Intervals
Time Limit: 10000/5000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 2931 Accepted Submission(s): 1067Problem Description
You are given n closed, integer intervals [ai, bi] and n integers c1, ..., cn.
Write a program that:
> reads the number of intervals, their endpoints and integers c1, ..., cn from the standard input,
>
computes the minimal size of a set Z of integers which has at least ci
common elements with interval [ai, bi], for each i = 1, 2, ..., n,> writes the answer to the standard output
Input
The
first line of the input contains an integer n (1 <= n <= 50 000) -
the number of intervals. The following n lines describe the intervals.
The i+1-th line of the input contains three integers ai, bi and ci
separated by single spaces and such that 0 <= ai <= bi <= 50
000 and 1 <= ci <= bi - ai + 1.Process to the end of file.
Output
The
output contains exactly one integer equal to the minimal size of set Z
sharing at least ci elements with interval [ai, bi], for each i = 1, 2,
..., n.Sample Input
5
3 7 3
8 10 3
6 8 1
1 3 1
10 11 1Sample Output
6
令1..i共选d[i]个数,那么有0 <= d[i + 1] - d[i] <= 1;
并且对于每个约束a, b, c, 都有 d[b] - d[a - 1] >= c
这样就是一个线性规划问题,可以用最短路求解。
对于求最大值,就将不等式转化为求最短路中的三角不等式,即:d[u] + w >= d[v], 然后求最短路即可。
对于求最小值,就将不等式转化为求最长路中的三角不等式,即:d[u] + w <= d[v], 然后求最长路即可。
当然求最小值时,也可以按求最大值的方法加边,只不过这时候加的边都是反向边,从终点到起点跑最短路,然后结果取反就可以了。
Accepted Code:
1 /************************************************************************* 2 > File Name: 1384.cpp 3 > Author: Stomach_ache 4 > Mail: [email protected] 5 > Created Time: 2014年08月26日 星期二 08时59分19秒 6 > Propose: 7 ************************************************************************/ 8 #include <queue> 9 #include <cmath> 10 #include <string> 11 #include <cstdio> 12 #include <vector> 13 #include <fstream> 14 #include <cstring> 15 #include <iostream> 16 #include <algorithm> 17 using namespace std; 18 /*Let‘s fight!!!*/ 19 20 const int INF = 0x3f3f3f3f; 21 const int MAX_N = 50050; 22 typedef pair<int, int> pii; 23 vector<pii> G[MAX_N]; 24 int n, d[MAX_N]; 25 bool inq[MAX_N]; 26 27 void AddEdge(int u, int v, int w) { 28 G[u].push_back(pii(v, w)); 29 } 30 31 void spfa(int s) { 32 queue<int> Q; 33 memset(d, 0x3f, sizeof(d)); 34 memset(inq, false, sizeof(inq)); 35 d[s] = 0; 36 inq[s] = true; 37 Q.push(s); 38 while (!Q.empty()) { 39 int u = Q.front(); Q.pop(); inq[u] = false; 40 for (int i = 0; i < G[u].size(); i++) { 41 int v = G[u][i].first, w = G[u][i].second; 42 if (d[u] + w < d[v]) { 43 d[v] = d[u] + w; 44 if (!inq[v]) Q.push(v), inq[v] = true; 45 } 46 } 47 } 48 } 49 50 int main(void) { 51 while (~scanf("%d", &n)) { 52 for (int i = 0; i <= 50005; i++) G[i].clear(); 53 int s = INF, t = -1; 54 for (int i = 0; i < n; i++) { 55 int a, b, c; 56 scanf("%d %d %d", &a, &b, &c); 57 b++; 58 s = min(s, a); t = max(t, b); 59 AddEdge(b, a, -c); 60 } 61 for (int i = s; i < t; i++) AddEdge(i, i + 1, 1), AddEdge(i + 1, i, 0); 62 63 spfa(t); 64 65 printf("%d\n", -d[s]); 66 } 67 68 return 0; 69 }