Algorithm to find the maximum sum in a sequence of overlapping intervals

Question

The problem I am trying to solve has a list of intervals on the number line, each with a pre-defined score. I need to return the maximum possible total score.

The c

Answer 1

I thought of this a bit and came up with something.

Interval Trees provide an efficient way of finding all the intervals that overlap a given interval. Walking through the entire set of intervals, we can find all the overlapping intervals for a given one. Once we have these, we can find the interval with the highest score, store it and move on.

Building the tree takes O(N Log N) time and lookup takes O(Log N) time. Because we do a lookup for all elements, the solution becomes O(N Log N).

However, if we face something like the example above where the highest score interval in one group reduces the total, the algorithm fails because we have no way of knowing that the highest score interval should not be used before hand. The obvious way around this would be to calculate both (or all) totals in case we are not sure, but that puts us back to a potentially O(N^2) or worse solution.