G = (V, E ) with |V | = n. Construct a graph G(p) on V by including each e E
with probability p = 12 log n/(c(/2)2 ). By max-ow/min-cut the s t min-cut of G has value v .
As in lecture
6.856
Problem Set 9
Nov. 5th, 2002
1. Run k times and take the median. For the median to be out of range, k /2 estimates must have
deviated by more than n. In expectation, only X = k /4 estimates are out of range. So by Cherno,
the median is out of range
6.856 Randomized Algorithms
David Karger
Handout #25, December 5th, 2002 Homework 12 Solutions
Problem 1
(a) Consider the vector = (1, 1, . . . , 1). Then P is just the sum of the rows of P , which
is just the vector of column sums, each of which is 1 by
6.856
Problem Set 13(Final)
Dec. 2nd, 2002
1. (a) The loops make it aperiodic, and so we merely need show irreducibility(strong connectedness)
to get unique stationary distribution. We need to show that from a pair (T1 , r1 ), we can get
to another pair
6.856 Randomized Algorithms
David Karger
Handout #16, October 27, 2002 Homework 7 Solutions
Problem 1
(a) We let each machine broadcast independently with probability 1/n. Then the probability
that exactly one machine tries to broadcast in a given timeslo
6.856 Randomized Algorithms
David Karger
Handout #4, September 17, 2002 Homework 1 Solutions
M.R. refers to this text:
Motwani, Rajeez, and Prabhakar Raghavan. Randomized Algorithms. Cambridge: Cambridge University Press, 1995.
Problem 1 MR 1.1.
(a) We re
6.856 Randomized Algorithms
David Karger
Handout #18, November 2, 2002 Homework 8 Solutions
Problem 1 First lets eliminate edges of length 0. Two vertices connected by an edge of
length 0 have the same distance to everything, so we can contract all such e
Administration:
Homework Grading signup.
Complexity note
model assumes source of random bits
we will assume primitives: biased coins, uniform sampling
in homework, saw equivalent
Review Game Tree
Changed presentation from book.
We used game tree w
Complexity.
What is a rand. alg?
What is an alg?
Turing Machines. RAM with large ints. log-cost RAM as TM.
language as decision problem (vs optimization problems) graphs with
small min-cut. algos accept/reject
complexity class as set of languages
6.856 Randomized Algorithms
David Karger
Handout #10, 2002 Homework 4 Solutions
M. R. refers to this text:
Motwani, Rajeez, and Prabhakar Raghavan. Randomized Algorithms. Cambridge: Cambridge
University Press, 1995.
Problem 1
(a) MR Exercise 4.2. Each nod
6.856 Randomized Algorithms
David Karger
Handout #4, September 21, 2002 Homework 1 Solutions
M. R. refers to this text:
Motwani, Rajeez, and Prabhakar Raghavan. Randomized Algorithms. Cambridge:
Cambridge University Press, 1995.
Problem 1 MR 1.8.
(a) The
6.856 Randomized Algorithms
David Karger
Handout #8, September 30, 2002 Homework 3 Solutions
Problem 1
We may think of asking a resident as ipping a coin with bias p=f. Flip the coin N times.
If you get k heads, set p = k /N . Note k has a binomial distri