Codeforces 341D Iahub and Xors (二維樹狀陣列 差分 推薦)
D. Iahub and Xors
time limit per test 1 second
memory limit per test 256 megabytes
Iahub does not like background stories, so he'll tell you exactly what this problem asks you for.
You are given a matrix a with n rows and n columns. Initially, all values of the matrix are zeros. Both rows and columns are 1-based, that is rows are numbered 1, 2, ..., n
We will call a submatrix (x0, y0, x1, y1) such elements ai, j for which two inequalities hold: x0 ≤ i ≤ x1, y0 ≤ j ≤ y1.
Write a program to perform two following operations:
- Query(x0, y0, x1, y1): print the xor sum of the elements of the submatrix (x0, y0, x1, y1).
- Update(x0, y0, x1, y1, v): each element from submatrix (x0, y0, x1, y1) gets xor-ed by value v.
Input
The first line contains two integers: n
If the i-th operation from the input is a query, the first number from i-th line will be 1. It will be followed by four integers x0, y0, x1, y1. If the i-th operation is an update, the first number from the i-th line will be 2. It will be followed by five integers x0, y0, x1, y1, v.
It is guaranteed that for each update operation, the following inequality holds: 0 ≤ v < 262. It is guaranteed that for each operation, the following inequalities hold: 1 ≤ x0 ≤ x1 ≤ n, 1 ≤ y0 ≤ y1 ≤ n.
Output
For each query operation, output on a new line the result.
Examples
input
Copy
3 5 2 1 1 2 2 1 2 1 3 2 3 2 2 3 1 3 3 3 1 2 2 3 3 1 2 2 3 2
output
Copy
3 2
Note
After the first 3 operations, the matrix will look like this:
1 1 2 1 1 2 3 3 3
The fourth operation asks us to compute 1 xor 2 xor 3 xor 3 = 3.
The fifth operation asks us to compute 1 xor 3 = 2.
題目連結:http://codeforces.com/contest/341/problem/D
題目大意:給一個n*n的矩陣,初始值全為0,有兩種操作,1表示查詢以(x0, y0)和(x1, y1)為對頂點的矩陣中所有值的異或和,2表示將以(x0, y0)和(x1, y1)為對頂點的矩陣中的所有值都異或v
題目分析:先考慮一維的情況,讓點i維護[1,i]的字首異或和,對於(l, r)這段區間
1)更新:通過後綴異或和的方式,更新l和r+1兩個點 =》update(l, v);update(r+1, v);
2)查詢:利用字首異或和(a^a=0) =》query(l-1)^query(r)
需要特別注意的一點是,異或操作與加法不同,a+a=2a而a^a=0,因此對於區間(l, r),更新時真正影響的其實只有l,l+2,l+4...這些,因此需要分奇偶討論,所以需要兩個樹狀陣列來維護。
將其拓展到二維即可
#include <cstdio>
#include <cstring>
#include <algorithm>
#define ll long long
using namespace std;
int const MAX = 1e3 + 5;
int n, m;
ll xorsum[2][2][MAX][MAX];
int lowbit(int x) {
return x & (-x);
}
void update(int x, int y, ll val) {
for (int i = x; i <= n; i += lowbit(i)) {
for (int j = y; j <= n; j += lowbit(j)) {
xorsum[x & 1][y & 1][i][j] ^= val;
}
}
}
ll query(int x, int y) {
ll ans = 0;
for (int i = x; i > 0; i -= lowbit(i)) {
for (int j = y; j > 0; j -= lowbit(j)) {
ans ^= xorsum[x & 1][y & 1][i][j];
}
}
return ans;
}
int main() {
int tp, x0, y0, x1, y1;
ll v;
scanf("%d %d", &n, &m);
while (m--) {
scanf("%d", &tp);
if (tp == 1) {
scanf("%d %d %d %d", &x0, &y0, &x1, &y1);
ll ans = 0;
ans ^= query(x0 - 1, y0 - 1);
ans ^= query(x0 - 1, y1);
ans ^= query(x1, y0 - 1);
ans ^= query(x1, y1);
printf("%I64d\n", ans);
} else {
scanf("%d %d %d %d %I64d", &x0, &y0, &x1, &y1, &v);
update(x0, y0, v);
update(x0, y1 + 1, v);
update(x1 + 1, y0, v);
update(x1 + 1, y1 + 1, v);
}
}
}