Consider the following algorithm for generating a uniform random permutation: PERMUTE-IN-CYCLES (A) n = A.length let B[1...n] be a new array offset = RANDOM(1, n) for index = 1 to n pos = index + offset if pos > n pos = pos - n B[pos] = A[index] return B a) (7 pts) Show that each element A[index] has a $\frac{1}{n}$ probability of ending up in any particular position in B. b) (3 pts) Explain why the resulting permutation is not uniformly random.

          Consider the following algorithm for generating a uniform random permutation:
PERMUTE-IN-CYCLES (A)
n = A.length
let B[1...n] be a new array
offset = RANDOM(1, n)
for index = 1 to n
pos = index + offset
if pos > n
pos = pos - n
B[pos] = A[index]
return B
a) (7 pts) Show that each element A[index] has a $\frac{1}{n}$ probability of ending up in any particular
position in B.
b) (3 pts) Explain why the resulting permutation is not uniformly random.

Consider the following algorithm for generating a uniform random permutation:
PERMUTE-IN-CYCLES (A)
n = A.length
let B[1...n] be a new array
offset = RANDOM(1, n)
for index = 1 to n
pos = index + offset
if pos > n
pos = pos - n
B[pos] = A[index]
return B
a) (7 pts) Show that each element A[index] has a (1)/(n) probability of ending up in any particular
position in B.
b) (3 pts) Explain why the resulting permutation is not uniformly random.

Added by Christian A.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Ct Turner

Step 1

The offset is a random number between 1 and n, generated by the RANDOM(1,n) function. Since the offset is randomly chosen for each iteration, each element A[index] has an equal chance of being assigned to any position in B. This is because the offset can take any Show more…

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Consider the following algorithm for generating a uniform random permutation: PERMUTE-IN-CYCLES(A) n = A.length let B[1...n] be a new array offset = RANDOM(1,n) for index = 1 to n pos = index + offset if pos > n pos = pos - n B[pos] = A[index] return B a) (7 pts) Show that each element A[index] has a probability of ending up in any particular position in B. b) (3 pts) Explain why the resulting permutation is not uniformly random.