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I have an algorithmic problem in which I have a highway that is a straight line of length $n$ and a set of unique respective costs for construction of a radio tower for each mile on the highway. I am trying to use dynamic programming to find a way to minimize the cost of ensuring that each point in the highway is covered by at least one tower. I am given a length $n$ for the highway, an array of costs of length $n$ + 1, and must assume that the towers all will have a 5 mile range (eg a tower built at point $i$ will have range [$i$-5,$i$+5] inclusive).

I believe that my subproblem is OPT($i$) where $i$ is the minimum cost to cover the first $i$ miles of highway, however am having trouble constructing my recursive formula or algorithm. I know that a greedy approach would not be optimal in this case. I think that perhaps the shortest path problem could be harnessed however am having trouble creating an answer that follows the steps of dynamic programming. So far this is my algorithm:

  1. Initialize an array C of length $n$+1, where C[$i$] represents the minimum cost to cover the first $i$ miles of the highway.
  2. Set C[0] = 0 and C[1] = cost[1].
  3. For $i$ = 2 to $n$, calculate C[$i$] using the recurrence relation: C[$i$] = min(cost[$i$] + C[$i$-5], C[$i$-1])
  4. Return C[$n$], which represents the minimum cost to cover the entire highway.

Any advice would be appreciated!

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    $\begingroup$ I find SE to work best with an explicit, answerable question near the end of a question post. There's How do I ask a Good Question? $\endgroup$
    – greybeard
    Dec 1, 2023 at 11:31

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