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: Cambri
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:
Cambr
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
assignme
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: Cambri
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/m
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
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
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 t
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 ex
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 sam
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Complexity note
model assumes source of random bits
we will assume primitives: biased coins, uniform sampling
in homework, saw equivalent
Review Game Tree
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
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
Universi
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 g