Farmer John has received a shipment of N large hay bales (1 \le N \le 4000) and placed them at various locations along the road leading to his barn. Unfortunately, he completely forgets that Bessie the cow is out grazing along the road, and she may now be trapped within the bales!

Each bale j has a size S_j and a distinct position P_j giving its location along the one-dimensional road. Bessie the cow starts at some location where there is no hay bale, and can move around freely along the road, even up to the position at which a bale is located, but she cannot cross through this position. As an exception, if she runs in the same direction for D units of distance, she builds up enough speed to break through and permanently eliminate any hay bale of size strictly less than D. Of course, after doing this, she might open up more space to allow her to make a run at other hay bales, eliminating them as well.

Bessie can escape to freedom if she can eventually break through either the leftmost or rightmost hay bale. Please compute the total area of the road consisting of real-valued starting positions from which Bessie cannot escape. For example, if Bessie cannot escape if she starts between hay bales at positions 1 and 5, then these encompass an area of size 4 from which she cannot escape.

[Problem credits: Brian Dean, 2015]

#3936 upload - done - how_to_reg -

Trapped in the Haybales timer 1s memory 256MB

#3936

Trapped in the Haybales 1s 256MB