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
3 3 112 1 57 10 012 6 4
Output
3
Input
3 4 21 3 3 30 3 3 23 0 1 1
Output
5
Input
3 4 10000000000000000001 3 3 30 3 3 23 0 1 1
Output
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) .
题目:G - Xor-Paths
思路:
折半思想,前一半:从位置(1,1)开始到x+y=(n+m)/2,可以看成函数,进行异或。
后一半:从(n,m)开始,一直到x+y=(n+m)/2,异或。
异或有交换律,还有其他运算法则。
由运算法则可以推出公式
k=a1^a2^...^an;
令i=(1+n)/2;
q=a1^a2^...^ai;
sum=ai^...^an;
k=sum^ai^q;
所以q=sum^ai^k;
然后你第一个函数在走到的点给sum值打个标记,然后第二个函数把sum^ai就是取消这一步的值(这个值在前一个函数)
再^k就是找到跟现在这个sum异或起来为k(也就是合成一条路径)标记的值,也就是方案有多少
#include <iostream> #include <stdio.h> #include <stdlib.h> #include <string.h> #include <map> #include <algorithm> using namespace std; typedef long long ll; const int maxn=25; ll a[maxn][maxn],ans,k; map<ll,ll>mp[maxn]; int dx[4]={0,1,0,-1}; int dy[4]={1,0,-1,0}; int n,m; void dfs_pre(int x,int y,ll sum) { if(x+y==(n+m+2)/2) {mp[x][sum]++;return ;}//x+y==(n+m+2)/2,有个+2是因为有两种特殊情况,一个是n=1,一个是m=1 for(int i=0;i<2;i++) { int tx=x+dx[i]; int ty=y+dy[i]; if(tx<1||ty<1||tx>n||ty>m) continue; dfs_pre(tx,ty,sum^a[tx][ty]); } } void dfs_end(int x,int y,ll sum) { if(x+y==(n+m+2)/2) {ans+=mp[x][sum^k^a[x][y]];return ;} for(int i=2;i<4;i++) { int tx=x+dx[i]; int ty=y+dy[i]; if(tx<1||ty<1||tx>n||ty>m) continue; dfs_end(tx,ty,sum^a[tx][ty]); } } int main() { scanf("%d%d%I64d",&n,&m,&k); for(int i=1;i<=n;i++) { for(int j=1;j<=m;j++) { scanf("%I64d",&a[i][j]); } } dfs_pre(1,1,a[1][1]); dfs_end(n,m,a[n][m]); printf("%I64d\n",ans); return 0; }
原文地址:https://www.cnblogs.com/EchoZQN/p/10351453.html