In a certain island of the Caribbean there are N cities, numbered from 1 to N. For each ordered pair of cities (u, v) you know the cost c[u][v] > 0 of flying directly from u to v. In particular, there is a flight between every pair of cities. Each such flight takes one day and flight costs are not necessarily symmetric.

Suppose you are in city u and you want to get to city v. You would like to use this opportunity to obtain frequent flyer status. In order to get the status, you have to travel on at least minDays consecutive days. What is the minimum total cost c(u, v) of a flight schedule that gets you from u to v in at least minDays days?

Design a dynamic programming algorithm to solve this problem. Assume you can access c[x][y] for any pair x, y in constant time. You are also given N, u, v and minDays N.

Hint: one way to solve this problem is using dynamic states similar to those on Bellman-Fords algorithm.

Please answer the following parts:

Define the entries of your table in words. E.g. T(i) or T(i, j) is ...

State a recurrence for the entries of your table in terms of smaller subproblems. Don't forget your base case(s).

Write pseudocode for your algorithm to solve this problem.

State and analyze the running time of your algorithm.

Faster (and correct) solutions are worth more credit.