You are considering opening a series of ice cream shops along a long, straight beach. There are n potential locations where you could open a shop, and the distance of these locations from the start of the beach are, in meters and in increasing order, d₁, d₂, d₃, ...., d_n; you may assume all of the d_i are integers. The constraints are as follows:

At each location you may open at most one ice cream shop.

The expected profit from opening an ice cream shop at location i is p_i , where p_i > 0 and i = 1, 2, 3, ..., n.

Any two ice cream shops should be at least x meters apart, where x is a positive integer. Give an efficient algorithm to compute the maximum expected total profit subject to these constraints.

For this question, please define any subproblems and definitions you are using, state the Recurrence Relation for the entries including all base cases (and explain why it's a correct recurrence relation), give the pseudocode for the alg, and analyze its Running time.