How can I randomly choose M points from N-dimensional hypercube ?

I am unable to understand the O(n log n) algorithm. I've found many examples, but there are too difficult for me. Could somebody explain it in simple words ? I saw that a lot of users use lower_bound, but I don't see why it works.

I know that there'll be answers like google it etc, but I've already spent many hours and I didn't find anything useful.

Let's say I have boundaries left and right, which are doubles and I want to do binary search. How long should I iterate to be sure that given error isn't too big.

For example, in the last contest there was a problem http://codeforces.com/contest/579/problem/E

I saw some codes and authors use 100, 200 etc iterations