Problem 1. (a) Describe an algorithm that sorts an input array A[1... n] by calling a subroutine SQRTSORT (k), which sorts the subarray A[k+1...k+ vn] in place, given an arbitrary integer k between 0 and n - Vn as input. (To simplify the problem, assume that Vn is an integer.) Your algorithm is only allowed to inspect or modify the input array by calling SQRISORT; in particular, your algorithm must not directly compare, move or copy array elements. How many times does your algorithm call SQRTSORT in the worst case? Give pseudocode. Remem- ber that you cannot use anything other than calling the SQRTSORT routine and maybe some loops. No formal proof of correctness is mandatory. But write a few sentences justifying your approach. Prove that your algorithm from part (a) is optimal up to constant factors. In (b) other words, if f(n) is the number of times your algorithm calls SQRTSORT, prove that no algorithm can sort using o(f (n)) calls to SQRTSORT. Note that here we are assuming that these algorithms cannot do anything other than calling SQRTSORT repeatedly. (Hint: Think of a typical worst case example for any sorting algorithm) (c) Now suppose SQRTSORT is implemented recursively, by calling your sorting algo- rithm from part (a). For example, at the second level of recursion, the algorithm is sorting arrays roughly of size n/4. What is the worst-case running time of the resulting sorting algorithm? (To simplify the analysis, assume that the array size n has the form 22"), so that repeated square roots are always integers.)

Coding in Python, we have the function to perform this task, V col_Hdeg whichinputs the adjacency matrix A of the graph and initializes the array vert_color s tohold vertex colors. It then calculates the degree of each vertex and stores it in thearray degrees.The while loop iterates until all vertices are colored and at each iteration, thevertex v with the highest degree is colored. The array temp is used to find the currentcolors of the neighbors of vertex v and an unused smallest color is determined aftersorting this array.

Given two sorted arrays, write a function to merge them into a single sorted array. What is the time complexity of your solution?

Write a loop in assembly that increments each element of array x as defined in the previous problem. The size of the array (6) may be hard-coded. Previous Problem array: x dq 43,12,13,53,-1,21