C. Coloring Trees

ZS the Coder and Chris the Baboon has arrived at Udayland! They walked in the park where n trees grow. They decided to be naughty and color the trees in the park. The trees are numbered with integers from 1 to n from left to right.

Initially, tree i has color ci. ZS the Coder and Chris the Baboon recognizes only m different colors, so 0 ≤ ci ≤ m, where ci = 0 means that tree i is uncolored.

ZS the Coder and Chris the Baboon decides to color only the uncolored trees, i.e. the trees with ci = 0. They can color each of them them in any of the m colors from 1 to m. Coloring the i-th tree with color j requires exactly pi, j litres of paint.

The two friends define the beauty of a coloring of the trees as the minimum number of contiguous groups (each group contains some subsegment of trees) you can split all the n trees into so that each group contains trees of the same color. For example, if the colors of the trees from left to right are 2, 1, 1, 1, 3, 2, 2, 3, 1, 3, the beauty of the coloring is 7, since we can partition the trees into 7 contiguous groups of the same color : {2}, {1, 1, 1}, {3}, {2, 2}, {3}, {1}, {3}.

ZS the Coder and Chris the Baboon wants to color all uncolored trees so that the beauty of the coloring is exactly k. They need your help to determine the minimum amount of paint (in litres) needed to finish the job.

Please note that the friends can't color the trees that are already colored.

The first line contains three integers, n, m and k (1 ≤ k ≤ n ≤ 100, 1 ≤ m ≤ 100) — the number of trees, number of colors and beauty of the resulting coloring respectively.

The second line contains n integers c1, c2, ..., cn (0 ≤ ci ≤ m), the initial colors of the trees. ci equals to 0 if the tree number i is uncolored, otherwise the i-th tree has color ci.

Then n lines follow. Each of them contains m integers. The j-th number on the i-th of them line denotes pi, j (1 ≤ pi, j ≤ 109) — the amount of litres the friends need to color i-th tree with color j. pi, j's are specified even for the initially colored trees, but such trees still can't be colored.

Print a single integer, the minimum amount of paint needed to color the trees. If there are no valid tree colorings of beauty k, print  - 1.

In the first sample case, coloring the trees with colors 2, 1, 1 minimizes the amount of paint used, which equals to 2 + 3 + 5 = 10. Note that 1, 1, 1 would not be valid because the beauty of such coloring equals to 1 ({1, 1, 1} is a way to group the trees into a single group of the same color).

In the second sample case, all the trees are colored, but the beauty of the coloring is 3, so there is no valid coloring, and the answer is  - 1.

In the last sample case, all the trees are colored and the beauty of the coloring matches k, so no paint is used and the answer is 0.

题意：给定N棵树，每棵树都有不同的颜色0-n，0代表没有涂色。如果这棵树没有颜色，那么你可以给他上色，如果已经有颜色，那你不能进行任何操作。接着给定一个数K，K代表着一排树的颜色有多少个相同的连续段。最后，给定每棵 树染上每种颜色的花费，求满足K的最小花费？

思路：比较明显的dp，但是当时并不会做，后来问了ZK大佬，才习得姿势，dp一直都不会，慢慢积累。

首先dp[i][j][k]代表，前i棵树的最后一棵树涂第j种颜色，并且涂成k个相同连续段的最小花费。首先初始状态dp[0][0][0]=0,然后分两种情况走。

1.color[i+1]!=0时，那么有2种情况：当前的j==color[i+1]时，dp[i+1][color[i+1]][k]=min(dp[i+1][color[i+1]][k],dp[i][j][k]),

当前的j!=color[i+1]时，dp[i+1][color[i+1]][k+1]=min(dp[i+1][color[i+1]][k+1],dp[i][j][k])。

2.color[i+1]==0时，那这棵树有1-p（m种）颜色可以选择，所以当p==j时，dp[i+1][p][k]=min(dp[i][j][k]+val[i+1][p],dp[i+1][p][k]),

当p!=j时，dp[i+1][p][k+1]=min(dp[i][j][k]+val[i+1][p],dp[i+1][p][k+1])。

代码如下：

#include <iostream>

#include <queue>

#include <stack>

#include <cstdio>

#include <vector>

#include <map>

#include <set>

#include <bitset>

#include <algorithm>

#include <cmath>

#include <cstring>

#include <cstdlib>

#include <string>

#include <sstream>

#define lson l,m,rt*2

#define rson m+1,r,rt*2+1

#define mod 1000000007

#define mt(A,B) memset(A,B,sizeof(A))

using namespace std;

typedef long long LL;

const int N=200000+10;

const LL INF=0x3f3f3f3f3f3f3f3fLL;

LL dp[105][105][105];

LL c[105],val[105][105],ans=INF;

int main()

{

#ifdef Local

    freopen("data.txt","r",stdin);

#endif

    int i,j,K,n,m,k;

    cin>>n>>m>>K;

    for(i=1;i<=n;i++)cin>>c[i];

    for(i=1;i<=n;i++){

        for(j=1;j<=m;j++){

            cin>>val[i][j];

        }

    }

    mt(dp,0x3f);

    dp[0][0][0]=0;

    for(i=0;i<n;i++){

        for(j=0;j<=m;j++){

            for(k=0;k<=i;k++){

                if(dp[i][j][k]!=INF){

                    if(c[i+1]){

                        dp[i+1][c[i+1]][k+(c[i+1]!=j)]=min(dp[i+1][c[i+1]][k+(c[i+1]!=j)],dp[i][j][k]);

                    }

                    else{

                        for(int p=1;p<=m;p++){

                            dp[i+1][p][k+(p!=j)]=min(dp[i][j][k]+val[i+1][p],dp[i+1][p][k+(p!=j)]);

                        }

                    }

                }

            }

        }

    }

    for(i=1;i<=m;i++)

    {

        ans=min(dp[n][i][K],ans);

    }

    if(ans==INF)cout<<-1<<endl;

    else cout<<ans<<endl;

}