浙江财经大学
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Codeforces Round #369 (Div. 2) C – Coloring Trees

本文由 Ocrosoft 于 2016-08-31 15:01:37 发表

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 ≤ cim, 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.

Input

The first line contains three integers, nm and k (1 ≤ kn ≤ 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 ≤ cim), 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 jpi, j‘s are specified even for the initially colored trees, but such trees still can’t be colored.

Output

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.

Examples
input
3 2 2
0 0 0
1 2
3 4
5 6

output
10

input
3 2 2
2 1 2
1 3
2 4
3 5

output
-1

input
3 2 2
2 0 0
1 3
2 4
3 5

output
5

input
3 2 3
2 1 2
1 3
2 4
3 5

output
0

Note

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.

Solution

题意:有一排n棵树,有些树数涂了颜色的,有些则没有。颜料有m种,一棵树0表示没有颜色,其他表示涂了第x种颜色。第i棵数涂第j种颜色需要c[i][j]数量的这种颜料。现在要给没有颜色的树涂上颜色,使:树的分组恰好为k,并且求出最少要使用多少数量的颜料。连续且颜色相同的树分为一组,如2, 1, 1, 1, 3, 2, 2, 3, 1, 3,分为{2}, {1, 1, 1}, {3}, {2, 2}, {3}, {1}, {3}这样7组。
思路:
dp[i][j][k]表示第i棵树,涂上第j种颜料,到这棵树前面有几组。转移方程分几种情况写就好了。

#include <set>
#include <map>
#include <list>
#include <cmath>
#include <stack>
#include <queue>
#include <ctime>
#include <string>
#include <cstdio>
#include <vector>
#include <cctype>
#include <climits>
#include <sstream>
#include <cstring>
#include <cstdlib>
#include <iostream>
#include <algorithm>
#include <functional>
#define ms(a) memset(a,0,sizeof(a))
typedef long long LL;
const LL LINF = 1e17;
const int INF = INT_MAX / 2;
const int MAXN = 100 + 10;
const int MOD = 100000;
using namespace std;
LL dp[MAXN][MAXN][MAXN];
int tree[MAXN];
int use[MAXN][MAXN];
int n, m, k;
void init()
{
	for (int i = 0; i <= n; i++)
	for (int j = 0; j <= m; j++)
	for (int l = 0; l <= k; l++)
	dp[i][j][l] = LINF;
}
void work(int i,int j)
{
	int sj = j;
	if (j == -1)j = tree[i];
	for (int l = 1; l <= i; l++)
	{
		LL tmp = LINF;
		tmp = min(tmp, dp[i - 1][j][l]);
		for (int p = 1; p <= m; p++)
		{
			if (p == j)continue;
			tmp = min(tmp, dp[i - 1][p][l - 1]);
		}
		if (tmp == LINF)continue;
		dp[i][j][l] = tmp + (sj == -1 ? 0 : use[i][j]);
	}
}
int main()
{
	cin >> n >> m >> k;
	for (int i = 1; i <= n; i++)
		scanf("%d", &tree[i]);
	for (int i = 1; i <= n; i++)
		for (int j = 1; j <= m; j++)
			scanf("%d", &use[i][j]);
	init();
	if (!tree[1])for (int i = 1; i <= m; i++)dp[1][i][1] = use[1][i];
	else dp[1][tree[1]][1] = 0;
	for (int i = 2; i <= n; i++){
		if (tree[i])work(i, -1);
		else for (int j = 1; j <= m; j++)work(i, j);}
	LL ans = LINF;
	for (int i = 1; i <= m; i++) ans = min(ans, dp[n][i][k]);
	printf("%I64d\n", ans == LINF ? -1 : ans);
	return 0;
}

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