(a) For each of the following recurrences, give an expression for theruntime T (n) if the recurrence can be solved with the Master Theorem.Otherwise, indicate that the Master Theorem does not apply.(i) T (n) = T (n/2) + T(n/2) + T(n/2) + n2(ii) T (n) = 0.5T (n/2)+ 1/n(iii) T (n) = 3T (n/3) + n(iv) T (n) = 4T (n/2) + nlogn

• Simplify the Boolean functions by means of the tabulation matching method: a. F(w,x,y,z) = Σm(2,3,12,13,14,15) b. P(m,n,o,p,q,r) =Em(6,9,13,18,19,25,27,29,41,45,57,61) c. P(A,B,C,D,E,F,G) =Σm(20,28,38,39,52,60,102,103)

4. Consider the recurrence: T(n) = T(n/2) + T(n/4) + n, T(m) = 1 for m <= 5. Use the substitution method to give a tight upper bound on the solution to the recurrence using O-notation.

how to use a histrogram to estimate the the size of selection of the form σA<=γ(r)?

Let ap = 1, an = an-1 + 3 be the recurrence relation of algorithm A and T(1) = 1, T(n) = T(n/5)+n be the recurrence relation of algorithm B. 1. Find the solution for both relations. Use any method.

analyze the running time, getting that M(n) = Ω(n log(n)) recurrence relation

Determine whether or not the function f : Z × Z ! Z is onto, if f((m, n)) =m-n.

Appendix A 10-Fold Cross Validation for Parameter Selection Cross Validation is the standard method for evaluation in empirical machine learning. It can also be used for parameter selection if we make sure to use the training set only. To select parameter A of algorithm A(X) over an enumerated range d E [A1,..., A] using dataset D, we do the following: 1. Split the data D into 10 disjoint folds. 2. For each value of A e (A1,..., Ar]: (a) For i = 1 to 10 Train A(A) on all folds but ith fold Test on ith fold and record the error on fold i (b) Compute the average performance of A on the 10 folds. 3. Pick the value of A with the best average performance Now, in the above, D only includes the training data and the parameter A is chosen without the knowledge of the test data. We then re-train on the entire train set D using the chosen A and evaluate the result on the test set.

Reduce the following function using k-map technique, F(A, B, C, D) = π(0, 2, 3, 8, 9, 12, 13, 15).