Answered: Randomized Algorithm - Basics (4 Questions in 1)

Tue, 21 Nov 2023 06:06:00 +0000

Option a: exit()

The given code runs a loop 'n' times, similar to flipping a coin at each iteration. If the coin shows heads (say, random() > 0.5), the program stops. In the worst case, it takes O(n) time, but it might finish earlier if the loop exits before 'n' iterations.

Option b: i = i + 1

The actual time complexity may vary due to the random(). In some cases, it might terminate early, and in others, it might run for the full of n iterations. On average, the loop will iterate approximately n/2 times because the random() function that generates values uniformly in the 0 to 1 range, and there's a probability of 0.5 that the condition will be true. Therefore, the expected time complexity is proportional to O(n).

Option c: i = i - 1

Similar to above condition(b) the average time complexity of this could be O(n). However, in some cases this may lead to an infinite loop if the condition consistently leads to i being decremented.

Option d: i = 2 * i

In the worst case, where the loop condition always evaluates to true due to the randomness of `random()`, the loop may iterate n times, resulting in a time complexity of (O(n)).

On average, the loop might double the variable i with iteration, and the loop will execute until 2^k>= n, where k is the number of iterations. On solving k = log₂n, yielding an average time complexity of O(log n).

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Answered: Randomized Algorithm - Basics (4 Questions in 1)