1s=10^6 μs 1m = 60 x 1s log n Vn nlogn n² 2. For each function f(n) and time t in the following table, determine the largest size n of a problem that can be solved in time t, assuming that the algorithm to solve the problem takes f(n) microseconds. n³ 2n n! 1hr 60 x 1m 1 day = 24 x 1hr 1 second 1 minute 1 hour 1 month= 28 x 1 day 1 day 1 month 1 year = 12 x 1 month 1 year 1 century = 100 x 1 year 1 century

The travel time function of an algorithm has the form: f(n) = 3n^2 + 4n + 8  Prove that Big Oh is n^2 which is denoted by f(n) = O(n^2);

You are going on holiday. There are n cities in the world. There are also m airlines, the ith of which operates flights in both directions between cities ai and bi. You live in city 1, and you would like to travel to city n and back. You don’t mind stopovers in other cities, but you don’t want to have more than one stopover in any city.Design an algorithm which runs in time polynomial in n and m and determines whether it is possible to travel from city 1 to city n and back to city 1 without going through any other city more than once.

5. You are given a set of n positive numbers A = {a₁,..., an} and a positive integer t. Design a dynamic programming algorithm running in O(nt) time that decides whether there exists a subset A' CA such that Σ x = t. Note that each element of A can be xЄA' used at most once. Is the run-time of your algorithm polynomial with respect to the size of the input?

Given a program designed to calculate rocket trajectories, the algorithm has an O(2n) complexity and takes 1 second for each iteration, and n = 15. How many hours will it take to calculate the trajectories, worst case?

Below is pseudocode representing the Compute-Opt algorithm: Compute-Opt(j):    If j = 0 then        Return 0    Else        Return max(vj+Compute-Opt(p(j)), Compute-Opt(j − 1))    Endif Memoization of this algorithm does what to its running time? Group of answer choices It reduces the running time from exponential to linear. It reduces the running time from O(n2) to O(n). It reduces the running time from exponential to O(n log n). It doesn't significantly reduce the running time, but it enables the algorithm to utilize less space.

Consider array A of n numbers. We want to design a dynamic programming algorithm to find the maximum sum of consecutive numbers with at least one number. Clearly, if all numbers are positive, the maximum will be the sum of all the numbers. On the other hand, if all of them are negative, the maximum will be the largest negative number. The complexity of your dynamic programming algorithm must be O(n2). However, the running time of the most efficient algorithm is O(n). Design the most efficient algorithm you can and answer the following questions based on it. To get the full points you should design the O(n) algorithm. However, if you cannot do that, still answer the following questions based on your algorithm and you will get partial credit. Write the recursion that computes the optimal solution recursively based on the solu- (a) tion(s) to subproblem(s). Briefly explain how it computes the solution. Do not forget the base case(s) of your recursion.

please answer all parts clearly with all work, thanks

Hello, hope you're having a great day. Can you please help me do exercise 1, 2 and 3 please? Thank you!

Consider the following recursive algorithm, where // denotes integer division: 3//2 = 1, 5//2 = 2, etc. F(n):if n <= 1:    returnF(n//2)for i from 0 to n    for j from 0 to n//2        print(i+j)    Let function T(n) denote the running time of this algorithm. Derive T(n) and prove its worst case timecomplexity