前缀和与差分
前缀和
一维前缀和
公式:
\[s[i] = s[i - 1] + a[i]
\]
模板:
const int N = 10000 + 10;
int n,m;
int a[N],s[N];
int main() {
scanf("%d%d",&n,&m);
for (int i = 1; i <= n; i++) {
scanf("%d",&a[i]);
s[i] = s[i - 1] + a[i];
}
while (m--) {
int l,r;
scanf("%d%d",&l,&r);
printf("%d\n",s[r] - s[l - 1]);
}
return 0;
}
二维前缀和
公式:
\[求前缀和:s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i][j]\\
算子子矩阵和:s[x2][y2] - s[x1 - 1][y2] - s[x2][y1 - 1] + s[x1 - 1][y1 - 1]
\]
模板:
const int N = 10000 + 10;
int n,m,q;
int a[N][N],s[N][N];
int main() {
scanf("%d%d%d",&n,&m,&q);
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
scanf("%d",&a[i][j]);
s[i][j] = s[i - 1][j] + s[i][j - 1] - s[i - 1][j - 1] + a[i][j];
}
}
while (q--) {
int x1,y1,x2,y2;
scanf("%d%d%d%d",&x1,&y1,&x2,&y2);
printf("%d\n",s[x2][y2] - s[x1 - 1][y2] - s[x2][y1 - 1] + s[x1 - 1][y1 - 1]);
}
return 0;
}
差分
一维差分
公式:
\[b[l]\ +=\ c\quad b[r + 1]\ -=\ c
\]
模板:
const int N = 10000 + 10;
int n,m;
int a[N],b[N];
void insert(int l,int r,int c) {
b[l] += c;
b[r + 1] -= c;
}
int main() {
scanf("%d%d",&n,&m);
for (int i = 1; i <= n; i++) {
scanf("%d",&a[i]);
insert(i,i,a[i]);
}
while (m--) {
int l,r,c;
scanf("%d%d%d",&l,&r,&c);
insert(l,r,c);
}
for (int i = 1; i <= n; i++) {
b[i] += b[i - 1];
printf("%d ",b[i]);
}
return 0;
}
二维差分
公式:
\[b[x1][y1]\ +=\ c\quad b[x1][y2 + 1]\ -=\ c\\
b[x2 + 1][y1]\ -=\ c\quad b[x2 + 1][y2 + 1]\ +=\ c
\]
模板:
const int N = 10000 + 10;
int n,m,q;
int a[N][N],b[N][N];
void insert(int x1,int y1,int x2,int y2,int c) {
b[x1][y1] += c;
b[x1][y2 + 1] -= c;
b[x2 + 1][y1] -= c;
b[x2 + 1][y2 + 1] += c;
}
int main() {
scanf("%d%d%d",&n,&m,&q);
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
scanf("%d",&a[i][j]);
insert(i,j,i,j,a[i][j]);
}
}
while (q--) {
int x1,y1,x2,y2,c;
cin >> x1 >> y1 >> x2 >> y2 >> c;
insert(x1,y1,x2,y2,c);
}
for (int i = 1; i <= n;i++) {
for (int j = 1; j <= m;j++) {
b[i][j] += b[i - 1][j] + b[i][j - 1] - b[i - 1][j - 1];
printf("%d ",b[i][j]);
}
puts("");
}
return 0;
}