Lighthouses

There is a straight line with $N$ distant points marked $1, 2, \ldots, N$. The point $i$ is exactly $1$ unit distant along the positive axis from $i - 1$. Some of the points have lighthouses. You are given a binary string $S$ where $S_i = 1$ means that the $i$-th point has a lighthouse, and $S_i = 0$ otherwise. Each of the lighthouses can be set to have any arbitrary non-negative integer power $P