P3611 [USACO17JAN] Cow Dance Show S

After several months of rehearsal, the cows are just about ready to put on their annual dance performance; this year they are performing the famous bovine ballet "Cowpelia". The only aspect of the show that remains to be determined is the size of the stage. A stage of size $K$ can support $K$ cows dancing simultaneously. The $N$ cows in the herd ($1 \leq N \leq 10,000$) are conveniently numbered $1 \ldots N$ in the order in which they must appear in the dance. Each cow $i$ plans to dance for a specific duration of time $d(i)$. Initially, cows $1 \ldots K$ appear on stage and start dancing. When the first of these cows completes her part, she leaves the stage and cow $K+1$ immediately starts dancing, and so on, so there are always $K$ cows dancing (until the end of the show, when we start to run out of cows). The show ends when the last cow completes her dancing part, at time $T$. Clearly, the larger the value of $K$, the smaller the value of $T$. Since the show cannot last too long, you are given as input an upper bound $T_{max}$ specifying the largest possible value of $T$. Subject to this constraint, please determine the smallest possible value of $K$.

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