Modify the closest pair of points algorithm so that the separating line L now separates the first n/4 points (sorted according to their r coordinates) from the remaining 3n/4 points. Write the recurrence relation that gives the running time of the modified algorithm. Is the running time of your algorithm still O(nlog n)? Specify the best asymptotic running time you can get for your algorithm and briefly justify.

Now let the line L separate the first √ n points (according to their x- coordinates) from the remaining n - √ n points. Write the recurrence relation that gives the running time of this modification of the algorithm. Is the running time of your algorithm still O(n log n)? If your answer is yes, provide a brief justification; if your answer is no, provide a (asymptotic) lower bound on the running time of the modified algorithm that should be enough to justify your answer.

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Write a program that uses a random number generator to generate a two digit positive integer and allows the user to perform one or more of the following operations: a. double the number. b. reverse the digits of the number. c. raise the number to the power of 2, 3, or 4. d. sum the digits of the number. e. if the number is a two-digit number, then raise the first digit to the power of the second digit. f. if the number is a three-digit number and the last digit is less than or equal to 4, then raise the first two digits to the power of the last digit. after performing an operation if the number is less than 10, add 10 to the number. also, after each operation determine if the number is prime. each successive operation should be performed on the number generated by the last operation. your program should not contain any global variables and each of these operations must be implemented by a separate function. also, your program should be menu driven. 7. (fraction calculator) write a program that