Question

Turn a Las Vegas algorithm to a Monte Carlo algorithm. Let A be a randomized algorithm for some decision problem P (e.g., to decide if a graph is connected or not). Suppose that for any given instance of P of size n, A runs in expected T(n) time and always outputs the correct answer. Use A to give a new randomized algorithm NEWALG for the problem P such that NEWALG always runs in 100T(n) time and • if the input is a 'Yes' instance¹, then it will be accepted with probability at least 0.999. • if the input is a 'No' instance, then it will always be rejected. Describe your algorithm and justify its correctness and running time. Hint: You may use Markov's inequality.

Name: turn a las vegas algorithm to a monte carlo algorithm let a be a randomized algorithm for some decision problem p eg to decide if a graph is connected or not suppose that for any given insta 48795
Uploaded: 2024-07-08T18:38:33-08:00
Duration: 3 min 32 s
Channel: Nick Johnson
Description: turn a las vegas algorithm to a monte carlo algorithm let a be a randomized algorithm for some decision problem p eg to decide if a graph is connected or not suppose that for any given insta 48795

          Turn a Las Vegas algorithm to a Monte Carlo algorithm.
Let A be a randomized algorithm for some decision problem P (e.g., to decide if a graph is connected or not).
Suppose that for any given instance of P of size n, A runs in expected T(n) time and always outputs the correct answer. Use A to give a new randomized algorithm NEWALG for the problem P such that NEWALG always runs in 100T(n) time and
• if the input is a 'Yes' instance¹, then it will be accepted with probability at least 0.999.
• if the input is a 'No' instance, then it will always be rejected.
Describe your algorithm and justify its correctness and running time.
Hint: You may use Markov's inequality.

turn a las vegas algorithm to a monte carlo algorithm let a be a randomized algorithm for some decision problem p eg to decide if a graph is connected or not suppose that for any given insta 48795

Added by Allen D.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Nick Johnson

07/08/2024

Step 1

The algorithm will run algorithm A repeatedly for at most 100 times. If A returns 'Yes' even once, NEWALG returns 'Yes'. Otherwise, if A returns 'No' in all 100 runs, NEWALG returns 'No'. Show more…

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Turn a Las Vegas algorithm to a Monte Carlo algorithm. Let A be a randomized algorithm for some decision problem P (e.g., to decide if a graph is connected or not). Suppose that for any given instance of P of size n, A runs in expected T(n) time and always outputs the correct answer. Use A to give a new randomized algorithm NEWALG for the problem P such that NEWALG always runs in 100T(n) time and • if the input is a 'Yes' instance¹, then it will be accepted with probability at least 0.999. • if the input is a 'No' instance, then it will always be rejected. Describe your algorithm and justify its correctness and running time. Hint: You may use Markov's inequality.