Briefly describe the subset-sum problem. Assuming the target value t is expressed in unary, is it possible to solve the problem in polynomial time? If yes, provide a formal proof. If not, explain why not.

Please no written by hand solution Please give a problem (example) in parameter learning (Bayesian) and solve it.

Discuss how you might extend the Star-Cabing algorithm to compute iceberg cubes where the iceberg condition tests for an avg that is no bigger than some value, e

I have been confused for a while on worst case, best case, and average case for algorithms. I get that big O noation is worst case, Omega notation is best case, and theta is average notation, but I don't get how are we supposed to be able to tell what the function (ex. (n), (log n), ( n log n), etc.) is for those algorithms and why are they those functions? I got an AI generated response earlier, which was helpful in pointing out what kinds of algorithms might have (n) or (log n), but what I really want to know is how can I tell if a function is going to specifically be (log n) vs (n log n), (n), etc., why is it that function and which functions (log n) (n), (n log n), are better or faster than others.

LOWER BOUNDS BY THE METHOD OF ADVERSARIES. We have n numbers that take only at most k ≥ 2 different values (as in exercise 2 above). We are interested in a lower bound for an algorithm that groups these numbers by value when we can only use an identity oracle, i.e., an oracle that decides whether two numbers x and y are equal or not. Convince yourself that there is an obvious O(kn) algorithm for finding the groups. Using the method of adversaries, show that (kn) is a lower bound.

If you are given a set S of integers and a number t, prove that this issue falls into the NP class. Is there a subset of S where the total number of items is t? Note: Complexity in Data Structures and Algorithms

Considering the problem of performing rapid COVID test among a large group of people. Assuming there are exactly 1 positive case among N people, and you are asked to find the positive case with the least number tests. 1. A straight forward approach is to test every single person, which requires N tests in total. Can you design a testing framework that works better than the greedy one under, even under the worst-case scenario? Write the pseudo code for your approach and briefly explain the time complexity of your approach. 2. Based on your framework, if there are 1,000 people, what are the maximum number of tests that need to be performed in order to identify the positive case (number of tests required in the worst case)?

Prove, by induction on S, that for any finite set S, P(S)] = 21s, explaining each step in plain words, briefly. First, what are you doing the induction on? Write each part out completely. (This is such a standard proof exercise that you can find it on the Web, hence the requirement to add your own words.)

2. Recurrences (Master Method). Solve each recurrence below using the Master Method and write your answer using e-notation. Make sure that you show all your work including the corresponding case and the values of e or k used. If it is not possible to solve a recurrence using the Master Method, prove it by showing that the form is inapplicable or by showing that all 3 cases cannot be satisfied. (a) T(n)=37(n/2) + n² (b) T(n)=25T(n/5)+n (e) T(n)= T(3n/10)+n (d) T(n)= n³T(n) + 2n (e) T(n)= 3T(n/3) +n¹/3 (f) T(n)=9T(n/3) + n² lg n (g) T(n)=9T(n/3) + nlg²n (h) T(n)= T(n/3)+27 (n/5)+n (i) Bonus T(n) = 27 (√n) +logn