by Nathan Pinsker

Let's start with a small case, to get a feel for the problem. If we have 4 cows with values 1, 2, 3, and 4, then we can either pair up cow 1 with cow 2, 3, or 4.

If we pair up cow 1 with cow 2, then milking will take 7 units of time total. If we pair up cow 1 with cow 3, milking will take 6 units. Finally, if we pair up cow 1 with cow 4, milking will take 5 units, which is the best we can do.

More generally, if we have 4 cows with values $A < B < C < D$, and we've paired off A with B and C with D, then it's always beneficial to swap B and D. This is because $min(A + D, B + C) < C + D$, since $A < C$ and $B < D$.

Similarly, if we've paired off A with C and B with D, then we should swap C and D. This is because $min(A + D, B + C) < B + D$ for similar reasons.

It follows directly that we should always pair the cow that takes the least amount of time with the cow that takes the most amount of time, and remove these two from the pool. We can then repeat this with the fastest and slowest cows to milk from the new set, and continue in this fashion until we have paired off all the cows.

One final wrinkle is that there can be a gigantic number of cows. To deal with this, we instead keep track of each possible (unique) time to milk each cow, as well as the number of such cows. If there are $A$ cows that take the minimum amount of time to milk and $B$ cows that take the maximum amount of time to milk, then we can pair off $min(A, B)$ cows with each other in a single step to make our algorithm more efficient. This guarantees that we eliminate either $A$ or $B$ cows, decreasing the number of unique values of milk output by one. The overall algorithm is thus $O(n)$.

Here's Brian Dean's solution:

#include <iostream>
#include <fstream>
#include <cmath>
#include <algorithm>
#include <vector>
using namespace std;

typedef pair<int,int> pii;
vector<pii> V;
int N;

int main(void)
{
ifstream fin ("pairup.in");
ofstream fout ("pairup.out");
fin >> N;
for (int i=0; i<N; i++) {
int x, y;
fin >> x >> y;
V.push_back(pii(y,x));
}
sort(V.begin(), V.end());
int M = 0, i=0, j=N-1;
while (i <= j) {
int x = min(V[i].second, V[j].second);
if (i==j) x /= 2;
M = max(M, V[i].first + V[j].first);
V[i].second -= x;
V[j].second -= x;
if (V[i].second == 0) i++;
if (V[j].second == 0) j--;
}
fout << M << "\n";
return 0;
}