P10189 [USACO24FEB] Maximizing Productivity B 题解
先说说暴力做法:
每次遍历一遍,看看是否满足 \(t_i + s \le c_i\),满足就计数,不满足就挂。单次时间复杂度显然为 \(O(N)\),总得时间复杂度约为 \(O(NQ)\),TLE是肯定的~
暴力代码
// Problem: Problem 3. Maximizing Productivity
// Contest: USACO - USACO 2024 February Contest, Bronze
// URL: https://usaco.org/index.php?page=viewproblem&cpid=1385
// Memory Limit: 256 MB
// Time Limit: 4000 ms
//
// Powered by CP Editor (https://cpeditor.org)
/*Code by Leo2011*/
#include <bits/stdc++.h>
#define log printf
#define EPS 1e-8
#define INF 0x3f3f3f3f
#define FOR(i, l, r) for (int(i) = (l); (i) <= (r); ++(i))
#define IOS \
ios::sync_with_stdio(false); \
cin.tie(nullptr); \
cout.tie(nullptr);
using namespace std;
typedef __int128 i128;
typedef long long ll;
typedef pair<int, int> PII;
const int N = 2e5 + 10;
int n, q, c[N], t[N];
template <typename T>
inline T read() {
T sum = 0, fl = 1;
char ch = getchar();
for (; !isdigit(ch); ch = getchar())
if (ch == '-') fl = -1;
for (; isdigit(ch); ch = getchar()) sum = sum * 10 + ch - '0';
return sum * fl;
}
template <typename T>
inline void write(T x) {
static T sta[35];
int top = 0;
do { sta[top++] = x % 10, x /= 10; } while (x);
while (top) putchar(sta[--top] + 48);
}
int main() {
n = read<int>(), q = read<int>();
FOR(i, 1, n) c[i] = read<int>();
FOR(i, 1, n) t[i] = read<int>();
FOR(i, 1, q) {
int v = read<int>(), s = read<int>(), cnt = 0;
FOR(j, 1, n) if (s + t[j] < c[j])++ cnt;
if (cnt >= v) puts("YES");
else puts("NO");
}
return 0;
}
评测记录,开了优化也只有 20pts。
暴力废了,开始整正解。
\(2 \times 10^5\)?二分法,了解一下?
然后,然后遇到了个问题:二分这东西要求有序,可是 \(c_i\) 与 \(t_i\) 是绑定在一起的啊!咋整?
注意 \(c_i\) 与 \(c_i - t_i\) 没关系,而这个式子能够求出啥啊?想要去到第 \(i\) 个农场最大的 \(S\) 嘛(\(c_i - t_i + t_i = c_i\),刚好卡点儿到,不可能再晚了)!
总结一下,问题相当于预处理出来一个数组 \(a\),使 \(a_i = c_i - t_i\),求数组中是否有至少 \(V\) 个数满足 \(a_i \ge S\)。
先排个序(不然还是 \(O(n)\) 嘛),然后就可以分出来两种做法:
做法一:二分,用 upper_bound
函数(因为是大于等于),时间复杂度约为 \(O(QlogN)\)。
做法二:排完序后显然满足 \(a_V \ge a_{V - 1} \ge a_{V - 2} ... a_1\),因此如果 \(a_V\) 都小于等于 \(S\) 了,那它前面的自然也都满足。因此只需要判断 \(a_V \ge S\) 成不成立即可。时间复杂度约为 \(O(Q)\)。
赛时ACCode(做法二)
// Problem: Problem 3. Maximizing Productivity
// Contest: USACO - USACO 2024 February Contest, Bronze
// URL: https://usaco.org/index.php?page=viewproblem&cpid=1385
// Memory Limit: 256 MB
// Time Limit: 4000 ms
//
// Powered by CP Editor (https://cpeditor.org)
/*Code by Leo2011*/
#include <iostream>
#include <algorithm> // 万能头CE
#define log printf
#define EPS 1e-8
#define INF 0x3f3f3f3f
#define FOR(i, l, r) for (int(i) = (l); (i) <= (r); ++(i))
#define IOS \
ios::sync_with_stdio(false); \
cin.tie(nullptr); \
cout.tie(nullptr);
using namespace std;
typedef __int128 i128;
typedef long long ll;
typedef pair<int, int> PII;
int N, Q;
int close[200010], t[200010], ans[200010], V, S;
template <typename T>
inline T read() {
T sum = 0, fl = 1;
char ch = getchar();
for (; !isdigit(ch); ch = getchar())
if (ch == '-') fl = -1;
for (; isdigit(ch); ch = getchar()) sum = sum * 10 + ch - '0';
return sum * fl;
}
template <typename T>
inline void write(T x) {
static T sta[35];
int top = 0;
do { sta[top++] = x % 10, x /= 10; } while (x);
while (top) putchar(sta[--top] + 48);
}
bool x(int a, int b) { return a > b; }
int main() {
cin >> N >> Q;
for (int i = 1; i <= N; i++) { cin >> close[i]; }
for (int i = 1; i <= N; i++) { cin >> t[i]; }
for (int i = 1; i <= N; i++) { ans[i] = close[i] - t[i]; }
sort(ans + 1, ans + N + 1, x);
while (Q--) {
cin >> V >> S;
if (ans[V] > S) cout << "YES" << endl;
else cout << "NO" << endl;
}
return 0;
}