Professor Olay is consulting for an oil company, which is planning a large pipeline running east to west through an oil fields of $n$ wells. The company wants to connect a spur pipeline from each well directly to the main pipeline along a shortest route (either north or south), as shown on Figure 9.2. Given the $x$- and $y$- coordinates of the wells, how should the professor pick the optimal location of the main pipeline, which would be the one that minimizes the total length of the spurs? Show how to determine the optimal location in linear time.
We just find the median of the $y$ coordinates. The $x$ coordinates are irrelevant.
Let's assume that $n$ is odd. There are $\lfloor n / 2 \rfloor$ south of the median and the same amount of wells north of the median. Let the pipeline pass through the median. We shall reason about why this location is optimal.
Suppose we move the pipeline one meter north. This reduces the total pipe length with $\lfloor n/2 \rfloor$ meters for the pipes north of the median, but adds another $\lceil n/2 \rceil$ for the pipes south of median, including the median itself. The more we move north, the more the total pipe length will increase. The same reasoning holds if we move the main pipeline south.
If $n$ is even, then any location between the lower and upper median is optimal.