How to minimize the average distance between pumps and cities?

Question

There are n cities [1, 2, 3, .... n] and k available pumps. These pumps can be installed in k cities. The cities are all in a straight line.

How do I need to install these so that the average distance between the cities and the nearest pumps is minimized?

What I am thinking is that dividing is this:

Suppose there are 6 cities [C1, C2, C3, C4, C5, C6] and 3 pumps [P1, P2, P3].

I divide the array of cities equally -> [C1, C2] , [C2, C3] .. and then each block gets one pump.

Then again break that [C1,C2] and let C2 have the pump. Same way C4 and C6.

So basically recursively calling and assigning the city a pump. Is this the right way to approach this problem?

Answer 1

Minimizing the average distance is equivalent to minimizing the sum of the distances, so let me work with that formulation of the problem.

Hint #1: When $k = 1$, the pump should be located at the median of the positions of the cities.

Hint #2: When $k > 1$, you can use a dynamic programming approach. Let $f(i, j)$ be the optimal cost of serving cities $1,...,i$ with $j$ pumps. Your goal is to compute $f(n, k)$. Now observe that $$ f(n,k) = \min_{i=1,...,n} \left( \sum_{i': i < i' \le n} d(\text{city}_{i'}, \text{pump}_k) + f(i,k-1) \right), $$ because if the rightmost pump (pump number $k$) serves cities $i+1$ through $n$, then the total cost is the sum of distances of those cities to pump $k$, plus the optimal cost of serving cities $1$ through $i$ with $k-1$ pumps. Here $\text{pump}_k$ is the position at the median of cities $i+1,...,n$.