Randomized Algorithms (600.464/664):
Assignment 1  Solutions
P.C. Shyamshankar
Due: September 13, 2012
1. (a) Two random variables X and Y are independent, if:
x, y : P [X = x, Y = y ] = P [X = x] P [Y = y ].
(1)
Conversely, X and Y are not independent i
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Randomized Algorithms (600.464/664)
MidSemester Exam: Solutions
P.C. Shyamshankar
October 25, 2012
1.
n2 les are stored into an n n array of storage devices independently and with the following probabilities. Every le
gets stored into device (i, j ) : 1
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Randomized Algorithms (600.464/664)
Assignment 8: Solutions
P.C. Shyamshankar
Due: November 29, 2012
1.
Let P be any irreducible Markov Chain. Prove that for any
states u and v , the GCDs of the cycle lengths passing through
u and v are equal.
Since P is
Randomized Algorithms (600.464/664):
Assignment 6 Solutions
P.C. Shyamshankar
December 4, 2012
1.
Let a universe U be of size m, and let R be an array of n
locations. Let p = m + 1 be a prime number. A set of functions
F from U to R is strongly 2universa
Randomized Algorithms (600.464/664)
Assignment 5: Solutions
P.C. Shyamshankar
Due: October 18, 2012
1.
Derandomize an algorithm for the version of the set discrepancy
problem where each Aij  1, using the conditional probability
method.
We start with the
Randomized Algorithms (600.464/664)
Assignment 4: Solutions
P.C. Shyamshankar
Due: October 11, 2012
1.
For any vector v cfw_0, 1n , v = 0n , a vector x cfw_1, +1n is
said to be a detector for v if v x = 0. Prove that there exists a
set S of O(n) vectors s
Randomized Algorithms (600.464/664)
Assignment 3: Solutions
P.C. Shyamshankar
Due: October 4, 2012
1.
For the Hamming Center Problem, after solving the LP relaxiation, if we deterministically round each xi to 0 if xi < 0.5 and
to 1 if xi 0.5, prove that t
Randomized Algorithms (600.464/664) Assignment 2: Solutions
P.C. Shyamshankar
Due: September 20
1. (a) Here, X, Y cfw_0, 1, 2. The joint distribution of X and Y is:
X \Y
0
1
2
0
1
2
1/4
1/4
1/16
1/4
1/8
1/16
0
0
0
(1)
(b) X and Y are not independent, sinc
s
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