Suppose you are given a very long array A, whose first n elements are integers (in arbitrary order) and the remaining elements are the special symbol 1. You can access any position I in A by referring to A[i].  However, you do not known.  Give a n algorithm for determining n. Analyze the number of accesses to A made by your algorithm ,in terms of n. For full credit, your algorithm must make O(logn) accesses to A

There’s a length n (n ≥ 3) array and we’d like to fill in each element of the array with an integer.The integers should be in {1, 2, . . . , k} and any 3 adjacency entries cannot have the same number. Given nand k , design a polynomial time algorithm to compute the number of ways to fill this array. Please alsojustify the correctness and the time complexity of your answers.

Show how quicksort can be made to run in O(nlogn) time in the worst case. Assume the input array is A[0:n-1] and all elements in A are distinct. Write your answer as pseudo-code and use plain language to explain the idea of your algorithm. (Hint: you can use any algorithm helper functions in parts of the designed algorithm. If you use an algorithm, you can directly call the function name in the pseudo-code without expanding the details of the helper function, as long as you clearly explain the meaning of the helper function.)

Question 1. The input is an array a[1],. ,..., a[n] of size n containing natural numbers between 1 and n. Describe an O(n) algorithm that determines if the array contains 2 consecutive integers.