Design a dynamic programming algorithm for the Longest Alternating Subsequence problem described below: Input: A sequence of n integers Output: The length of the longest subsequence where the numbers alternate between being larger and smaller than their predecessor The algorithm must take O(n²) time. You must also write and explain the recurrence. Example 1: Input: [3, 5, 4, 1, 3, 6, 5, 7, 3, 4] Output: 8 ([3, 5, 4, 6, 5, 7, 3, 4]) Example 2: Input: [4,7,2,5,8, 3, 8, 0, 4, 7, 8] Output: 8 ([4, 7, 2, 5, 3, 8, 0,4]) (Take your time with this for the subproblem for this one)

(Program) Write a program that tests the effectiveness of the rand() library function. Start by initializing 10 counters, such as zerocount, onecount, twocount, and so forth, to 0. Then generate a large number of pseudorandom integers between 0 and 9. Each time 0 occurs, increment zerocount; when 1 occurs, increment onecount; and so on. Finally, display the number of 0s, 1s, 2s, and so on that occurred and the percentage of time they occurred.

The implementation of a queue in an array, as given in this chapter, uses the variable count to determine whether the queue is empty or full. You can also use the variable count to return the number of elements in the queue. On the other hand, class linkedQueueType does not use such a variable to keep track of the number of elements in the queue. Redefine the class linkedQueueType by adding the variable count to keep track of the number of elements in the queue. Modify the definitions of the functions addQueue and deleteQueue as necessary. Add the function queueCount to return the number of elements in the queue. Also, write a program to test various operations of the class you defined.

(Numerical) Write a program that tests the effectiveness of the rand() library function. Start by initializing 10 counters to 0, and then generate a large number of pseudorandom integers between 0 and 9. Each time a 0 occurs, increment the variable you have designated as the zero counter; when a 1 occurs, increment the counter variable that’s keeping count of the 1s that occur; and so on. Finally, display the number of 0s, 1s, 2s, and so on that occurred and the percentage of the time they occurred.