可持久化线段树
数组一般开maxn<<5，但有的时候也会不够，不知道怎么判断得到的建议是“贴着内存开”。
最套路的应用就是各种形式的区间k小：
K小数
保存一下模板
code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
const int maxn = 2e5 + 3;
int n, m, a[maxn], b[maxn], rt[maxn], cnt, ans;
inline int read()
{
int x = 0, f = 1;
char ch = getchar();
while(ch < '0' || ch > '9')
{
if(ch == '-')
{
f = -1;
}
ch = getchar();
}
while(ch >= '0' && ch <= '9')
{
x = (x << 1) + (x << 3) + (ch^48);
ch = getchar();
}
return x * f;
}
struct seg
{
struct node
{
int sum;
}t[maxn<<5];
int lc[maxn<<5], rc[maxn<<5], tot;
void build(int &x, int l, int r)
{
x = ++tot;
if(l == r) return;
int mid = (l + r) >> 1;
build(lc[x], l, mid);
build(rc[x], mid+1, r);
}
void update(int &x, int pre, int l, int r, int pos)
{
x = ++tot; lc[x] = lc[pre]; rc[x] = rc[pre];
t[x].sum = t[pre].sum + 1;
if(l == r) return;
int mid = (l + r) >> 1;
if(pos <= mid) update(lc[x], lc[x], l, mid, pos);
else update(rc[x], rc[x], mid+1, r, pos);
}
int query(int pre, int x, int l, int r, int k)
{
if(l == r) return l;
int mid = (l + r) >> 1;
int res = t[lc[x]].sum - t[lc[pre]].sum;
if(res >= k) return query(lc[pre], lc[x], l, mid, k);
else return query(rc[pre], rc[x], mid+1, r, k-res);
}
}t;
int main()
{
n = read(); m = read();
for(int i=1; i<=n; i++)
{
a[i] = read(); b[i] = a[i];
}
sort(b+1, b+1+n);
cnt = unique(b+1, b+1+n)-b-1;
t.build(rt[0], 1, cnt);
for(int i=1; i<=n; i++)
{
int x = lower_bound(b+1, b+1+cnt, a[i])-b;
t.update(rt[i], rt[i-1], 1, cnt, x);
}
while(m--)
{
int l = read(), r = read(), k = read();
ans = t.query(rt[l-1], rt[r], 1, cnt, k);
printf("%d\n", b[ans]);
}
return 0;
}
花神的嘲讽计划
可以预处理特定长度的hash然后对存在性进行查找，直接定点前缀和相减、
code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
const int maxn = 1e5 + 3;
int n, m, k, cnt, rt[maxn];
ull p[maxn], h[maxn], a[maxn];
inline int read()
{
int x = 0, f = 1;
char ch = getchar();
while(ch < '0' || ch > '9')
{
if(ch == '-')
{
f = -1;
}
ch = getchar();
}
while(ch >= '0' && ch <= '9')
{
x = (x << 1) + (x << 3) + (ch^48);
ch = getchar();
}
return x * f;
}
struct seg
{
struct node
{
int sum;
}t[maxn<<5];
int lc[maxn<<5], rc[maxn<<5], tot;
void build(int &x, int l, int r)
{
x = ++tot;
if(l == r) return;
int mid = (l + r) >> 1;
build(lc[x], l, mid);
build(rc[x], mid+1, r);
}
void update(int &x, int pre, int l, int r, int pos)
{
x = ++tot; lc[x] = lc[pre], rc[x] = rc[pre];
t[x].sum = t[pre].sum + 1;
if(l == r) return;
int mid = (l + r) >> 1;
if(pos <= mid) update(lc[x], lc[x], l, mid, pos);
else update(rc[x], rc[x], mid+1, r, pos);
}
int query(int x, int pre, int l, int r, int pos)
{
if(l == r)
{
if(t[x].sum - t[pre].sum) return 1;
else return 0;
}
int mid = (l + r) >> 1;
if(pos <= mid) return query(lc[x], lc[pre], l, mid, pos);
else return query(rc[x], rc[pre], mid+1, r, pos);
}
}t;
int main()
{
n = read(); m = read(); k = read();
p[0] = 1;
for(int i=1; i<=k; i++) p[i] = p[i-1] * 131;
for(int i=1; i<=n; i++)
{
a[i] = read();
h[i] = h[i-1] * 131 + a[i];
if(i >= k) a[i] = h[i] - h[i-k] * p[k];
else a[i] = h[i];
}
for(int i=1; i<=n; i++) h[i] = a[i];
sort(h+1, h+1+n);
cnt = unique(h+1, h+1+n)-h-1;
t.build(rt[0], 1, cnt);
for(int i=1; i<=n; i++)
{
int x = lower_bound(h+1, h+1+cnt, a[i])-h;
t.update(rt[i], rt[i-1], 1, cnt, x);
}
while(m--)
{
int l = read(), r = read();
ull tmp = 0;
for(int i=1; i<=k; i++)
{
int x = read();
tmp = tmp * 131 + x;
}
int x = lower_bound(h+1, h+1+cnt, tmp)-h;
if(h[x] == tmp && t.query(rt[l+k-2], rt[r], 1, cnt, x)) printf("No\n");
else printf("Yes\n");
}
return 0;
}
森林
跟线段树分治和LCT什么的没有关系，那些动态图数据结构最主要的应用是删边。
如果把它放到了树上，发现它在数组上的时候前缀和相减是答案，在树上就可以使用树上差分的套路继续减法，可持久化树的每一个状态(时间)都是一些权值的累计，减掉需要减的东西就好，减谁都行。
路径不路径的，对建权值树没有影响，其它的，减法照常做就完了……
因为树链剖分涉及大小不方便修改，所以倍增求LCA，发现倍增LCA支持加边！！
code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
const int maxn = 8e4 + 1;
int dep[maxn], f[maxn][17], fa[maxn], son[maxn], cnt, root[maxn];
int T, n, m, q, a[maxn], ans, b[maxn];
bool vis[maxn];
char op[3];
inline int read()
{
int x = 0, f = 1;
char ch = getchar();
while(ch < '0' || ch > '9')
{
if(ch == '-')
{
f = -1;
}
ch = getchar();
}
while(ch >= '0' && ch <= '9')
{
x = (x << 1) + (x << 3) + (ch^48);
ch = getchar();
}
return x * f;
}
struct edge
{
int next, to;
}e[maxn<<2];
int head[maxn], len;
void add(int x, int y)
{
e[++len].to = y; e[len].next = head[x];
head[x] = len;
}
struct seg
{
struct node
{
int sum;
}t[maxn*600];
int lc[maxn*600], rc[maxn*600], tot;
void build(int &x, int l, int r)
{
x = ++tot;
if(l == r) return;
int mid = (l + r) >> 1;
build(lc[x], l, mid);
build(rc[x], mid+1, r);
}
void update(int &x, int pre, int l, int r, int pos)
{
x = ++tot; lc[x] = lc[pre]; rc[x] = rc[pre];
t[x].sum = t[pre].sum + 1;
if(l == r) return;
int mid = (l + r) >> 1;
if(pos <= mid) update(lc[x], lc[x], l, mid, pos);
else update(rc[x], rc[x], mid+1, r, pos);
}
//我又一次在query的时候忘了传入k-lsize，而是传了个k
int query(int x, int y, int pre1, int pre2, int l, int r, int k)
{
if(l == r) return b[l];
int lsize = t[lc[x]].sum + t[lc[y]].sum - t[lc[pre1]].sum - t[lc[pre2]].sum;
int mid = (l + r) >> 1;
if(lsize >= k) return query(lc[x], lc[y], lc[pre1], lc[pre2], l, mid, k);
else return query(rc[x], rc[y], rc[pre1], rc[pre2], mid+1, r, k-lsize);
}
}t;
int find(int x) {return fa[x] == x ? x : fa[x] = find(fa[x]);}
void dfs(int x, int fat, int rt)
{
f[x][0] = fat;
for(int i=1; i<=16; i++)
{
f[x][i] = f[f[x][i-1]][i-1];
}
son[rt]++;
dep[x] = dep[fat] + 1;
fa[x] = fat;
vis[x] = 1;
int k = lower_bound(b+1, b+1+cnt, a[x])-b;
t.update(root[x], root[fat], 1, cnt, k);
for(int i=head[x]; i; i=e[i].next)
{
int u = e[i].to;
if(u == fat) continue;
dfs(u, x, rt);
}
}
int LCA(int x, int y)
{
if(x == y) return x;
if(dep[x] > dep[y]) swap(x, y);
for(int i=16; i>=0; i--)
{
if(dep[f[y][i]] >= dep[x])
{
y = f[y][i];
}
}
if(x == y) return x;
for(int i=16; i>=0; i--)
{
if(f[x][i] != f[y][i])
{
x = f[x][i]; y = f[y][i];
}
}
return f[x][0];
}
int main()
{
read();
n = read(); m = read(); q = read();
for(int i=1; i<=n; i++)
{
a[i] = read(); b[i] = a[i]; fa[i] = i;
//这里的并查集为什么不直接染成rt？
}
sort(b+1, b+1+n);
cnt = unique(b+1, b+1+n)-b-1;
for(int i=1; i<=m; i++)
{
int x = read(), y = read();
add(x, y); add(y, x);
}
t.build(root[0], 1, cnt);
for(int i=1; i<=n; i++)
{
if(!vis[i])
{
dfs(i, 0, i); fa[i] = i;
}
}
while(q--)
{
scanf("%s", op);
int x = read()^ans, y = read()^ans;
if(op[0] == 'Q')
{
int k = read()^ans;
int lca = LCA(x, y);
ans = t.query(root[x], root[y], root[lca], root[f[lca][0]], 1, cnt, k);
printf("%d\n", ans);
}
else
{
add(x, y); add(y, x);
int u = find(x), v = find(y);
if(son[u] < son[v])
{
swap(u, v); swap(x, y);
}
dfs(y, x, u);
}
}
return 0;
}
P2633 Count on a tree
是上一题没有加边操作的简化版。
code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
const int maxn = 1e5 + 3;
int n, m, a[maxn], b[maxn], ans, cnt, f[maxn][18], root[maxn], dep[maxn];
inline int read()
{
int x = 0, f = 1;
char ch = getchar();
while(ch < '0' || ch > '9')
{
if(ch == '-')
{
f = -1;
}
ch = getchar();
}
while(ch >= '0' && ch <= '9')
{
x = (x << 1) + (x << 3) + (ch^48);
ch = getchar();
}
return x * f;
}
struct edge
{
int next, to;
}e[maxn<<1];
int head[maxn], len;
void add(int x, int y)
{
e[++len].to = y; e[len].next = head[x];
head[x] = len;
}
struct seg
{
struct node
{
int sum;
}t[maxn<<8];
int lc[maxn<<8], rc[maxn<<8], tot;
void build(int &x, int l, int r)
{
x = ++tot;
if(l == r) return;
int mid = (l + r) >> 1;
build(lc[x], l, mid);
build(rc[x], mid+1, r);
}
void update(int &x, int pre, int l, int r, int pos)
{
x = ++tot; lc[x] = lc[pre]; rc[x] = rc[pre];
t[x].sum = t[pre].sum + 1;
if(l == r) return;
int mid = (l + r) >> 1;
if(pos <= mid) update(lc[x], lc[x], l, mid, pos);
else update(rc[x], rc[x], mid+1, r, pos);
}
int query(int x, int y, int pre1, int pre2, int l, int r, int k)
{
if(l == r) return b[l];
int mid = (l + r) >> 1;
int lsize = t[lc[x]].sum + t[lc[y]].sum - t[lc[pre1]].sum - t[lc[pre2]].sum;
if(lsize >= k) return query(lc[x], lc[y], lc[pre1], lc[pre2], l, mid, k);
else return query(rc[x], rc[y], rc[pre1], rc[pre2], mid+1, r, k-lsize);
}
}t;
void dfs(int x, int fa)
{
f[x][0] = fa;
for(int i=1; i<=17; i++)
{
f[x][i] = f[f[x][i-1]][i-1];
}
dep[x] = dep[fa] + 1;
int k = lower_bound(b+1, b+1+cnt, a[x])-b;
t.update(root[x], root[fa], 1, cnt, k);
for(int i=head[x]; i; i=e[i].next)
{
int u = e[i].to;
if(u == fa) continue;
dfs(u, x);
}
}
int LCA(int x, int y)
{
if(x == y) return x;
if(dep[x] > dep[y]) swap(x, y);
for(int i=17; i>=0; i--)
{
if(dep[f[y][i]] >= dep[x])//>=居然被我写成了<=
{
y = f[y][i];
}
}
if(x == y) return x;
for(int i=17; i>=0; i--)
{
if(f[x][i] != f[y][i])
{
x = f[x][i]; y = f[y][i];
}
}
return f[x][0];
}
int main()
{
n = read(); m = read();
for(int i=1; i<=n; i++)
{
a[i] = read(); b[i] = a[i];
}
sort(b+1, b+1+n);
cnt = unique(b+1, b+1+n)-b-1;
t.build(root[0], 1, cnt);
for(int i=1; i<n; i++)
{
int x = read(), y = read();
add(x, y); add(y, x);
}
dfs(1, 0);
while(m--)
{
int u = read()^ans, v = read(), k = read();
int lca = LCA(u, v);
//经典树上差分
ans = t.query(root[u], root[v], root[lca], root[f[lca][0]], 1, cnt, k);
printf("%d\n", ans);
}
return 0;
}
疯狂的颜色序列
这种数颜色的好像也是常见的套路了，然而我老是忘……
记录每个位置的nxt，表示这个位置上的颜色下一次出现的位置，区间内出现的颜色种类就是nxt在右端点右边的颜色个数，这得到的是前缀和，还是以前缀和相减的形式。
code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
const int maxn = 5e5 + 3;
int n, m, a[maxn], pos[maxn], nxt[maxn], ans, rt[maxn];
inline int read()
{
int x = 0, f = 1;
char ch = getchar();
while(ch < '0' || ch > '9')
{
if(ch == '-')
{
f = -1;
}
ch = getchar();
}
while(ch >= '0' && ch <= '9')
{
x = (x << 1) + (x << 3) + (ch^48);
ch = getchar();
}
return x * f;
}
struct seg
{
struct node
{
int sum;
}t[maxn<<5];
int lc[maxn<<5], rc[maxn<<5], tot;
void update(int &x, int pre, int l, int r, int pos)
{
x = ++tot; lc[x] = lc[pre], rc[x] = rc[pre];
t[x].sum = t[pre].sum + 1;
if(l == r) return;
int mid = (l + r) >> 1;
if(pos <= mid) update(lc[x], lc[x], l, mid, pos);
else update(rc[x], rc[x], mid+1, r, pos);
}
//查询的不是单点，而是以这个点为左端点的后缀和！
int query(int pre, int x, int l, int r, int pos)
{
if(l == r)
{
return t[x].sum - t[pre].sum;
}
int mid = (l + r) >> 1;
if(pos <= mid) return query(lc[pre], lc[x], l, mid, pos)+t[rc[x]].sum-t[rc[pre]].sum;
else return query(rc[pre], rc[x], mid+1, r, pos);
}
}t;
int main()
{
n = read(); m = read();
for(int i=1; i<=n; i++)
{
a[i] = read();
}
for(int i=1; i<=n; i++) pos[a[i]] = n + 1;
for(int i=n; i>=1; i--)
{
nxt[i] = pos[a[i]];
pos[a[i]] = i;
}
for(int i=1; i<=n; i++)
{
t.update(rt[i], rt[i-1], 1, n+1, nxt[i]);
}
while(m--)
{
int l = read(), r = read();
l = (l + ans) % n + 1; r = (r + ans) % n + 1;
if(l > r) swap(l, r);
ans = t.query(rt[l-1], rt[r], 1, n+1, r+1);
printf("%d\n", ans);
}
return 0;
}
影魔
转化成平面点对，感觉这个是主席树的另一个应用类型了，就是有两个限制条件都是区间的形式，想到了二维前缀和，主席树的优点应该是省空间？
code
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int maxn = 2e5 + 7;
int n, st[maxn], top, p1, p2, a[maxn], L[maxn], R[maxn], m, root[maxn];
inline int read()
{
int x = 0, f = 1;
char ch = getchar();
while(ch < '0' || ch > '9')
{
if(ch == '-')
{
f = -1;
}
ch = getchar();
}
while(ch >= '0' && ch <= '9')
{
x = (x << 1) + (x << 3) + (ch^48);
ch = getchar();
}
return x * f;
}
struct node
{
int l, r, w;
};
struct edge
{
int next; node to;
}e[maxn<<2];
int head[maxn], len;
void add(int x, node y)
{
e[++len].to = y; e[len].next = head[x];
head[x] = len;
}
struct seg
{
struct tree
{
ll sum, tag;
}t[maxn<<5];
int lc[maxn<<5], rc[maxn<<5], tot;
void update(int &x, int l, int r, int L, int R, int w)
{
if(!x) x = ++tot;
t[x].sum += 1ll * (min(R,r)-max(L,l)+1) * w;
if(L <= l && r <= R)
{
t[x].tag += 1ll * w; return;
}
int mid = (l + r) >> 1;
if(L <= mid) update(lc[x], l, mid, L, R, w);
if(R > mid) update(rc[x], mid+1, r, L, R, w);
}
int merge(int x, int y)
{
if(!x || !y) return x+y;
t[x].sum += t[y].sum; t[x].tag += t[y].tag;
lc[x] = merge(lc[x], lc[y]);
rc[x] = merge(rc[x], rc[y]);
return x;
}
ll query(int x, int l, int r, int L, int R)
{
if(!x) return 0;
if(L <= l && r <= R) return t[x].sum;
int mid = (l + r) >> 1; ll res = 0;
if(L <= mid) res += query(lc[x], l, mid, L, R);
if(R > mid) res += query(rc[x], mid+1, r, L, R);
return res + 1ll * (min(R,r)-max(L,l)+1) * t[x].tag;
}
}t;
int main()
{
n = read(); m = read(); p1 = read(); p2 = read();
for(int i=1; i<=n; i++)
{
a[i] = read();
while(top && a[st[top]] < a[i]) R[st[top--]] = i;
L[i] = st[top];
st[++top] = i;
}
while(top) R[st[top--]] = n + 1;
for(int i=1; i<=n; i++)
{
if(i != n) add(i, (node){i+1, i+1, p1});
if(L[i] && R[i] <= n) add(L[i], (node){R[i], R[i], p1});
if(L[i] && i+1<=R[i]-1) add(L[i], (node){i+1, R[i]-1, p2});
if(R[i]<=n && L[i]+1<=i-1) add(R[i], (node){L[i]+1, i-1, p2});
}
for(int i=1; i<=n; i++)
{
for(int j=head[i]; j; j=e[j].next)
{
node v = e[j].to;
int l = v.l, r = v.r, w = v.w;
t.update(root[i], 1, n, l, r, w);
}
root[i] = t.merge(root[i], root[i-1]);
}
for(int i=1; i<=m; i++)
{
int l = read(), r = read();
printf("%lld\n", t.query(root[r], 1, n, l, r)-t.query(root[l-1], 1, n, l, r));
}
return 0;
}
时光花火，水月星辰
