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数学原理:
利用上面的性质,在树状数组的尾部插入数据,来建立一个树状数组
void push(int pos){ int i,lb = lowbit(pos); c[pos] = a[pos]; for(i=1;i<<=1){ c[pos] = max(c[pos],c[pos-i]); }}
void update(int pos,int v){ int i,lb; c[pos] = a[pos] = v; lb = lowbit(pos); for(i=1;i<<=1){ //利用孩子更新自己 c[pos] = c[pos] > c[pos-i] ? c[pos] : c[pos-i]; } int pre = c[pos]; pos+=lowbit(pos);//父亲的位置 /* 更新父亲 */ while(pos <= n){ if( c[pos] < pre){ //更新的父亲 c[pos] = pre; pos +=lowbit(pos); } //没有更新父亲 else break; } }
设 query(x,y)query(x,y) 求区间 [x,y] 之间的最值, 已知 c[x] 表示 [x−lowbit(x)+1,x] 之间的最值,那如何求区间 [x,y] 的最值呢?
我们不难发现:
所以,我们发现下面的规律,因为 y−lowbit(y)+1y−lowbit(y)+1 表示 c[y]c[y] 结点所管辖范围的最左边的点若
query(x,y)=max(c[y],query(x,y−lowbit(y)))query(x,y)=max(c[y],query(x,y−lowbit(y)));
query(x,y)=max(a[y],query(x,y−1))query(x,y)=max(a[y],query(x,y−1));
int query(int x,int y){ int res = -1; while(x <= y){ int nx = y - lowbit(y)+1; //最左边的点 if(nx >= x ){ res = res < c[y] ? c[y] :res; //判断是否最优 y = nx-1; // 下一个求解区间 } else { // nx < x res = res < a[y] ? a[y] :res; //判断是否最优 y--; } } return res;}
特点:
所以,树状数组求区间最值特别适合那些:一边在尾部添加数据,一边查询的题目
const int maxn = 1e6 + 5, maxe = 1e6 + 5; //点与边的数量int n, m;int N = maxn;int a[maxn], c[maxn]; // a是原数组inline int lowbit(int x) { return x & -x; }inline int fa(int p) { return p + lowbit(p); }inline int left(int p) { return p - lowbit(p); }inline int g(int a, int b) { return a>b ? a : b; }void update_by_child(int p, int v) { //alias push c[p] = a[p] = v; int lb = lowbit(p); for (int i = 1; i < lb; i <<= 1) c[p] = g(c[p], c[p - i]);}void update(int p, int v) { update_by_child(p, v); int t = c[p]; for (p = fa(p); p <= N; p = fa(p)) { if (g(t, c[p])) c[p] = t; else break; }}int query(int l, int r) { // 求区间最值 int ret = a[l]; for (; l <= r; ) { int next = left(r) + 1; if (next >= l) ret = g(ret, c[r]), r = next - 1; else ret = g(ret, a[r]), r--; } return ret;}
思路: 利用树状数组求区间最值
#include#include #include #include using namespace std;typedef long long ll;const int maxn = 1e6 + 5, maxe = 1e6 + 5; //点与边的数量int n, m;int N = maxn;int a[maxn], c[maxn]; // a是原数组inline int lowbit(int x) { return x & -x; }inline int fa(int p) { return p + lowbit(p); }inline int left(int p) { return p - lowbit(p); }inline int g(int a, int b) { return a>b ? a : b; }void update_by_child(int p, int v) { //alias push c[p] = a[p] = v; int lb = lowbit(p); for (int i = 1; i < lb; i <<= 1) c[p] = g(c[p], c[p - i]);}void update(int p, int v) { update_by_child(p, v); int t = c[p]; for (p = fa(p); p <= N; p = fa(p)) { if (g(t, c[p])) c[p] = t; else break; }}int query(int l, int r) { // 求区间最值 int ret = a[l]; for (; l <= r; ) { int next = left(r) + 1; if (next >= l) ret = g(ret, c[r]), r = next - 1; else ret = g(ret, a[r]), r--; } return ret;}int main() { while (1) { memset(a, 0, sizeof(a)); memset(c, 0, sizeof(c)); if (scanf("%d%d", &n, &m) == EOF) break; for (int i = 1; i <= n; ++i) { int t; scanf("%d", &t); update_by_child(i, t); //初始化原数组与树状数组 } char s[10]; for (int i = 1; i <= m; ++i) { scanf("%s", s); int x, y; if (s[0] == 'Q') { scanf("%d%d", &x, &y); ll ans = query(x, y); printf("%lld\n", ans); } else { scanf("%d%d", &x, &y); update(x, y); } } } return 0;}
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