Randomized Algorithms u0026 Monte Carlo Quiz Quiz

Assess your understanding of randomized algorithms, the Monte Carlo method, expected outcomes, and error analysis. This quiz provides practical examples and core concepts for learners interested in probabilistic computing techniques.

1. Monte Carlo Method Application

   In simulating the value of π using the Monte Carlo method, why does increasing the number of random points lead to a more accurate estimate?

   1. Because the points are always located closer to the circle center
   2. Because the sample average converges to the expected value due to the law of large numbers
   3. Because more points decrease the randomness of outcomes
   4. Because the algorithm becomes deterministic

   Explanation: Increasing the number of random points in a Monte Carlo simulation utilizes the law of large numbers, helping the sample average approach the true expected value. This leads to a better estimate of π. More points do not decrease randomness (option B), the algorithm remains probabilistic rather than deterministic (option C), and the locations of points are uniformly distributed, not always closer to the circle center (option D).

2. Las Vegas vs. Monte Carlo Algorithms

   Which statement best distinguishes a Las Vegas algorithm from a Monte Carlo algorithm?

   1. A Las Vegas algorithm always produces a correct result, but its runtime varies
   2. A Las Vegas algorithm sometimes produces incorrect outputs
   3. A Las Vegas algorithm is only used for integer multiplication
   4. A Las Vegas algorithm always runs in constant time

   Explanation: Las Vegas algorithms are randomized algorithms that produce correct results with varying runtime, but never incorrect answers. Option B is wrong because Las Vegas algorithms do not always run in constant time. Option C incorrectly describes Monte Carlo algorithms, which may give wrong answers but with bounded runtime. Option D is unrelated and not a property of Las Vegas algorithms.

3. Randomized Quicksort Analysis

   When using a randomized pivot in Quicksort, how does this impact the algorithm's expected runtime on a sorted array as compared to using a fixed pivot?

   1. It guarantees the best case runtime in every run
   2. It increases space complexity to O(n^3)
   3. It reduces the expected runtime to O(n log n) instead of O(n^2)
   4. It makes the algorithm unstable

   Explanation: Randomly choosing the pivot in Quicksort avoids worst-case scenarios caused by sorted input, bringing the expected runtime down to O(n log n). Option B is wrong because space complexity remains O(log n) due to recursion. The algorithm does not become unstable as in option C. Option D is incorrect, as randomness ensures good expected performance but does not guarantee the best case every time.

4. Error Probability in Monte Carlo Algorithms

   Suppose a Monte Carlo algorithm has an individual error probability of 0.2. If the algorithm is run independently five times and outputs the majority result, what is the main benefit of this approach?

   1. It reduces the overall error probability compared to a single run
   2. It guarantees zero probability of error
   3. It increases the error probability to 1
   4. It makes each run deterministic

   Explanation: Running the Monte Carlo algorithm multiple times and using the majority output exploits the law of large numbers, statistically lowering the chance of an overall wrong answer. Option B is incorrect because the error is reduced, not eliminated. Option C is wrong as each run remains probabilistic. Option D is the opposite of the actual outcome.

5. Randomized Min-Cut Algorithm

   In the randomized algorithm for finding the minimum cut in a connected undirected graph, what is the main risk associated with the contraction step?

   1. It guarantees finding the maximum cut instead
   2. Contracting an edge in the minimum cut could merge important partitions and miss the optimal cut
   3. The algorithm could remove all vertices
   4. The contraction step takes exponential time

   Explanation: Contracting an edge from the minimum cut merges nodes that should stay in separate partitions, possibly causing the optimal cut to be missed. Option B is incorrect because the algorithm never eliminates all vertices. Option C is false as it does not find the maximum cut. Option D is also incorrect since the contraction operation is efficient, not exponential in time.