Given a set of ranges S, and an overlapping range R, find the smallest subset in S that encompases R

Question

The following is a practice interview question that was given to me by someone, and I\'m not sure what the best solution to this is:

Given a set of ranges

Answer 1

How about using an interval tree for queries? (https://en.m.wikipedia.org/wiki/Interval_tree) I'm not sure if greedy could work here or not. If we look at the last set of choices, overlapping with the high point in R, there's a possibility of overlap between the earlier choices for each one of those, for example:

R = (2,10) and we have (8,10) and (7,10) both overlapping with (6,8)

In that case, we only need to store one value for (6,8) as a second leg of the path; and visiting (6,8) again as we make longer paths towards the low point in R would be superfluous since we already know (6,8) was visited with a lower leg count. So your idea of eliminating intervals as we go makes sense. Could something like this work?

leg = 1
start with the possible end (or beginning) intervals
label these intervals with leg
until end of path is reached:
  remove the intervals labeled leg from the tree
  for each of those intervals labeled leg:
    list overlapping intervals in the chosen direction
  leg = leg + 1
  label the listed overlapping intervals with leg