USACO 2014 December Contest, Bronze Problem 3. Cow Jog

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USACO2014-DEC-B3

答案

(Analysis by Jonathan Paulson - jonathanpaulson@gmail.com)

If a faster cow is behind a slower cow, the faster cow is eventually going to catch up and join the slower cow's group (or join a group in the middle).

So we want to count the number of cows who have no slower cows ahead of them; this is the number of cows who won't join another group (and hence will start their own group).

The most obvious way to to go through each cow and check if any of the cows ahead of them are faster. But this is too slow: there are about N2N2 pairs of cows, and N = 105105. So N2N2 is about 10101010, and computers only do about 109109 operations per second (which usually translates to about 107107iterations of a simple loop in one second of time).

Luckily, there is a faster way: start from the back, and keep track of the slowest cow as you go. This only takes about NN operations, which is very fast.

Here is my C++ code for the fast approach:

#include <cstdio>
#include <vector>

using namespace std;

typedef long long ll;

int main() {
  ll n;
  scanf("%lld", &n);
  vector<ll> S;
  for(ll i=0; i<n; i++) {
    ll x, s;
    scanf("%lld %lld\n", &x, &s);
    S.push_back(s);
  }

  ll ans = 1;
  ll slow = S[n-1];
  for(ll i=n-2; i>=0; i--) {
    if(S[i] > slow) {
      // cows group up
    } else {
      ans++;
    }
    slow = min(slow, S[i]);
  }
  printf("%lld\n", ans);
}