Use a map container to store the minimum length of any range whose endpoint is on or before the key value.
In other words, $$$map[i]$$$ will denote the minimum length of a range with it's endpoint less than or equal to $$$i$$$. This map can be computed in $$$O(nlogn)$$$ time.
Now for every range $$$[l_i,r_i]$$$, find the key value less than $$$l_i$$$ in the map and pair up the minimum length obtained so with the length of the range number $$$i$$$. Just like you are searching for the best companion to this current range. Note that you don't need to search for ranges ahead of $$$r_i$$$ as they will ultimately be taken into consideration.

And your answer is the minimum sum you obtain while doing so for all ranges from $$$1$$$ to $$$n$$$.