UVa 348 最优矩阵链乘 (DP)
来源:互联网 发布:ce系统导航软件 编辑:程序博客网 时间:2024/05/16 18:04
Optimal Array Multiplication Sequence
Optimal Array Multiplication Sequence
Given two arrays A and B, we can determine the array C = A B using the standard definition of matrix multiplication:
The number of columns in the A array must be the same as the number of rows in the B array. Notationally, let's say that rows(A) and columns(A) are the number of rows and columns, respectively, in the A array. The number of individual multiplications required to compute the entire C array (which will have the same number of rows as A and the same number of columns as B) is then rows(A)columns(B) columns(A). For example, if A is a array, and B is a array, it will take , or 3000 multiplications to compute the C array.
To perform multiplication of more than two arrays we have a choice of how to proceed. For example, if X, Y, and Z are arrays, then to compute X Y Z we could either compute (X Y) Z or X (Y Z). Suppose Xis a array, Y is a array, and Z is a array. Let's look at the number of multiplications required to compute the product using the two different sequences:
(X Y) Z
- multiplications to determine the product (X Y), a array.
- Then multiplications to determine the final result.
- Total multiplications: 4500.
X (Y Z)
- multiplications to determine the product (Y Z), a array.
- Then multiplications to determine the final result.
- Total multiplications: 8750.
Clearly we'll be able to compute (X Y) Z using fewer individual multiplications.
Given the size of each array in a sequence of arrays to be multiplied, you are to determine an optimal computational sequence. Optimality, for this problem, is relative to the number of individual multiplications required.
Input
For each array in the multiple sequences of arrays to be multiplied you will be given only the dimensions of the array. Each sequence will consist of an integer N which indicates the number of arrays to be multiplied, and then N pairs of integers, each pair giving the number of rows and columns in an array; the order in which the dimensions are given is the same as the order in which the arrays are to be multiplied. A value of zero for N indicates the end of the input. N will be no larger than 10.
Output
Assume the arrays are named . Your output for each input case is to be a line containing a parenthesized expression clearly indicating the order in which the arrays are to be multiplied. Prefix the output for each case with the case number (they are sequentially numbered, starting with 1). Your output should strongly resemble that shown in the samples shown below. If, by chance, there are multiple correct sequences, any of these will be accepted as a valid answer.
Sample Input
31 55 2020 135 1010 2020 35630 3535 1515 55 1010 2020 250
Sample Output
Case 1: (A1 x (A2 x A3))Case 2: ((A1 x A2) x A3)Case 3: ((A1 x (A2 x A3)) x ((A4 x A5) x A6))
矩阵链乘在小白P170有讲解 也是经典的动态规划 有了转移方程题目就好做了
设d[i][j]为矩阵i到矩阵j的最优链乘 初始状态d[i][i]=0 k为i,j之间的一个矩阵
x[i],y[i]分别表示第i个矩阵的行和列 则有d[i][j]=min{d[i][k]+d[k+1][j]+x[i]*y[k]*y[j]}
感觉这题比较难的就是路径打印了 仔细观察一下结构 其实也不难 记录每个k值 直接递归就行了
#include<cstdio>#include<cstring>using namespace std;#define maxn 11#define INF 0x3f3f3f3f#define t dp(i,k)+dp(k+1,j)+x[i]*y[k]*y[j]int d[maxn][maxn],pre[maxn][maxn],x[maxn],y[maxn],n;int dp(int i,int j){ if(d[i][j]<INF) return d[i][j]; for(int k=i; k<j; ++k) if(d[i][j]>t) { d[i][j]=t; pre[i][j]=k; } return d[i][j];}void print(int i,int j){ int k=pre[i][j]; if(k) { printf("("); print(i,k); printf(" x "); print(k+1,j); printf(")"); } else { printf("A%d",i); }}int main(){ int k=1; while(scanf("%d",&n),n) { memset(d,0x3f,sizeof(d)); memset(pre,0,sizeof(pre)); for(int i=1; i<=n; ++i) { scanf("%d%d",&x[i],&y[i]); d[i][i]=0; } dp(1,n); printf("Case %d: ",k); print(1,n); printf("\n"); k++; } return 0;}
- UVa 348 最优矩阵链乘 (DP)
- 最优矩阵链乘(经典DP)
- UVA 348 最优矩阵链乘(线性动态规划)
- UVA 348 Optimal Array Multiplication Sequence 最优矩阵链乘 dp经典
- UVA 348 Optimal Array Multiplication Sequence 最优矩阵链乘 区间DP
- UVa 348 Optimal Array Multiplication Sequence (DP 最优矩阵链乘)
- UVa 348 Optimal Array Multiplication Sequence (DP 最优矩阵链乘)
- 区间DP-最优矩阵链乘
- UVA 348 Optimal Array Multiplication Sequence(最优矩阵链乘)
- UVA 10003 Cutting Sticks [线性结构dp] [最优矩阵链乘]
- 最优矩阵链乘
- 最优矩阵链乘
- 最优矩阵链乘
- 最优矩阵链乘
- 最优矩阵链乘
- 最优矩阵链乘
- UVA - 348 Optimal Array Multiplication Sequence 最优矩阵连乘
- POJ1651:Multiplication Puzzle(区间DP 最优矩阵链乘)
- 360更新补丁之后,程序无法运行解决历程
- hibernate中cascade属性
- nyoj 90 整数划分 【深搜】
- HDU 4652 Dice(概率dp)
- Objective-C中的Protocol
- UVa 348 最优矩阵链乘 (DP)
- [翻译自mos文章]当job_queue_processes=0 时,cjq 进程依然启动
- IOS编程 之 数据库
- Windows Server 2003下安装PHP +mssql2000
- Java中的数据类型
- 无向连通图求割点(tarjan算法去掉改割点剩下的联通分量数目)
- 深圳云计算公司有哪些
- 委托 I
- Mac OS X Yosemite Beta版本登录界面卡死解决方案