Codeforces#498F. Xor-Paths(折半搜索)

time limit per test

3 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

There is a rectangular grid of size n×mn×m. Each cell has a number written on it; the number on the cell (i,ji,j) is ai,jai,j. Your task is to calculate the number of paths from the upper-left cell (1,11,1) to the bottom-right cell (n,mn,m) meeting the following constraints:

• You can move to the right or to the bottom only. Formally, from the cell (i,ji,j) you may move to the cell (i,j+1i,j+1) or to the cell (i+1,ji+1,j). The target cell can't be outside of the grid.
• The xor of all the numbers on the path from the cell (1,11,1) to the cell (n,mn,m) must be equal to kk (xor operation is the bitwise exclusive OR, it is represented as '^' in Java or C++ and "xor" in Pascal).

Find the number of such paths in the given grid.

Input

The first line of the input contains three integers nn, mm and kk (1≤n,m≤201≤n,m≤20, 0≤k≤10180≤k≤1018) — the height and the width of the grid, and the number kk.

The next nn lines contain mm integers each, the jj-th element in the ii-th line is ai,jai,j (0≤ai,j≤10180≤ai,j≤1018).

Output

Print one integer — the number of paths from (1,11,1) to (n,mn,m) with xor sum equal to kk.

Examples

input

Copy

3 3 11
2 1 5
7 10 0
12 6 4

output

Copy

3

input

Copy

3 4 2
1 3 3 3
0 3 3 2
3 0 1 1

output

Copy

5

input

Copy

3 4 1000000000000000000
1 3 3 3
0 3 3 2
3 0 1 1

output

Copy

0

Note

All the paths from the first example:

• (1,1)→(2,1)→(3,1)→(3,2)→(3,3)(1,1)→(2,1)→(3,1)→(3,2)→(3,3);
• (1,1)→(2,1)→(2,2)→(2,3)→(3,3)(1,1)→(2,1)→(2,2)→(2,3)→(3,3);
• (1,1)→(1,2)→(2,2)→(3,2)→(3,3)(1,1)→(1,2)→(2,2)→(3,2)→(3,3).

All the paths from the second example:

• (1,1)→(2,1)→(3,1)→(3,2)→(3,3)→(3,4)(1,1)→(2,1)→(3,1)→(3,2)→(3,3)→(3,4);
• (1,1)→(2,1)→(2,2)→(3,2)→(3,3)→(3,4)(1,1)→(2,1)→(2,2)→(3,2)→(3,3)→(3,4);
• (1,1)→(2,1)→(2,2)→(2,3)→(2,4)→(3,4)(1,1)→(2,1)→(2,2)→(2,3)→(2,4)→(3,4);
• (1,1)→(1,2)→(2,2)→(2,3)→(3,3)→(3,4)(1,1)→(1,2)→(2,2)→(2,3)→(3,3)→(3,4);
• (1,1)→(1,2)→(1,3)→(2,3)→(3,3)→(3,4)(1,1)→(1,2)→(1,3)→(2,3)→(3,3)→(3,4).

题意：从$(1, 1)$走到$(n, m)$，路径上权值异或起来为$k$的有几条

昨晚前五题都1A之后有点上天qwq。。想了很久才发现这是个思博题不过没时间写了qwq。

考虑如果直接dfs的话是$2^{n + m}$

然后meet in the middle 一下就好了

#include<cstdio>
#include<cstring>
#include<algorithm>
#include<vector>
#include<queue>
#include<map>
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/hash_policy.hpp>
using namespace __gnu_pbds;
#define MP(x, y) make_pair(x, y)
#define Pair pair<int, int>
#define int long long
using namespace std;
const int MAXN  =  * 1e5 + , INF = 1e9 + ;
inline int read() {
    char c = getchar(); int x = , f = ;
    while(c < '' || c > '') {if(c == '-') f = -; c = getchar();}
    while(c >= '' && c <= '') x = x *  + c - '', c = getchar();
    return x * f;
}
int N, M, K;
int a[][];
cc_hash_table<int, int> mp[];
int dfs(int x, int y, int now) {
    if(x <  || x > N || y <  || y > M) return ;
    if(x + y == (N + M + ) / ) return mp[x][now ^ a[x][y]];
    int ans = ;
    ans += dfs(x - , y, now ^ a[x - ][y]);
    ans += dfs(x, y - , now ^ a[x][y - ]);
    return ans;
}
void fuck(int x, int y, int now) {
    if(x <  || x > N || y <  || y > M) return ;
    if(x + y == (N + M + ) / ) {mp[x][now]++; return ;}
    fuck(x + , y, now ^ a[x + ][y]);
    fuck(x, y + , now ^ a[x][y + ]);
}
main() {
#ifdef WIN32
    freopen("a.in", "r", stdin);
#endif
    N = read(); M = read(); K = read();
    for(int i = ; i <= N; i++)
        for(int j = ; j <= M; j++)
            a[i][j] = read();
    fuck(, , a[][]);
    printf("%lld", dfs(N, M, K ^ a[N][M]));
}
/*
1 1 1000000000000000000
1000000000000000000
*/