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Each of Farmer John's N cows (1 <= N <= 1,000) produces milk at a different positive rate, and FJ would like to order his cows according to these rates from the fastest milk producer to the slowest.
FJ has already compared the milk output rate for M (1 <= M <= 10,000) pairs of cows. He wants to make a list of C additional pairs of cows such that, if he now compares those C pairs, he will definitely be able to deduce the correct ordering of all N cows. Please help him determine the minimum value of C for which such a list is possible.
5 5 2 1 1 5 2 3 1 4 3 4
FJ is comparing 5 cows and has already determined that cow 2 > cow 1, cow 1 > cow 5, cow 2 > cow 3, cow 1 > cow 4, and cow 3 > cow 4 (where the '>' notation means "produces milk more quickly").
From the information in the 5 test results, Farmer John knows that since cow 2 > cow 1 > cow 5 and cow 2 > cow 3 > cow 4, cow 2 has the highest rank. However, he needs to know whether cow 1 > cow 3 to determine the cow with the second highest rank. Also, he will need one more question to determine the ordering between cow 4 and cow 5. After that, he will need to know if cow 5 > cow 3 if cow 1 has higher rank than cow 3. He will have to ask three questions in order to be sure he has the rankings: "Is cow 1 > cow 3? Is cow 4 > cow 5? Is cow 5 > cow 3?"