Burn the Linked Camp
It is well known that, in the period of The Three Empires, Liu Bei, the emperor of the Shu Empire, was defeated by Lu Xun, a general of the Wu Empire. The defeat was due to Liu Bei‘s wrong decision that he divided his large troops into a number of camps, each of which had a group of armies, and located them in a line. This was the so-called "Linked Camps".
Let‘s go back to that time. Lu Xun had sent many scouts to obtain the information about his enemy. From his scouts, he knew that Liu Bei had divided his troops into n camps, all of which located in a line, labeled by 1..n from left to right. The ith camp had a maximum capacity of Ci soldiers. Furthermore, by observing the activities Liu Bei‘s troops had been doing those days, Lu Xun could estimate the least total number of soldiers that were lived in from the ith to the jth camp. Finally, Lu Xun must estimate at least how many soldiers did Liu Bei had, so that he could decide how many troops he should send to burn Liu Bei‘s Linked Camps.
Input
There are multiple test cases! On the first line of each test case, there are two integers n (0<n<=1,000) and m (0<=m<=10,000). On the second line, there are n integers C1。。。Cn. Then m lines follow, each line has three integers i, j, k (0<i<=j<=n, 0<=k<2^31), meaning that the total number of soldiers from the ith camp to the jth camp is at least k.
Output:
For each test case, output one integer in a single line: the least number of all soldiers in Liu Bei‘s army from Lu Xun‘s observation. However, Lu Xun‘s estimations given in the input data may be very unprecise. If his estimations cannot be true, output "Bad Estimations" in a single line instead.
Sample Input:
3 2
1000 2000 1000
1 2 1100
2 3 1300
3 1
100 200 300
2 3 600
Sample Output:
1300
Bad Estimations
大意：三国时期刘备被陆逊火烧连营的故事。陆逊派了侦察兵，得知刘备把他的部队划分成了n个营，它们都在一条线上，从左到右分别标记为1,...,n。第i个营寨的最大士兵容量为Ci。通过一段时间观察，陆逊可以估计出[i,j]这些营的士兵和的最小值。最后问估计出的所有士兵的最小数。若估算数据与已知最大容量抵触，则认为估计错误。
分析：记v[i]为前i个兵营中的士兵数。
由题目得以下3个约束条件。
1.v[a]-v[a-1]>=0;要确保每个营中人数不为负。
2.对于容量Ci，有v[i]-v[i-1]<=Ci;
3.对于三元组输入 a b k,有v[b]-v[a-1]>=k。
全部整理成<=类型的不等式，满足差分约束条件即可。我们要求最大的k，也就是最小的-k,所以最短路解决问题。
差分约束-zoj-2770