Given a sorted array of integers, write a program to implement the Binary Search Algorithm and find the index of a target element in the array. If the target element is not found, return -1.  a) Explain the working principles and steps involved in the Binary Search Algorithm. b) Implement the Binary Search Algorithm in a programming language of your choice. c) Analyze the time complexity of the Binary Search Algorithm. d) Provide a test case to demonstrate the correctness of your implementation.

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Randomized Select Algorithm

A randomized algorithm is defined as one that uses a certain amount of randomization in its logic or operation. It is said to be an algorithm that depends on the random number to perform its operation. It is used for reducing the overall time complexity, space complexity in the standard algorithm. The randomized algorithm uses the concept of random numbers to find the solutions to the different kinds of problems and also to improve the problem solutions.