algorithm

Q1) Show, by applying the limit test, that each of the following is true. a) The functions f(n)= n(n-1)/2 and g(n)= n^2 grow asymptotically at equal rate. b) The functions f(n)=log n grow asymptotically at slower rate than g(n)=n. Q2) Show that log (n!) = Θ (nlog n); Q3) Design an algorithm that uses comparisons to select the largest and the second largest of n elements. Find the time complexity of your algroithm (expressed using the big-O notation). Q4) Given an a binary array or list of n elements, where each element is either a 0 or 1, we would like to arrange the elements so that all of those that are equal to 0’s appear first followed by all the elements that are equal to 1’s. a) Write an algroithm or a function that uses comparisons to arrange the elements as given above. Do not use any extra arrays in your algorithm. b) Find the time, T(n), needed by your algorithm in the worst-case and then express it using the big-O notation. c) Find the time, T(n), needed by your algorithm in the best-case and then express it using the big-Ω notation. d) Find the time, T(n), needed by your algorithm in the average-case and express it using the big-Θ notation.