题目链接:http://poj.org/problem?id=1201
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 end points 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 <= 50000) -- 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 <= 50000 and 1 <= ci <= bi - ai+1.
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 1
Sample Output
6
Source
题意:(转)
[ai, bi]区间内和点集Z至少有ci个共同元素,那也就是说如果我用Si表示区间[0,i]区间内至少有多少个元素的话,那么Sbi - Sai >= ci,这样我们就构造出来了一系列边,权值为ci,但是这远远不够,因为有很多点依然没有相连接起来(也就是从起点可能根本就还没有到终点的路线),此时,我们再看看Si的定义,也不难写出0<=Si
- Si-1<=1的限制条件,虽然看上去是没有什么意义的条件,但是如果你也把它构造出一系列的边的话,这样从起点到终点的最短路也就顺理成章的出现了。
我们将上面的限制条件写为同意的形式:
Sbi - Sai >= ci
Si - Si-1 >= 0
Si-1 - Si >= -1
这样一来就构造出了三种权值的边,而最短路自然也就没问题了。
但要注意的是,由于查分约束系统里常常会有负权边,所以为了避免负权回路,往往用Bellman-Ford或是SPFA求解(存在负权回路则最短路不存在)。
PS:
因为求的是[ai,bi]区间,所以我们添加边的时候需要(u-1, v, w)!
把距离dis初始化为负无穷, if(dis[v] < dis[u] + w)即可!
代码如下:
#include <cstdio> #include <cstring> #include <stack> #include <iostream> #include <algorithm> using namespace std; #define INF 0x3f3f3f3f #define N 50017 #define M 50017 int n, m, k; int Edgehead[N], dis[N]; struct Edge { int v,w,next; } Edge[3*M]; bool vis[N]; //int cont[N]; int minn, maxx; int MIN(int a, int b) { if(a < b) return a; return b; } int MAX(int a, int b) { if(a > b) return a; return b; } void Addedge(int u, int v, int w) { Edge[k].next = Edgehead[u]; Edge[k].w = w; Edge[k].v = v; Edgehead[u] = k++; } int SPFA( int start)//stack { int sta[N]; int top = 0; for(int i = 1 ; i <= n ; i++ ) dis[i] = -INF; dis[start] = 0; //++cont[start]; memset(vis,false,sizeof(vis)); sta[++top] = start; vis[start] = true; while(top) { int u = sta[top--]; vis[u] = false; for(int i = Edgehead[u]; i != -1; i = Edge[i].next)//注意 { int v = Edge[i].v; int w = Edge[i].w; if(dis[v] < dis[u] + w) { dis[v] = dis[u]+w; if( !vis[v] )//防止出现环 { sta[++top] = v; vis[v] = true; } // if(++cont[v] > n)//有负环 // return -1; } } } return dis[maxx]; } int main() { int u, v, w; while(~scanf("%d",&n))//n为目的地 { k = 1; memset(Edgehead,-1,sizeof(Edgehead)); minn = INF; maxx = -1; for(int i = 1 ; i <= n ; i++ ) { scanf("%d%d%d",&u,&v,&w); Addedge(u-1,v,w); maxx = MAX(v,maxx); minn = MIN(u-1,minn); } for(int i = minn; i <= maxx; i++)//新边,保证图的连通性还必须添加每相邻两个整数点i,i+1的边 { Addedge(i,i+1,0); Addedge(i+1,i,-1); } int ans = SPFA(minn);//从点minn开始寻找最短路 printf("%d\n",ans); } return 0; }
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