K - Candies（差分约束）

最新推荐文章于 2019-08-14 20:39:58 发布

原创最新推荐文章于 2019-08-14 20:39:58 发布 · 375 阅读

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#差分约束

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本文介绍了一个经典的差分约束系统问题，通过一个幼儿园老师分配糖果的故事背景引入。文章详细阐述了如何利用SPFA算法来解决该问题，以确保每个孩子得到的糖果数量符合他们之间的相对公平要求，同时最大化两个特定孩子之间糖果数量的差距。

During the kindergarten days, flymouse was the monitor of his class. Occasionally the head-teacher brought the kids of flymouse’s class a large bag of candies and had flymouse distribute them. All the kids loved candies very much and often compared the numbers of candies they got with others. A kid A could had the idea that though it might be the case that another kid B was better than him in some aspect and therefore had a reason for deserving more candies than he did, he should never get a certain number of candies fewer than B did no matter how many candies he actually got, otherwise he would feel dissatisfied and go to the head-teacher to complain about flymouse’s biased distribution.

snoopy shared class with flymouse at that time. flymouse always compared the number of his candies with that of snoopy’s. He wanted to make the difference between the numbers as large as possible while keeping every kid satisfied. Now he had just got another bag of candies from the head-teacher, what was the largest difference he could make out of it?

Input

The input contains a single test cases. The test cases starts with a line with two integers N and M not exceeding 30 000 and 150 000 respectively. N is the number of kids in the class and the kids were numbered 1 through N. snoopy and flymouse were always numbered 1 and N. Then follow M lines each holding three integers A, B and c in order, meaning that kid A believed that kid B should never get over c candies more than he did.

Output

Output one line with only the largest difference desired. The difference is guaranteed to be finite.

Sample Input

2 2
1 2 5
2 1 4

Sample Output

Hint
32-bit signed integer type is capable of doing all arithmetic.

这是第一道差分约束题目，用spfa求最短路径。
约束条件：
dis[v]

#include <iostream>
#include <cstdio>
#include <cstring>
#include <cstdlib>
#include <algorithm>
#include <cmath>
#include <queue>
#include <stack>
#define INF 0x3f3f3f3f
using namespace std;
struct node
{
   int to, v, next;
}t[152121];
int n, m;
int dis[34567];
int head[34567];
bool v[30980];
int top;
void Initial()//初始化
{
   for(int i=0;i<=n;i++)
   {
     v[i] = false;
     dis[i] = INF;
     head[i] = -1;
   }
   top = 1;
}
void spfa()//最短路
{
  stack<int>q;//××××注意，这里用到栈，刚开始的时候用的是queue,但是超时了。两者的区别不是很大，视情况而定，一般是使用队列
  q.push(1);
  v[1] = true;
  dis[1] = 0;
  while(!q.empty())
  {
      int k = q.top();
      q.pop();
      v[k] = false;
      for(int i=head[k];i!=-1;i=t[i].next)
      {
          if(dis[t[i].to]>t[i].v+dis[k])//更新（差分约束条件）
          {
              dis[t[i].to]=t[i].v+dis[k];
              if(!v[t[i].to])
              {
                q.push(t[i].to);
                v[t[i].to] = true;
              }
          }
      }
  }
  cout<<dis[n]<<endl;
}
int main()
{
     int a, b, c;
   scanf("%d %d", &n, &m);
   Initial();
   for(int i=1;i<=m;i++)
   {
     scanf("%d %d %d", &a, &b, &c);
     t[top].to = b;
   t[top].v = c;
   t[top].next = head[a];
   head[a] = top++;
   }
   spfa();
   return 0;
}