Also interested

I think it's a good idea to put the problem link.

WLOG assume that $$$A_j < B_j$$$. Cost of walking is annoying so we can get rid of it by setting $$$[L_i, R_i]$$$ to $$$[L_i-i, R_i-i-1]$$$.

Now at each point we can either increase time or decrease it by paying $$$1$$$. Brute force solution would be trivial: don't change time for as long as possible.

Create a segment tree which will be able to answer the following query: Given a starting time $$$t$$$, find ending time and total cost to walk on a segment $$$[l, r]$$$.

We can get our answer for any $$$t$$$ by setting $$$t=L_l$$$ or $$$t=R_l$$$ depending on the direction our first move in time would be when starting at $$$t$$$.

All we need is a segment tree that for each node stores answer for the $$$2$$$ starting cases.

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