Question

I am looking for an algorithm I could use to solve this, not the code. I wondered about using linear programming with relaxation, but maybe there are more efficient ways for solving this?

The problem

I have set of intervals with weights. Intervals can overlap. I need to find maximal sum of weights of disjunctive intervals subset.

Example

Intervals with weights :

|--3--|          |---1-----|
    |----2--| |----5----|

Answer: 8

Answer 1

I have an exact O(nlog n) DP algorithm in mind. Since this is homework, here is a clue:

Sort the intervals by right edge position as Saeed suggests, then number them up from 1. Define f(i) to be the highest weight attainable by using only intervals that do not extend to the right of interval i’s right edge.

EDIT: Clue 2: Calculate each f(i) in increasing order of i. Keep in mind that each interval will either be present or absent. To calculate the score for the “present” case, you’ll need to hunt for the “rightmost” interval that is compatible with interval i, which will require a binary search through the solutions you’ve already computed.

That was a biggie, not sure I can give more clues without totally spelling it out 😉