POJ 1837-Balance Balance Time Limit: 1000MS Memory Limit: 30000K Total Submissions: 10845 Accepted: 6742 Description Gigel has a strange “balance” and he wants to poise it. Actually, the device is different from any other ordinary balance. It orders two arms of negligible weight and each arm’s length is 15. Some hooks are attached to these arms and Gigel wants to hang up some weights from his collection of G weights (1 <= G <= 20) knowing that these weights have distinct values in the range 1..25. Gigel may droop any weight of any hook but he is forced to use all the weights. Finally, Gigel managed to balance the device using the experience he gained at the National Olympiad in Informatics. Now he would like to know in how many ways the device can be balanced. Knowing the repartition of the hooks and the set of the weights write a program that calculates the number of possibilities to balance the device. It is guaranteed that will exist at least one solution for each test case at the evaluation. Input The input has the following structure: • the first line contains the number C (2 <= C <= 20) and the number G (2 <= G <= 20); • the next line contains C integer numbers (these numbers are also distinct and sorted in ascending order) in the range -15..15 representing the repartition of the hooks; each number represents the position relative to the center of the balance on the X axis (when no weights are attached the device is balanced and lined up to the X axis; the absolute value of the distances represents the distance between the hook and the balance center and the sign of the numbers determines the arm of the balance to which the hook is attached: ‘-‘ for the left arm and ‘+’ for the right arm); • on the next line there are G natural, distinct and sorted in ascending order numbers in the range 1..25 representing the weights’ values. Output The output contains the number M representing the number of possibilities to poise the balance. Sample Input 2 4 -2 3 3 4 5 8 Sample Output 2 Source Romania OI 2002

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这一题题意为：给定一个天平，在天平两个臂上有若干个挂钩，给定若干砝码，问将所有砝码都挂上挂钩并使天平平衡的方法数有多少种？这一题有一定的难度，如果只有两个挂钩，问题比较好解决，可以看我的另一篇博客从一个数字序列中取若干个数字使得和为某个数，问共有多少种取数方案；但是这一题的挂钩有若干个，情况比较多，想过用动态规划、背包问题，但是一直想不出状态转换方程。 后来看了某前辈的解题报告，他提出了一个平衡度的概念，并推导出如下的状态转换方程：①dp[i][ j+ w[i]*c[k] ]= ∑(dp[i-1][j])。dp[i][j]表示前i个砝码使平衡度为j共有多少种方案数。 按照常理，我们得到的状态转换方程为：②dp[i][j] =∑(dp[i – 1][j – c[i] * w[i]])，表示dp[i][j]等于前i-1个砝码使平衡度为j-x的方案数之和，这个x就是第i个砝码挂在c个挂钩中的某一个产生的力臂。稍加推导就能得出①等价于②。 有了状态转换方程，我们可以很快的写出代码： [cpp] #include<iostream> using namespace std; int c[21]; int w[21]; int dp[21][15001]; int main() { int n,g; cin>>n>>g; for(int i=1;i<=n;i++) cin>>c[i]; for(int i=1;i<=g;i++) cin>>w[i]; dp[0][7500]=1;//不挂任何砝码的时候它本身就平衡了，所以是一种“挂法” for(int i=1;i<=g;i++)//对于每一个砝码 { for(int j=0;j<=15000;j++)//挂上去之后可能使天平出现0-15000中的任意一种状态 { if(dp[i-1][j])//如果这个状态之前出现过，则直接用 { for(int k=1;k<=n;k++) { dp[i][j+w[i]*c[k]]+=dp[i-1][j]; } } } } cout<<dp[g][7500]<<endl; return 0; } [/cpp] 代码提交AC，用时0MS，内存1060K。]]>