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2011/01/27

「5×5のマス目に6個の○を次の条件を満たすように全部書く」を解いてみた

00:03

404 Blog Not Found:Math - 5×5のマス目に6個の○を次の条件を満たすように全部書く

5×5のマス目に6個の○を次の条件を満たすように書きます。

条件:各行・各列に少なくとも1個は○を書く。同じマスには2個以上の○を書くことはできない。

このとき、6個の○を書く方法は全部で何通りありますか。

RSS を眺めているときにこの記事を見つけて dankogai ならきっと...!と思って「続きを読む」を押したら総当り6重 for ループでがっかりしたので,変数を可変にして自分で書いてみた.combination 関数をちゃんと書いてないのででかい数を渡すとオーバーフローします.

典型的な DP ですよね.20分ぐらいで書けました.間違ってたら指摘お願いします.

弾さんも書いている通り総当りでも 25_P_6 = 127,512,000 なんで TopCoder でもなければ総当たったほうが楽で,時と場合と目的に依ってはそれもベストなコードの1つですね(TopCoderだとTLEする).

/Users/yuyarin% time ./a.out       
n=5 m=5 c=6
answer: 4200
./a.out  0.00s user 0.00s system 18% cpu 0.012 total
/Users/yuyarin% time ./a.out 7 7 9 
n=7 m=7 c=9
answer: 13003620
./a.out 7 7 9  0.00s user 0.00s system 17% cpu 0.013 total
/Users/yuyarin% time ./a.out 9 7 13
n=9 m=7 c=13
answer: 1016446631327
./a.out 9 7 13  0.00s user 0.00s system 31% cpu 0.008 total

どっかでオーバーフローしてるかもしれない.

イカソース


#include <iostream>
#include <cstdlib>
#include <vector>

using namespace std;

long long int combination(long long int n, long long int m)
{
	if(n<m)
		return 0;
	long long int r = 1;
	// warning! r may overflow with large n and m
	for(int i=0; i<m; i++)
		r *= n-i;
	for(int i=0; i<m; i++)
		r /= i+1;
	return r;
}

int main(int argc, char *argv[])
{
	int n=5, m=5, c=6;
	if(argc==4)
	{
		n = atoi(argv[1]);
		m = atoi(argv[2]);
		c = atoi(argv[3]);
		if(n<1||m<1||c<1)
		{
			cerr << "usage: ./a.out n m c (n>0,m>0,c>0)" << endl;
			return 1;
		}
	}
	
	cout << "n=" << n << " m=" << m << " c=" << c << endl;
	
	vector<vector<long long int> > dp(n+1, vector<long long int>(m+1, 0));
	
	for(int i=1; i<=n; i++) for(int j=1; j<=m; j++)
	{
		dp[i][j] = combination(i*j, c);
		for(int h=0; h<=i; h++) for(int v=0; v<=j; v++)
		{
			if(h==0&&v==0) continue;
			dp[i][j] -= combination(i,h)*combination(j,v)*dp[i-h][j-v];
		}
	}
	
	cout << "answer: " << dp[n][m] << endl;
	
	return 0;
}

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