Question

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 (example: C(i,j) is the cost of multiplying matrices Ai...Aj) you are using, state the Recurrence 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.

Answer 1

Let us assume that A[i] => expected maximum profit This is obtained by opening some locations for I then A[1]-pi wherej -> max value dj S di - x this meanse that the location is closest to previous location let this be some x miles away If di < x then A[i] = A[i-1] This means that if we get a maximum profit then there is no requirement to open in another location which means max profit = A[i-1] otherwise the profit will be pi +AU] which indicates that there is no probability to open shops at any other locations now binary search is performed to find the time complexity ofj O(log n) similarly A[i]= 0(log n) A[n] = 0 (n log n) Which can be reduced to O (n)