Removal of billboards from given ones

Question

I came across this question

ADZEN is a very popular advertising firm in your city. In every road you can see their advertising billboards. Recentl

Answer 1

It's a typical DP problem. Lets say that P(n,k) is the maximum profit of having k billboards up to the position n on the road. Then you have following formula:

 P(n,k) = max(P(n-1,k), P(n-1,k-1) + C(n))
 P(i,0) = 0 for i = 0..n

Where c(n) is the profit from putting the nth billboard on the road. Using that formula to calculate P(n, k) bottom up you'll get the solution in O(nk) time.

I'll leave up to you to figure out why that formula holds.

edit

Dang, I misread the question.

It still is a DP problem, just the formula is different. Let's say that P(v,i) means the maximum profit at point v where last cluster of billboards has size i. Then P(v,i) can be described using following formulas:

P(v,i) = P(v-1,i-1) + C(v) if i > 0 
P(v,0) = max(P(v-1,i) for i = 0..min(k, v)) 
P(0,0) = 0

You need to find max(P(n,i) for i = 0..k)).