P3662 [USACO17FEB] Why Did the Cow Cross the Road II S

The long road through Farmer John's farm has $N$ crosswalks across it, conveniently numbered $1 \ldots N$ ($1 \leq N \leq 100,000$). To allow cows to cross at these crosswalks, FJ installs electric crossing signals, which light up with a green cow icon when it is ok for the cow to cross, and red otherwise. Unfortunately, a large electrical storm has damaged some of his signals. Given a list of the damaged signals, please compute the minimum number of signals that FJ needs to repair in order for there to exist some contiguous block of at least $K$ working signals.

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