Question

Dynamic Programming Algorithm (20 points) You are planning to establish a chain of electric vehicle charging stations along the Pacific Coast Highway. There are n potential locations along the highway, and the distance between the starting point and location k is dk ≥ 0, where k = 1, 2, . . . , n. It is assumed that di < dk for all i < k. These are important constraints: • At each location k you can open only one charging station with the expected profit pk. • You must open at least one charging station along the whole highway. • Any two stations should be at least M miles apart. Design a dynamic programming algorithm to maximize the profit by following these steps: a) Define (in plain English) the sub-problems to be solved (5 points). b) Write a recurrence relation for the sub-problems. Make sure you specify base cases. (7 points). c) Using the recurrence formula in part b, write an iterative pseudocode to find the solution. (5 points). d) What is the complexity of your solution? Explain your answer. (3 points)

Answer 1

To maximize the profit of establishing electric vehicle charging stations along the Pacific Coast Highway, a dynamic programming algorithm can be designed. The sub-problems involve finding the maximum profit at each location, considering constraints such as having at least one station along the whole highway and a minimum distance between stations. The solution involves defining sub-problems, writing recurrence relations and iteratively solving them.

Step-by-step explanation:

In this problem, we need to design a dynamic programming algorithm to maximize the profit of establishing electric vehicle charging stations along the Pacific Coast Highway. Here are the steps:

a) The sub-problems to be solved can be defined as finding the maximum profit of opening charging stations at each location, considering the constraint of having at least one station along the whole highway and a minimum distance of M miles between any two stations.

b) The recurrence relation for the sub-problems can be defined as follows:

Let dp[i] represent the maximum profit at location i. We can calculate it using the following formula:

dp[i] = max(dp[j] + pk), for all j where dk - dj >= M

Base case: dp[1] = p1

c) Here is the iterative pseudocode to find the solution:

dp[1] = p1
define dp as an array of size n
for i from 2 to n:
max_profit = -inf
for j from 1 to i-1:
if dk - dj >= M:
max_profit = max(max_profit, dp[j] + pk)
dp[i] = max_profit
return dp[n]

d) The complexity of this solution is O(n^2), where n is the number of potential locations. This is because we have nested loops to iterate over the locations and check the distances between them.