UVa_00861_Little_Bishops.cpp

Problem Links:

uva00861,

Problem:

Little Bishops

A bishop is a piece used in the game of chess which is played on a board of square grids. A bishop can only move diagonally from its current position and two bishops attack each other if one is on the path of the other. In the following figure, the dark squares represent the reachable locations for bishop B₁ form its current position.  The figure also shows that the bishops B₁ and B₂ are in attacking positions whereas B₁ and B₃ are not. B₂ and B₃ are also in non-attacking positions.

Now, given two numbers n and k, your job is to determine the number of ways one can put k bishops on an n × n chessboard so that no two of them are in attacking positions.

Input

 The input file may contain multiple test cases. Each test case occupies a single line in the input file and contains two integers n (1 ≤ n ≤ 8) and k (0 ≤ k ≤ n²).

A test case containing two zeros for n and k terminates the input and you won’t need to process this particular input.

Output

For each test case in the input print a line containing the total number of ways one can put the given number of bishops on a chessboard of the given size so that no two of them are in attacking positions. You may safely assume that this number will be less than 10¹⁵.

Sample Input
8 6
4 4
0 0

Sample Output
5599888
260

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Problem setter: Rezaul Alam Chowdhury

Solution:

Rotate the chess board 45 degree. Then no bishops should stay in the same row or same column. 
Using dynamic programming skills;
dp[i][j] representss the solution of j bishops filled in the first i rows.
dp[i][j] = dp[i-1][j] + dp[i-1][j-1] * (color[i] - (j-1))

Source Code:

//Tue Apr  5 00:46:15 CDT 2011
#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <cctype>
#include <string>
#include <cstring>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <ctime>

using namespace std;

void init(int n, vector<int> &black, vector<int> &white)
{
    for (int i = 1; i <= n; i++)
    {
        for (int j = 1; j <= n; j++)
        {
            if ((i + j) % 2)
                white[(i + j) / 2]++;
            else
                black[(i + j) / 2]++;
        }
    }
}

void bishops(int n, int k, vector<vector<int> > &dp, vector<int> color)
{
    for (int i = 0; i <= n; i++)
        dp[i][0] = 1;
    for (int i = 1; i <= k; i++)
        dp[0][i] = 0;
    for (int i = 1; i <= n; i++)
        for (int j = 1; j <= color[i] && j<=k; j++)
            dp[i][j] = dp[i - 1][j] + dp[i - 1][j - 1] * (color[i] - j + 1);
}

int main(int argc, char* argv[])
{
    //freopen("input.in", "r", stdin);
    //freopen("output.out", "w", stdout);
    int n, k;
    while (cin >> n >> k)
    {
        if (n == 0 && k == 0) return 0;
        vector<vector<int> > dp1(n + 1, vector<int>(k + 1, 0));
        vector<vector<int> > dp2(n, vector<int>(k + 1, 0));
        vector<int> white(n, 0);
        vector<int> black(n + 1, 0);
        init(n, black, white);
        sort(black.begin() + 1, black.end());
        sort(white.begin() + 1, white.end());
        bishops(n, k, dp1, black);
        bishops(n - 1, k, dp2, white);
        int count = 0;
        for (int i = 0; i <= k; i++)
            count += dp1[n][i] * dp2[n-1][k - i];
        cout << count << endl;
    }

    //fclose(stdin);
    //fclose(stdout);
    return 0;
}