标签：树状 sum 二维 数组 基本操作 ll getchar
单点修改区间查询
根据二维前缀和的思想对普通树状数组优化：
const int N = 4106;
inline ll Read()
{
ll x = 0, f = 1;
char c = getchar();
while (c != '-' && (c < '0' || c > '9')) c = getchar();
if (c == '-') f = -f, c = getchar();
while (c >= '0' && c <= '9') x = (x << 3) + (x << 1) + c - '0', c = getchar();
return x * f;
}
int n, m;
ll t[N][N];
void modify(int x, int y, ll val)
{
for (int i = x; i <= n; i += i & -i)
for (int j = y; j <= m; j += j & -j)
t[i][j] += val;
}
ll query(int x, int y)
{
ll ans = 0;
for (int i = x; i; i -= i & -i)
for (int j = y; j; j -= j & -j)
ans += t[i][j];
return ans;
}
int main()
{
n = Read(), m = Read();
for (int op, a, b, c, d; scanf ("%d", &op) != EOF; )
{
if(op == 2) a = Read(), b = Read(), c = Read(), d = Read(),
printf ("%lld\n", query(c, d) - query(a - 1, d) - query(c, b - 1) + query(a - 1, b - 1));
else a = Read(), b = Read(), c = Read(), modify(a, b, c);
}
return 0;
}
单点修改区间查询
一般区间修改的树状数组维护的都是差分数组，那么二维的也应该维护二维差分数组，接着是求和：
\[\begin{aligned}&\sum_{x=1}^{a}\sum_{y=1}^{b}\sum_{i=1}^{x}\sum_{j=1}^{y}t_{i,j}\\ =&\sum_{i=1}^{a}\sum_{j=1}^{b}(a-i+1)(b-j+1)t_{i,j}\\ =&(a+1)(b+1)\sum_{i=1}^{a}\sum_{j=1}^{b}t_{i,j}-\\ &(b+1)\sum_{i=1}^{a}\sum_{j=1}^{b}t_{i,j}\cdot i-\\ &(a+1)\sum_{i=1}^{a}\sum_{j=1}^{b}t_{i,j}\cdot j+\\ &\sum_{i=1}^{a}\sum_{j=1}^{b}t_{i,j}\cdot ij\end{aligned}\]const int N = 4106;
inline ll Read()
{
ll x = 0, f = 1;
char c = getchar();
while (c != '-' && (c < '0' || c > '9')) c = getchar();
if (c == '-') f = -f, c = getchar();
while (c >= '0' && c <= '9') x = (x << 3) + (x << 1) + c - '0', c = getchar();
return x * f;
}
int n, m;
ll t[4][N][N];
void modify(int x, int y, ll val)
{
for (int i = x; i <= n; i += i & -i)
for (int j = y; j <= m; j += j & -j)
t[0][i][j] += val,
t[1][i][j] += val * x,
t[2][i][j] += val * y,
t[3][i][j] += val * x * y;
}
ll query(int x, int y)
{
ll ans = 0;
for (int i = x; i; i -= i & -i)
for (int j = y; j; j -= j & -j)
ans += (x + 1) * (y + 1) * t[0][i][j] -
(y + 1) * t[1][i][j] -
(x + 1) * t[2][i][j] +
t[3][i][j];
return ans;
}
int main()
{
n = Read(), m = Read();
for (int op, a, b, c, d, k; scanf ("%d", &op) != EOF; )
{
if(op == 2) a = Read(), b = Read(), c = Read(), d = Read(),
printf ("%lld\n", query(c, d) - query(a - 1, d) - query(c, b - 1) + query(a - 1, b - 1));
else a = Read(), b = Read(), c = Read(), d = Read(), k = Read(),
modify(a, b, k), modify(a, d + 1, -k), modify(c + 1, b, -k), modify(c + 1, d + 1, k);
}
return 0;
}
标签：树状,sum,二维,数组,基本操作,ll,getchar 来源： https://www.cnblogs.com/GJY-JURUO/p/14876365.html
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