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一维习题：hdu1541 bzoj3211(hdu4027)
二维习题：hdu2642 1892 5517
一维树状数组：
struct FenwickTree {
int BIT[MXN];
int lowbit(int x) {return x&(-x);}
void add_bit(int x, int val, int N) {for(;x <= N; x += lowbit(x)) BIT[x] += val;}
int query_bit(int x) {int ans = 0; for(; x; x -= lowbit(x)) ans += BIT[x]; return ans;}
}bit;
改点求段：
void add(int x,int v){
while(x <= n){
ar[x] += v;
x += lowbit(x);
}
}
int query(int x){
int sum = 0;
while(x > 0){
sum += ar[x];
x -= lowbit(x);
}
return sum;
}
int range(int l, int r){
return query(r) - query(l-1);
}
改段求点：
void add(int x,int v){
while(x <= n){
delta[x] += v;
x += lowbit(x);
}
}
int query(int x){
int sum = 0;
while(x > 0){
sum += delta[x];
x -= lowbit(x);
}
return sum;
}
void init(){
for(int i=1;i<=n;++i){
scanf("%d", &ar[i]);
add(i, ar[i]-ar[i-1]);
}
}
int get_pos(int x){
return query(x);
}
改段求段：
//sum[i] = sigma(ar[x])+(i+1)*sigma(delta[x])-sigma(x*delta[x])
//delta[]是差分数组
void add(LL *a, int x, LL v){
while(x <= n){
a[x] += v;
x += lowbit(x);
}
}
LL query(LL *a, int x){
LL sum = 0;
while(x > 0){
sum += a[x];
x -= lowbit(x);
}
return sum;
}
void init(){
pre[0] = 0;
for(int i = 1; i <= n; ++i){
scanf("%lld", &ar[i]);
pre[i] = pre[i-1] + ar[i];
}
}
void update(int l, int r, LL x){
add(delta, l, x);add(delta, r+1, -x);
add(deltai, l, l*x);add(deltai, r+1, -x*(r+1));
}
LL range(int l, int r){
LL sum1 = pre[l-1]+l*query(delta, l-1)-query(deltai, l-1);
LL sum2 = pre[r]+(r+1)*query(delta, r)-query(deltai, r);
return sum2-sum1;
}
二维树状数组：
改点求段：
void add(int x, int y, int z){
int tmp = y;
while(x<=n){
y = tmp;
while(y<=n){
cw[x][y] += z, y += lowbit(y);
}
x += lowbit(x);
}
}
int query(int x, int y){
int res = 0, tmp = y;
while(x){
y = tmp;
while(y){
res += cw[x][y], y -= lowbit(y);
}
x -= lowbit(x);
}
return res;
}
改段求点：
//d[i][j]表示 a[i][j]与a[i−1][j]+a[i][j−1]−a[i−1][j−1]的差
//delta[][]是差分数组
void add(int x, int y, int z){
int tmp = y;
while(x <= n){
y = tmp;
while(y <= n){
delta[x][y] += z, y += lowbit(y);
}
x += lowbit(x);
}
}
void update(int xa,int ya,int xb,int yb,int z){
add(xa,ya,z);add(xa,yb+1,-z);add(xb+1,ya,-z);add(xb+1,yb+1,z);
}
int query(int x, int y){
int res = 0, tmp = y;
while(x){
y = tmp;
while(y){
res += delta[x][y], y -= lowbit(y);
}
x -= lowbit(x);
}
return res;
}
void init(){
for(int i = 1; i <= n; ++i){
for(int j = 1; j <= n; ++j){
int tmp = ar[i][j]-ar[i-1][j]-ar[i][j-1]+ar[i-1][j-1];
add(i,j,tmp);
}
}
}
改段求段：
sum[x][y] = (x+1)(y+1) (Sigma) (d[i][j]) - (y+1)(Sigma)(id[i][j]) - (x+1)(Sigma)(jd[i][j]) + (Sigma)(ij*d[i][j])
//sum[x][y] = (x+1)(y+1)sigma(d[i][j])-(y+1)sigma(i*d[i][j])-(x+1)sigma(j*d[i][j])+sigma(i*j*d[i][j])
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<assert.h>
#include<bitset>
#define lson rt<<1
#define rson rt<<1|1
#define lowbit(x) (x)&(-(x))
#define all(x) (x).begin(),(x).end()
using namespace std;
typedef long long LL;
const int INF = 0x3f3f3f3f;
const int N = (int)1e3 +107;
int ar[N][N], da[N][N], di[N][N], dj[N][N],dij[N][N];
int n, m, q;
//sumxy = (x+1)(y+1)sigma(d[i][j])-(y+1)sigma(i*d[i][j])-(x+1)sigma(j*d[i][j])+sigma(i*j*d[i][j])
void add(int x, int y, int z){
for(int i=x;i<=n;i+=lowbit(i)){
for(int j=y;j<=n;j+=lowbit(j)){
da[i][j] += z; di[i][j] += z*x; dj[i][j] += z*y; dij[i][j] += z*x*y;
}
}
}
void update(int xa,int ya,int xb,int yb,int z){
add(xa,ya,z);add(xa,yb+1,-z);add(xb+1,ya,-z);add(xb+1,yb+1,z);
}
int query(int x, int y){
int res = 0;
for(int i = x; i>0; i -= lowbit(i)){
for(int j = y; j>0; j -= lowbit(j)){
res += (x+1)*(y+1)*da[i][j] - (y+1)*di[i][j] - (x+1)*dj[i][j] + dij[i][j];
}
}
return res;
}
int ask(int xa,int ya,int xb,int yb){
return query(xb,yb)-query(xb,ya-1)-query(xa-1,yb)+query(xa-1,ya-1);
}
void init(){
for(int i = 1; i <= n; ++i){
for(int j = 1; j <= n; ++j){
int tmp = ar[i][j]-ar[i-1][j]-ar[i][j-1]+ar[i-1][j-1];
add(i,j,tmp);
//update(i,j,i,j,ar[i][j]);
}
}
}
int main(){
while(~scanf("%d", &n)){
memset(ar,0,sizeof(ar));
for(int i=1;i<=n;++i){
for(int j=1;j<=n;++j){
scanf("%d",&ar[i][j]);
}
}
init();
scanf("%d",&q);
while(q--){
int op,xa,xb,ya,yb,c;
scanf("%d%d%d%d%d",&op,&xa,&ya,&xb,&yb);
if(op==1){
scanf("%d",&c);
update(xa,ya,xb,yb,c);
}else{
printf("%d
", ask(xa,ya,xb,yb));
}
if(q<=0)break;
}
}
return 0;
}
习题答案：
HDU1892
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<assert.h>
#include<bitset>
#define lson rt<<1
#define rson rt<<1|1
#define lowbit(x) (x)&(-(x))
#define all(x) (x).begin(),(x).end()
using namespace std;
typedef long long LL;
const int INF = 0x3f3f3f3f;
const int N = (int)1e3 +107;
int ar[N][N], da[N][N], di[N][N], dj[N][N],dij[N][N];
int n, m, q;
//sumxy = (x+1)(y+1)sigma(d[i][j])-(y+1)sigma(i*d[i][j])-(x+1)sigma(j*d[i][j])+sigma(i*j*d[i][j])
void add(int x,int y,int c){
for(int i=x;i<=n;i+=lowbit(i)){
for(int j=y;j<=n;j+=lowbit(j)){
da[i][j]+=c;
}
}
}
int query(int x,int y){
int sum=0;
for(int i=x;i;i-=lowbit(i)){
for(int j=y;j;j-=lowbit(j)){
sum+=da[i][j];
}
}
return sum;
}
int ask(int xa,int ya,int xb,int yb){
return query(xb,yb)-query(xa-1,yb)-query(xb,ya-1)+query(xa-1,ya-1);
}
int main(){
int tim;
int tc=0;
scanf("%d",&tim);
while(tim--){
n=1002;
memset(da,0,sizeof(da));
for(int i=1;i<=n;++i){
for(int j=1;j<=n;++j){
add(i,j,1);
}
}
printf("Case %d:
", ++tc);
scanf("%d",&q);
while(q--){
char op[2];
int xa=0,xb=0,ya=0,yb=0,c;
scanf("%s",op);
if(op[0]=='S'){
scanf("%d%d%d%d",&xa,&ya,&xb,&yb);
xa++;ya++;xb++;yb++;
if(xa>xb)swap(xa,xb);
if(ya>yb)swap(ya,yb);
printf("%d
", ask(xa,ya,xb,yb));
}else if(op[0]=='A'){
scanf("%d%d%d",&xa,&ya,&c);
xa++;ya++;xb++;yb++;
add(xa,ya,c);
}else if(op[0]=='D'){
scanf("%d%d%d",&xa,&ya,&c);
xa++;ya++;xb++;yb++;
c=min(c,ask(xa,ya,xa,ya));
add(xa,ya,-c);
}else{
scanf("%d%d%d%d%d",&xa,&ya,&xb,&yb,&c);
xa++;ya++;xb++;yb++;
c=min(c,ask(xa,ya,xa,ya));
add(xa,ya,-c);
add(xb,yb,c);
}
}
}
return 0;
}
HDU2642:
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<assert.h>
#include<bitset>
#define lson rt<<1
#define rson rt<<1|1
#define lowbit(x) (x)&(-(x))
#define all(x) (x).begin(),(x).end()
using namespace std;
typedef long long LL;
const int INF = 0x3f3f3f3f;
const int N = (int)1e3 +107;
int ar[N][N], da[N][N], di[N][N], dj[N][N],dij[N][N];
int n, m, q;
//sumxy = (x+1)(y+1)sigma(d[i][j])-(y+1)sigma(i*d[i][j])-(x+1)sigma(j*d[i][j])+sigma(i*j*d[i][j])
void add(int x, int y, int z){
for(int i=x;i<=n;i+=lowbit(i)){
for(int j=y;j<=n;j+=lowbit(j)){
da[i][j] += z;
}
}
}
int query(int x, int y){
int res = 0;
for(int i = x; i>0; i -= lowbit(i)){
for(int j = y; j>0; j -= lowbit(j)){
res += da[i][j];
}
}
return res;
}
int ask(int xa,int ya,int xb,int yb){
return query(xb,yb)-query(xb,ya-1)-query(xa-1,yb)+query(xa-1,ya-1);
}
int main(){
while(~scanf("%d", &q)){
n=1001;
memset(da,0,sizeof(da));
memset(ar,0,sizeof(ar));
while(q--){
char op[2];
int xa,xb,ya,yb,c;
scanf("%s",op);
if(op[0]=='B'){
scanf("%d%d",&xa,&ya);
xa++;ya++;
if(ar[xa][ya]==0)add(xa,ya,1);
ar[xa][ya]=1;
}else if(op[0]=='D'){
scanf("%d%d",&xa,&ya);
xa++;ya++;
if(ar[xa][ya]==1)add(xa,ya,-1);
ar[xa][ya]=0;
}else{
scanf("%d%d",&xa,&xb);
scanf("%d%d",&ya,&yb);
xa++;ya++;
xb++;yb++;
if(xa>xb)swap(xa,xb);
if(ya>yb)swap(ya,yb);
printf("%d
", ask(xa,ya,xb,yb));
}
}
}
return 0;
}
hdu1541
#include<iostream>
#include<algorithm>
#include<cstring>
#include<cstdio>
using namespace std;
const int N=1e5+10;
int a[N],sum[N];
int lowbit(int x){return x&(-x);}
int Sum(int n){
int sum=0;
while(n>0){
sum+=a[n];
n-=lowbit(n);
}
return sum;
}
void add(int x){
while(x<=N){
++a[x];
x+=lowbit(x);
}
}
int main() {
int x,y,n;
while(~scanf("%d",&n)){
memset(a,0,sizeof(a));
memset(sum,0,sizeof(sum));
for(int i=0;i<n;++i){
scanf("%d %d",&x,&y);
//1 5 7 3 5
//2 6 8 4 6
//0 1 2 1 3
x++;
sum[Sum(x)]++;
add(x);
}
for(int i=0;i<n;++i){
printf("%d
",sum[i]);
}
}
return 0;
}
具体请访问这个博客：here
树状数组求区间最值
(bit[x]) 是区间([x-lowbit(x)+1, x])的最值
能转移到(x)的状态是 (x-2^0, x-2^1 ... x-2^k) 且 (2^k < lowbit(x))
若 (y - lowbit(y) >= x)，则(query(x,y) = max(bit[y], query(x, y-lowbit(y))))
若 (y - lowbit(y) < x)，则(query(x,y) = max(ar[y], query(x, y-1)))
#include<iostream>
#include<cstdio>
#include<assert.h>
#include<ctime>
#include<algorithm>
#include<cstring>
//#include<bits/stdc++.h>
#define lowbit(x) (x)&(-(x))
#define all(x) x.begin(),x.end()
#define iis std::ios::sync_with_stdio(false)
#define mme(a,b) memset((a),(b),sizeof((a)))
using namespace std;
typedef long long LL;
const int MXN = 2e5+7;
const int INF = 0x3f3f3f3f;
const int MOD = 1e9 + 7;
int n, m;
int ar[MXN], bit[MXN];
int lowbit(int x){return x & (-x);}
inline void mymax(int &a,int b){a = a > b? a: b;}
void update(int x){
while(x <= n){
bit[x] = ar[x];
int tmp = lowbit(x);
for(int i = 1; i < tmp; i <<= 1){
//bit[x] = max(bit[x], bit[x-i]);
mymax(bit[x], bit[x-i]);
}
x += lowbit(x);
}
}
int query(int x,int y){
int ans = 0;
while(y >= x){
//ans = max(ans, ar[y]);
mymax(ans, ar[y]);
--y;
for( ; y - lowbit(y) >= x; y -= lowbit(y)){
//ans = max(ans, bit[y]);
mymax(ans, bit[y]);
}
}
return ans;
}
int main(){
char op;
int a, b;
while(scanf("%d%d", &n, &m)!=EOF){
for(register int i = 0; i <= n; ++i)bit[i] = 0;
for(register int i = 1; i <= n; ++i){
scanf("%d", &ar[i]);
update(i);
}
while(m--){
scanf("%c", &op);
scanf("%c", &op);
scanf("%d%d", &a, &b);
if(op == 'U') {
ar[a] = b;
update(a);
}else {
a = query(a, b);
printf("%d
", a);
}
}
}
return 0;
}