Compare the (a) Linear search and (b) Binary search algorithms on an input array from which you are supposed to find a specific element. Consider three different cases for (a) (Best case, Average case, Worst case) Arrive at the computational complexity of (a) (all three cases) and (b) in terms of Big O(.) notation.

Game of 8 queens.Algorithmic solutions to the problems must be generated, with each of the following algorithm design techniques, in the Python language. Greedy Algorithms, Dynamic Programming, Bactracking, Branch and Bound.Calculate and analyze the complexity t(n) of each algorithm.Perform average comparison (minimum 3 executions with the same data pool) of time and memory consumption for each technique and algorithm.Given a chess board of size nxn, find the way(s) to place n queens, without any of them threatening each other.

Submit solutions to the following problems in a single pdf document preferably completed using LaTeX. It is likely that you will need to use programming (Python) or a matrix calculator to solve most of the problems. You do not need to submit the code, but you should include explanations of all your conclusions. 3. Consider the cat polygon with vertices at (0, 0), (3, 3), (3, 6), (2, 7), (1, 6), (-1, 6), (-2, 7), (-3, 6), and (-3, 3). a. Write a single matrix that will make the cat's face twice as tall, half as wide, and rotate anti- clockwise 45 degrees. What is the determinant of the transformation matrix? b. Calculate the area of the cat's face prior to transformation and after the transformation. How does the change in area relate to the determinant of the transformation matrix?

Consider the following functions as time complexities of some algorithms. (a) First write the worst-case runtime of each algorithm in Big-O notation. (b) Then arrange functions from low to high (Consider their Big-O notations as you arrange them; as n grows, the function that grows slower should come sooner than the one that grows faster). (c) Arbitrary, select TWO of arrangements in part (b) and for each one of them justify why you have chosen such an ordering for the corresponding function. Provide formal proof. For example if fl<=f2<= f3<= f4<=f5 you may choose to formally justify why (1) fl<=f2 and why (2) f4<=f5 . fI(n)=n², f2(n)=210 nlogn, f3(n)=3", f4(n)=n, f5(n)=1, f6(n)=5*logn, f7(n)= n", f8(n)=4 n*n, f9(n)=2log(n³)

given an array with repeating elements each element repeats thrice except one.find that one in O(1) space and O(n) time complexity in machine independent language.