【POJ 1275】 Cashier Employment（差分约束系统的建立和求解）

【POJ 1275】 Cashier Employment（差分约束系统的建立和求解）

Cashier Employment

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 7569		Accepted: 2856

Description

A supermarket in Tehran is open 24 hours a day every day and needs a number of cashiers to fit its need. The supermarket manager has hired you to help him, solve his problem. The problem is that the supermarket needs different number of cashiers at different times of each day (for example, a few cashiers after midnight, and many in the afternoon) to provide good service to its customers, and he wants to hire the least number of cashiers for this job.

The manager has provided you with the least number of cashiers needed for every one-hour slot of the day. This data is given as R(0), R(1), ..., R(23): R(0) represents the least number of cashiers needed from midnight to 1:00 A.M., R(1) shows this number for duration of 1:00 A.M. to 2:00 A.M., and so on. Note that these numbers are the same every day. There are N qualified applicants for this job. Each applicant i works non-stop once each 24 hours in a shift of exactly 8 hours starting from a specified hour, say ti (0 <= ti <= 23), exactly from the start of the hour mentioned. That is, if the ith applicant is hired, he/she will work starting from ti o'clock sharp for 8 hours. Cashiers do not replace one another and work exactly as scheduled, and there are enough cash registers and counters for those who are hired.

You are to write a program to read the R(i) 's for i=0..23 and ti 's for i=1..N that are all, non-negative integer numbers and compute the least number of cashiers needed to be employed to meet the mentioned constraints. Note that there can be more cashiers than the least number needed for a specific slot.

Input

The first line of input is the number of test cases for this problem (at most 20). Each test case starts with 24 integer numbers representing the R(0), R(1), ..., R(23) in one line (R(i) can be at most 1000). Then there is N, number of applicants in another line (0 <= N <= 1000), after which come N lines each containing one ti (0 <= ti <= 23). There are no blank lines between test cases.

Output

For each test case, the output should be written in one line, which is the least number of cashiers needed.
If there is no solution for the test case, you should write No Solution for that case.

Sample Input

1
1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
5
0
23
22
1
10

Sample Output

1

Source

Tehran 2000

做到如今为止 做的最麻烦的一个差分约束……

麻烦不是麻烦在求解上。而是建图……而且绝大多数差分约束的难点。都是建图。。毕竟求解就是个裸最短路

这题题意是 一个商店要招聘员工 商店有一个客流表 表示0~23点每点须要的最少员工数

之后一个整数n 表示有n个应聘员工

后面n行 每行一个0~23的整数 表示第i个员工工作的起始时间 每一个员工工作8小时

题目分析完成 条件不多 所以这就是这题的难点……要挖掘非常多隐含条件

设t[i]为商店第i小时须要的员工数 r[i]表示第i小时来应聘的员工数 数组S[i]表示0～i小时总共应聘的员工数

第一个条件 0 <= S[i]-S[i-1] <= r[i]

第二个条件 S[i]-S[i-8] >= t[i] （i-1时i-8招聘的员工刚好下班 因此S[i]-S[i-8]即为第i小时招聘的所有员工）

第三个条件 S[24]-S[0] <= sum（因为要找最少的总招聘人数，所以这里出现了一个未知量，而这个问题就转化成了求最小sum）

把三个问题化成同号 再细化一下 就变成了

S[i-1]-S[i] <= 0

S[i]-S[i-1] <= r[i]

S[i-8]-S[i] <= -t[i]（0 < i <= 8）

S[i+16]-S[i] <= sum-t[i]（8 < i <= 24)

S[24]-S[0] <= sum

找到不等关系就好办一点了 对sum二分 每次更新与sum有关的边权 然后跑最短路 推断一下能不能跑出来（有无负环）

代码例如以下：

#include <iostream>
#include <cmath>
#include <vector>
#include <cstdlib>
#include <cstdio>
#include <cstring>
#include <queue>
#include <list>
#include <algorithm>
#include <map>
#include <set>
#define LL long long
#define Pr pair<int,int>
#define fread() freopen("in.in","r",stdin)
#define fwrite() freopen("out.out","w",stdout)

using namespace std;
const int INF = 0x3f3f3f3f;
const int msz = 10000;
const double eps = 1e-8;

struct Edge
{
	int u,v,w;
};

Edge eg[233333];
int w[25],r[25];
int dis[25];
int tp;

bool bellman()
{
	memset(dis,INF,sizeof(dis));
	dis[0] = 0;

	for(int i = 0; i <= 24; ++i)
		for(int j = 0; j < tp; ++j)
			if(dis[eg[j].v] > dis[eg[j].u]+eg[j].w)
				dis[eg[j].v] = dis[eg[j].u]+eg[j].w;

	//判负环
	for(int j = 0; j < tp; ++j)
		if(dis[eg[j].v] > dis[eg[j].u]+eg[j].w)
			return false;

	return true;
}

void Add(int u,int v,int w)
{
	eg[tp].u = u;
	eg[tp].v = v;
	eg[tp++].w = w;
}

//重建与sum有关的边
void buildgraph(int sum)
{
	tp = 64;
	for(int i = 1; i < 9; ++i)
		Add(i,i+16,sum-w[i]);
	Add(24,0,-sum);
}

int main()
{
	int t,x,n,low,high;

	scanf("%d",&t);
	while(t--)
	{
		tp = 0;

		high = 0;
		for(int i = 1; i <= 24; ++i)
			scanf("%d",&w[i]);

		memset(r,0,sizeof(r));
		scanf("%d",&n);
		high = n;

		while(n--)
		{
			scanf("%d",&x);
			r[x+1]++;
		}

		//边权不变的边(与sum无关的边)
		for(int i = 1; i <= 24; ++i)
		{
			Add(i-1,i,r[i]);
			Add(i,i-1,0);
			if(i >= 9) Add(i,i-8,-w[i]);
		}

		low = 0;
		int ans = INF;
		while(low <= high)
		{
			int mid = (low+high)>>1;

			buildgraph(mid);
			if(bellman())
			{
				ans = mid;
				high = mid-1;
			}else low = mid+1;
		}

		if(ans == INF) puts("No Solution");
		else printf("%d
",ans);
	}

	return 0;
}