A fundamental problem in machine learning and data analytics is to partition or classify

objects in a $$natural$$ way based on their $$distances$$ to each other. In this question, you

are given an integer N$,$ and n points $1,2,...,$ in a horizontal line. We want to

partition the n points into N groups so that intuitively within each group the points are

$$close$$ together. The following is an example of what may be a good way of partitioning

the following n$=8$ points into N$=3$ groups, and what may be a bad way:

More precisely, each point $$

is a number on the x$-$axis. To measure how $$good$$ a