Find non-overlapping ranges with largest total sum of lengths

Question

I search algorithm, that will help me in the realization of a problem! The problem is as follows:

There is a list of ranges, I need to build a list with a subset of non-overlapping ranges (not necessarily adjacent), such that the sum of their lengths is maximum possible.

For example, for an input list

[(-1, 3), (2, 4), (0, 5), (-4, -1)]

the desired output is

[(0, 5), (-4, -1)]

with the sum of lengths (5 - 0) + ((-1) - (-4)) = 5 + 3 = 8

What is 'a maximum value'? What value do you want to maximize? — CiaPan, Oct 18 '16 at 11:29
What sum? What is the 'sum' in example [(1,2), (4,6)]: is it 3=[2-1]+[6-4] or 5=[6-1]...? — CiaPan, Oct 18 '16 at 11:37
He meant the difference between the min/max values of an interval list (3 - (-4)) in this case — Treycos, Oct 18 '16 at 11:38
@Treycos In *this* case it was pretty ambiguous, because the intervals have a common end, so sum of the intervals lengths *equals* their total span. That's why I asked about another example, with a hole. — CiaPan, Oct 18 '16 at 11:53
@deniska369 Okay, so if you meant [(1,2), (4,6)] -> [2-1]+[6-4] = 3, then why is your example result [(-1, 3), (-4, -1)] with the sum of 7, instead of [(0, 5), (-4, -1)] with the sum of 8...?! — CiaPan, Oct 18 '16 at 11:56
Intervals do not necessarily need to have a common beginning or end. I just need to keep only those intervals that do not intersect and it is maximum the total length of the ranges. — deniska369, Oct 18 '16 at 11:59
@deniska369 I have edited your question a bit, please verify if the new version correctly reflects your needs. — CiaPan, Oct 18 '16 at 12:11
Similar problem: [Maximum sum of the range non-overlapping intervals in a list of Intervals](http://stackoverflow.com/questions/18262121/maximum-sum-of-the-range-non-overlapping-intervals-in-a-list-of-intervals) — CiaPan, Oct 20 '16 at 08:34

score 2 · Accepted Answer · answered Oct 18 '16 at 14:05

This is the maximum weighted independent set problem with the weights being equal to the length of the intervals.

This can be solved using dynamic programming. Let the intervals be sorted by starting times.

Then define DP[I_j] = maximum weighted set of intervals such that I_j is chosen and only intervals previous to I_j are considered. This means the intervals intersecting with I_j shall not be considered.

DP[I_j] = MAX(DP[I_r]) + Wt(I_j)

Where I_r are the intervals that occur before I_j.

The time complexity is O(n^2) where n is the number of intervals.

Find non-overlapping ranges with largest total sum of lengths

1 Answers1