目录
NOIP知识点&&模板整理
数据结构
线段树1
#include<bits/stdc++.h>
using namespace std;
const int N = 100005;
typedef long long ll;
ll read()
{
char c = getchar(), last = ' ';
ll ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
ll a[N];
namespace seg
{
ll sum[N << 2], lazy[N << 2];
void pushup(int cnt)
{
sum[cnt] = sum[cnt << 1] + sum[cnt << 1 | 1];
}
void pushdown(int cnt, int l, int r)
{
int mid = l + r >> 1;
//sum[cnt] += lazy[cnt] * (r - l + 1);
sum[cnt << 1] += lazy[cnt] * (mid - l + 1);
sum[cnt << 1 | 1] += lazy[cnt] * (r - mid);
lazy[cnt << 1] += lazy[cnt];
lazy[cnt << 1 | 1] += lazy[cnt];
lazy[cnt] = 0;
}
void build(int cnt, int l, int r)
{
if(l == r)
{
sum[cnt] = a[l];
return;
}
int mid = l + r >> 1;
build(cnt << 1, l, mid);
build(cnt << 1 | 1, mid + 1, r);
pushup(cnt);
}
void add(int cnt, int l, int r, int nl, int nr, ll k)
{
if(l >= nl && r <= nr)
{
sum[cnt] += (r - l + 1) * k;
lazy[cnt] += k;
return;
}
int mid = l + r >> 1;
pushdown(cnt, l, r);
if(nl <= mid) add(cnt << 1, l, mid, nl, nr, k);
if(nr > mid) add(cnt << 1 | 1, mid + 1, r, nl, nr, k);
pushup(cnt);
}
ll query(int cnt, int l, int r, int nl, int nr)
{
if(l >= nl && r <= nr) return sum[cnt];
int mid = l + r >> 1;
ll ans = 0;
pushdown(cnt, l, r);
if(nl <= mid) ans += query(cnt << 1, l, mid, nl, nr);
if(nr > mid) ans += query(cnt << 1 | 1, mid + 1, r, nl, nr);
return ans;
}
}
int n, m;
int main()
{
scanf("%d%d", &n, &m);
for(int i = 1; i <= n; i ++) a[i] = read();
seg::build(1, 1, n);
int opt, x, y;
ll k, ans = 0;
for(int i = 1; i <= m; i ++)
{
opt = read(), x = read(), y = read();
if(opt == 1)
{
k = read();
seg::add(1, 1, n, x, y, k);
}
else if(opt == 2)
{
ans = seg::query(1, 1, n, x, y);
printf("%lld
", ans);
}
}
}
线段树2
#include<bits/stdc++.h>
using namespace std;
const int N = 100005;
typedef long long ll;
ll read()
{
char c = getchar(), last = ' ';
ll ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
int n, m, p;
ll a[N];
namespace seg
{
ll sum[N << 2], lazy1[N << 2], lazy2[N << 2];
void pushup(int cnt)
{
sum[cnt] = (sum[cnt << 1] + sum[cnt << 1 | 1]) % p;
}
void pushdown(int cnt, int l, int r)
{
int mid = l + r >> 1;
lazy1[cnt << 1] = ((lazy1[cnt << 1] * lazy2[cnt] % p) + lazy1[cnt] % p) % p;
lazy1[cnt << 1 | 1] = ((lazy1[cnt << 1 | 1] * lazy2[cnt] % p) + lazy1[cnt]) % p;
lazy2[cnt << 1] = lazy2[cnt << 1] * lazy2[cnt] % p;
lazy2[cnt << 1 | 1] = lazy2[cnt << 1 | 1] * lazy2[cnt] % p;
sum[cnt << 1] = ((lazy1[cnt] * (mid - l + 1) % p) + lazy2[cnt] * sum[cnt << 1] % p) % p;
sum[cnt << 1 | 1] = ((lazy1[cnt] * (r - mid) % p) + lazy2[cnt] * sum[cnt << 1 | 1] % p) % p;
lazy1[cnt] = 0;
lazy2[cnt] = 1;
}
void build(int cnt, int l, int r)
{
lazy2[cnt] = 1;
if(l == r)
{
sum[cnt] = a[l] % p;
return;
}
int mid = l + r >> 1;
build(cnt << 1, l, mid);
build(cnt << 1 | 1, mid + 1, r);
pushup(cnt);
}
void add1(int cnt, int l, int r, int nl, int nr, ll k)
{
if(l >= nl && r <= nr)
{
sum[cnt] = (sum[cnt] + (r - l + 1) * k % p) % p;
lazy1[cnt] = (lazy1[cnt] + k % p) % p;
return;
}
int mid = l + r >> 1;
pushdown(cnt, l, r);
if(nl <= mid) add1(cnt << 1, l, mid, nl, nr, k);
if(nr > mid) add1(cnt << 1 | 1, mid + 1, r, nl, nr, k);
pushup(cnt);
}
void add2(int cnt, int l, int r, int nl, int nr, ll k)
{
if(l >= nl && r <= nr)
{
sum[cnt] = sum[cnt] * k % p;
lazy1[cnt] = lazy1[cnt] * k % p;
lazy2[cnt] = lazy2[cnt] * k % p;
return;
}
int mid = l + r >> 1;
pushdown(cnt, l, r);
if(nl <= mid) add2(cnt << 1, l, mid, nl, nr, k);
if(nr > mid) add2(cnt << 1 | 1, mid + 1, r, nl, nr, k);
pushup(cnt);
}
ll query(int cnt, int l, int r, int nl, int nr)
{
if(l >= nl && r <= nr) return sum[cnt] % p;
int mid = l + r >> 1;
ll ans = 0;
pushdown(cnt, l, r);
if(nl <= mid) ans += query(cnt << 1, l, mid, nl, nr);
if(nr > mid) ans += query(cnt << 1 | 1, mid + 1, r, nl, nr);
return ans;
}
}
int main()
{
scanf("%d%d%d", &n, &m, &p);
for(int i = 1; i <= n; i ++) a[i] = read();
seg::build(1, 1, n);
int opt, x, y;
ll k, ans = 0;
for(int i = 1; i <= m; i ++)
{
opt = read(), x = read(), y = read();
if(opt == 1)
{
k = read();
seg::add2(1, 1, n, x, y, k);
}
else if(opt == 2)
{
k = read();
seg::add1(1, 1, n, x, y, k);
}
else if(opt == 3)
{
ans = seg::query(1, 1, n, x, y);
printf("%lld
", ans % p);
}
}
}
树状数组1
#include<bits/stdc++.h>
using namespace std;
const int N = 500005;
typedef long long ll;
ll read()
{
char c = getchar(), last = ' ';
ll ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
int n, m;
ll a[N];
namespace bit
{
ll s[N];
int low(int x){return x & -x;}
void add(int x, ll k){for(; x <= n; x += low(x)) s[x] += k;}
ll query(int x){ll ans = 0; for(; x; x -= low(x)) ans += s[x]; return ans;}
}
int main()
{
scanf("%d%d", &n, &m);
for(int i = 1; i <= n; i ++) bit::add(i, read());
ll opt, x, y;
for(int i = 1; i <= m; i ++)
{
opt = read(), x = read(), y = read();
if(opt == 1) bit::add(x, y);
else if(opt == 2) printf("%lld
", bit::query(y) - bit::query(x - 1));
}
}
树状数组2
#include<bits/stdc++.h>
using namespace std;
const int N = 500005;
typedef long long ll;
ll read()
{
char c = getchar(), last = ' ';
ll ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
int n, m;
ll a[N];
namespace bit
{
ll s[N];
int low(int x){return x & -x;}
void add(int x, ll k){for(; x <= n; x += low(x)) s[x] += k;}
ll query(int x){ll ans = 0; for(; x; x -= low(x)) ans += s[x]; return ans;}
}
int main()
{
scanf("%d%d", &n, &m);
for(int i = 1; i <= n; i ++) a[i] = read();
ll opt, x, y, k;
for(int i = 1; i <= m; i ++)
{
opt = read(), x = read();
if(opt == 1)
{
y = read(), k = read();
bit::add(x, k);
bit::add(y + 1, -k);
}
else if(opt == 2) printf("%lld
", bit::query(x) + a[x]);
}
}
st表
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 100005;
int n, m;
int st[N][21], Log[N];
int find(int x, int y)
{
int t = y - x + 1;
return max(st[x][Log[t]], st[y - (1 << Log[t]) + 1][Log[t]]);
}
int main()
{
scanf("%d%d", &n, &m);
for(int i = 1; (1 << i) < N; i ++) Log[1 << i] = i;
for(int i = 1; i <= n; i ++) if(Log[i] == 0) Log[i] = Log[i - 1];
for(int i = 1; i <= n; i ++) st[i][0] = read();
for(int i = 1; (1 << i) <= N; i ++)
for(int j = 1; (j + (1 << i - 1)) <= n; j ++)
st[j][i] = max(st[j][i - 1], st[j + (1 << (i - 1))][i - 1]);
for(int i = 1, x, y; i <= m; i ++)
{
x = read(), y = read();
printf("%d
", find(x, y));
}
}
单调队列（滑动窗口）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 1000005;
int n, k, a[N];
int q[N], l, r;
int main()
{
n = read(), k =read();
for(int i = 1; i <= n; i ++) a[i] = read();
for(int i = 1; i <= n; i ++)
{
while(l <= r && i - q[l] + 1 > k) l ++;
while(l <= r && a[q[r]] >= a[i]) r --;
q[++r] = i;
if(i >= k)
printf("%d ", a[q[l]]);
}
printf("
");
l = 0, r = 0;
for(int i = 1; i <= n; i ++)
{
while(l <= r && i - q[l] + 1 > k) l ++;
while(l <= r && a[q[r]] <= a[i]) r --;
q[++r] = i;
if(i >= k)
printf("%d ", a[q[l]]);
}
}
平衡树（fhq treap）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 1100005;
int read()
{
int ans = 0;
char c = getchar(), last = ' ';
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = (ans << 1) + (ans << 3) + c - '0', c = getchar();
if(last == '-') ans = - ans;
return ans;
}
int ecnt;
struct tree
{
int l, r, siz, rad, val;
}t[N<<2];
unsigned long long seed = 1;
int Rand()
{
seed *= 260817;
return int(seed);
}
void update(int cnt)
{
t[cnt].siz = t[t[cnt].l].siz + t[t[cnt].r].siz + 1;
}
int New(int x)
{
t[++ecnt].val = x;
t[ecnt].rad = Rand();
t[ecnt].siz = 1;
return ecnt;
}
void split(int cnt, int k, int &x, int &y)
{
if(!cnt)
{
x = y = 0;
return;
}
if(t[cnt].val <= k)
{
x = cnt;
split(t[cnt].r, k, t[cnt].r, y);
}
if(t[cnt].val > k)
{
y = cnt;
split(t[cnt].l, k, x, t[cnt].l);
}
update(cnt);
}
int merge(int x, int y)
{
if(x == 0) return y;
if(y == 0) return x;
if(t[x].rad <= t[y].rad)
{
t[x].r = merge(t[x].r, y);
update(x);
return x;
}
else if(t[x].rad > t[y].rad)
{
t[y].l = merge(x, t[y].l);
update(y);
return y;
}
}
int kth(int cnt, int k)
{
if(t[t[cnt].l].siz + 1 == k) return t[cnt].val;
if(t[t[cnt].l].siz >= k) return kth(t[cnt].l, k);
else return kth(t[cnt].r, k - t[t[cnt].l].siz - 1);
}
int n, m, rt;
int last = 0, ans = 0;
int main()
{
//freopen("1.in", "r", stdin);
//freopen("1.out", "w", stdout);
//srand(time(0));
scanf("%d%d", &n, &m);
int x, y, z, k, opt;
for(int i = 1; i <= n; i ++)
{
k = read();
split(rt, k, x, y);
rt = merge(merge(x, New(k)), y);
}
for(int i = 1; i <= m; i ++)
{
opt = read(), k = read();
k ^= last;
if(opt == 1)
{
split(rt, k, x, y);
rt = merge(merge(x, New(k)), y);
}
if(opt == 2)
{
split(rt, k, x, y);
split(x, k - 1, x, z);
z = merge(t[z].l, t[z].r);
rt = merge(merge(x, z), y);
}
if(opt == 3)
{
split(rt, k - 1, x, y);
last = t[x].siz + 1;
ans ^= last;
//printf("%lld
", last);
rt = merge(x, y);
}
if(opt == 4)
{
last = kth(rt, k);
ans ^= last;
//printf("%lld
", last);
}
if(opt == 5)
{
split(rt, k - 1, x, y);
last = kth(x, t[x].siz);
ans ^= last;
//printf("%lld
", last);
rt = merge(x, y);
}
if(opt == 6)
{
split(rt, k, x, y);
last = kth(y, 1);
ans ^= last;
//printf("%lld
", last);
rt = merge(x, y);
}
}
printf("%d", ans);
}
图论
最小生成树
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 5005, M = 200005;
int n, m, fa[N], ans;
int ecnt;
struct edge
{
int to, dis, from;
}edg[M << 1];
int find(int x)
{
while(x != fa[x]) x = fa[x] = fa[fa[x]];
return x;
}
bool cmp(edge x, edge y)
{
return x.dis < y.dis;
}
void kruscal()
{
sort(edg + 1, edg + 1 + m, cmp);
int cnt = 0;
for(int i = 1; i <= m; i ++)
{
int u = edg[i].from, v = edg[i].to;
if(find(u) == find(v)) continue;
cnt ++;
ans += edg[i].dis;
fa[find(v)] = find(u);
if(cnt == n - 1) break;
}
}
int main()
{
n = read(), m = read();
for(int i = 1; i <= n; i ++) fa[i] = i;
for(int i = 1; i <= m; i ++)
{
edg[++ecnt].from = read();
edg[ecnt].to = read();
edg[ecnt].dis = read();
}
kruscal();
printf("%d", ans);
}
单源最短路（dijkstra）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 100005;
int head[N], ecnt;
struct edge
{
int to, dis, nxt;
}edg[N << 2];
void add(int u, int v, int w)
{
edg[++ecnt].to = v;
edg[ecnt].dis = w;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, m, s;
int dis[N], vis[N];
priority_queue<pair<int, int>, vector<pair<int, int> >, greater<pair<int, int> > > q;
void dij()
{
memset(dis, 0x3f, sizeof(dis));
memset(vis, 0, sizeof(vis));
dis[s] = 0;
q.push(make_pair(0, s));
while(!q.empty())
{
int u = q.top().second;
q.pop();
if(vis[u]) continue;
vis[u] = 1;
for(int i = head[u]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(dis[v] > dis[u] + edg[i].dis)
{
dis[v] = dis[u] + edg[i].dis;
q.push(make_pair(dis[v], v));
}
}
}
}
int main()
{
n = read(), m = read(), s = read();
for(int i = 1, u, v, w; i <= m; i ++)
{
u = read(), v = read(), w = read();
add(u, v, w);
}
dij();
for(int i = 1; i <= n; i ++) printf("%d ", dis[i]);
}
单源最短路（SPFA）
#include<bits/stdc++.h>
using namespace std;
inline int read()
{
int ans=0;
char last=' ',ch=getchar();
while(ch<'0'||ch>'9') last=ch,ch=getchar();
while(ch>='0'&&ch<='9') ans=(ans<<3)+(ans<<1)+ch-'0',ch=getchar();
if(last=='-')ans=-ans;
return ans;
}
const int N = 10005, M = 500005;
int head[N], ecnt;
struct edge
{
int to, dis, nxt;
}edg[M];
void add(int u, int v, int w)
{
edg[++ecnt].to = v;
edg[ecnt].dis = w;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, m, s;
queue<int> q;
int dis[N], in[N];
void spfa()
{
memset(dis, 0x3f, sizeof(dis));
q.push(s);
dis[s] = 0;
in[s] = 1;
while(!q.empty())
{
int u = q.front();
q.pop();
in[u] = 0;
for(int i = head[u]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(dis[v] > dis[u] + edg[i].dis)
{
dis[v] = dis[u] + edg[i].dis;
if(!in[v]) q.push(v), in[v] = 1;
}
}
}
}
int main()
{
n = read(), m = read(), s = read();
for(int i = 1, u, v, w; i <= m; i ++)
{
u = read(), v = read(), w = read();
add(u, v, w);
}
spfa();
for(int i = 1; i <= n; i ++)
{
if(dis[i] == 0x3f3f3f3f) printf("%d ", (1 << 31) - 1);
else printf("%d ", dis[i]);
}
}
负环
#include<bits/stdc++.h>
using namespace std;
inline int read()
{
int ans=0;
char last=' ',ch=getchar();
while(ch<'0'||ch>'9') last=ch,ch=getchar();
while(ch>='0'&&ch<='9') ans=(ans<<3)+(ans<<1)+ch-'0',ch=getchar();
if(last=='-')ans=-ans;
return ans;
}
const int N = 2005;
int head[N], ecnt;
struct edge
{
int to, dis, nxt;
}edg[N << 2];
void add(int u, int v, int w)
{
edg[++ecnt].to = v;
edg[ecnt].dis = w;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, m;
queue<int> q;
int dis[N], vis[N], in[N];
bool spfa()
{
memset(dis, 0x3f, sizeof(dis));
memset(vis, 0, sizeof(vis));
memset(in, 0, sizeof(in));
q.push(1);
dis[1] = 0;
in[1] = 1;
while(!q.empty())
{
int u = q.front();
q.pop();
in[u] = 0;
if(vis[u] == n - 1) return 1;
vis[u] ++;
for(int i = head[u]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(dis[v] > dis[u] + edg[i].dis)
{
dis[v] = dis[u] + edg[i].dis;
if(!in[v]) q.push(v), in[v] = 1;
}
}
}
return 0;
}
int t;
int main()
{
t = read();
while(t --)
{
ecnt = 0;
memset(head, 0, sizeof(head));
n = read(), m = read();
for(int i = 1, u, v, w; i <= m; i ++)
{
u = read(), v = read(), w = read();
if(w >= 0) add(u, v, w), add(v, u, w);
else add(u, v, w);
}
if(spfa()) printf("YES
");
else printf("NO
");
}
}
差分约束
#include<bits/stdc++.h>
using namespace std;
inline int read()
{
int ans=0;
char last=' ',ch=getchar();
while(ch<'0'||ch>'9') last=ch,ch=getchar();
while(ch>='0'&&ch<='9') ans=(ans<<3)+(ans<<1)+ch-'0',ch=getchar();
if(last=='-')ans=-ans;
return ans;
}
const int N = 100005;
int head[N], ecnt;
struct edge
{
int to, dis, nxt;
}edg[N << 2];
void add(int u, int v, int w)
{
edg[++ecnt].to = v;
edg[ecnt].dis = w;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, m;
queue<int> q;
int dis[N], vis[N], in[N];
bool spfa()
{
q.push(0);
in[0] = 1;
while(!q.empty())
{
int u = q.front();
q.pop();
in[u] = 0;
if(vis[u] == n - 1) return 1;
vis[u] ++;
for(int i = head[u]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(dis[v] < dis[u] + edg[i].dis)
{
dis[v] = dis[u] + edg[i].dis;
if(!in[v]) q.push(v), in[v] = 1;
}
}
}
return 0;
}
int main()
{
n = read(), m = read();
for(int i = 1; i <= n; i ++) add(0, i, 1);
for(int i = 1, u, v, w; i <= m; i ++)
{
u = read(), v = read(), w = read();
add(u, v, -w);
}
if(spfa()) printf("NO");
else for(int i = 1; i <= n; i ++) printf("%d ", dis[i]);
}
LCA（最近公共祖先）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 500005;
int head[N], ecnt;
struct edge
{
int to, nxt;
}edg[N << 2];
void add(int u, int v)
{
edg[++ecnt].to = v;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, m, s;
int fa[N][24], dep[N], cnt;
void dfs(int x, int f)
{
for(int i = head[x]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(v == f) continue;
dep[v] = dep[x] + 1;
fa[v][0] = x;
dfs(v, x);
}
}
int find(int x, int y)
{
if(dep[x] < dep[y]) swap(x, y);
int d = dep[x] - dep[y];
for(int i = 22; i >= 0; i --)
{
if(d & (1 << i)) x = fa[x][i];
}
if(x == y) return x;
for(int i = 22; i >= 0; i --)
{
if(fa[x][i] != fa[y][i])
{
x = fa[x][i];
y = fa[y][i];
}
}
return fa[x][0];
}
int main()
{
n = read(), m = read(), s = read();
for(int i = 1, x, y; i < n; i ++)
{
x = read(), y = read();
add(x, y);
add(y, x);
}
dep[s] = 1;
dfs(s, s);
for(int i = 1; i <= 23; i ++)
{
for(int j = 1; j <= n; j ++)
{
fa[j][i] = fa[fa[j][i - 1]][i - 1];
}
}
for(int i = 1, x, y; i <= m; i ++)
{
x = read(), y = read();
printf("%d
", find(x, y));
}
}
树的直径（dfs）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 100005;
int head[N], ecnt;
struct edge
{
int to, nxt, dis;
}edg[N << 2];
void add(int u, int v, int w)
{
edg[++ecnt].to = v;
edg[ecnt].dis = w;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, k, s, maxn;
int dis[N];
void dfs(int x, int fa)
{
for(int i = head[x]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(v == fa) continue;
dis[v] = dis[x] + edg[i].dis;
if(dis[v] > maxn) maxn = dis[v], s = v;
dfs(v, x);
}
}
int main()
{
n = read();
for(int i = 1, u, v, w; i < n; i ++)
{
u = read(), v = read(), w = read();
add(u, v, w);
add(v, u, w);
}
dis[1] = 0, maxn = 0;
dfs(1, 1);
dis[s] = 0, maxn = 0;
dfs(s, s);
printf("%d", maxn);
}
树的直径（dp）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int read()
{
char c = getchar(), last = ' ';
int ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
const int N = 100005;
int head[N], ecnt;
struct edge
{
int to, nxt, dis;
}edg[N << 2];
void add(int u, int v, int w)
{
edg[++ecnt].to = v;
edg[ecnt].dis = w;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
int n, k, s, ans;
int f[N];
void dp(int x, int fa)
{
for(int i = head[x]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(v == fa) continue;
dp(v, x);
ans = max(ans, f[x] + edg[i].dis + f[v]);
f[x] = max(f[x], f[v] + edg[i].dis);
}
}
int main()
{
n = read();
for(int i = 1, u, v, w; i < n; i ++)
{
u = read(), v = read(), w = read();
add(u, v, w);
add(v, u, w);
}
dp(1, 0);
printf("%d", ans);
}
缩点（tarjan）
#include<bits/stdc++.h>
using namespace std;
const int N = 10005, M = 100005;
int n, m;
int a[N];
int head[N], head1[N], ecnt, ecnt1;
struct edge
{
int to, nxt;
}edg[M], edg1[M];
void add(int u, int v)
{
edg[++ecnt].to = v;
edg[ecnt].nxt = head[u];
head[u] = ecnt;
}
namespace tar
{
int low[N], dep[N], s[N], in[N], scc[N], val[N], top, cnt, ct;
void tarjan(int x)
{
low[x] = dep[x] = ++cnt;
s[top++] = x;
in[x] = 1;
for(int i = head[x]; i; i = edg[i].nxt)
{
int v = edg[i].to;
if(!dep[v])
{
tarjan(v);
low[x] = min(low[x], low[v]);
}
else if(in[v])
{
low[x] = min(low[x], dep[v]);
}
}
if(low[x] == dep[x])
{
ct ++;
while(s[top] != x)
{
top --;
in[s[top]] = 0;
scc[s[top]] = ct;
val[ct] += a[s[top]];
}
}
}
}
namespace top
{
void add1(int u, int v)
{
edg1[++ecnt1].to = v;
edg1[ecnt1].nxt = head1[u];
head1[u] = ecnt1;
}
int in[N], f[N];
void build()
{
for(int i = 1; i <= n; i ++)
{
for(int j = head[i]; j; j = edg[j].nxt)
{
int v = edg[j].to;
if(tar::scc[i] != tar::scc[v])
{
add1(tar::scc[i], tar::scc[v]);
in[tar::scc[v]] ++;
}
}
}
}
int topo()
{
queue<int> q;
for(int i = 1; i <= tar::ct; i ++)
if(in[i] == 0) q.push(i), f[i] = tar::val[i];
while(!q.empty())
{
int u = q.front();
q.pop();
for(int i = head1[u]; i; i = edg1[i].nxt)
{
int v = edg1[i].to;
f[v] = max(f[v], f[u] + tar::val[v]);
in[v] --;
if(in[v] == 0) q.push(v);
}
}
int ans = 0;
for(int i = 1; i <= tar::ct; i ++) ans = max(ans, f[i]);
return ans;
}
}
int main()
{
scanf("%d%d", &n, &m);
for(int i = 1; i <= n; i ++) scanf("%d", &a[i]);
for(int i = 1, u, v; i <= m; i ++)
{
scanf("%d%d", &u, &v);
add(u, v);
}
for(int i = 1; i <= n; i ++)
if(!tar::dep[i]) tar::tarjan(i);
top::build();
printf("%d", top::topo());
}
次短路
数论
gcd
#include<bits/stdc++.h>
using namespace std;
int gcd(int a, int b)
{
if(b == 0) return a;
return gcd(b, a % b);
}
int m, n;
int main()
{
scanf("%d%d", &n, &m);
printf("%d", gcd(n, m));
}
exgcd
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
ll read()
{
char c = getchar(), last = ' ';
ll ans = 0;
while(c < '0' || c > '9') last = c, c = getchar();
while(c >= '0' && c <= '9') ans = ans * 10 + c - '0', c = getchar();
if(last == '-') ans = -ans;
return ans;
}
ll gcd(ll a, ll b)
{
if(b == 0) return a;
return gcd(b, a % b);
}
void exgcd(ll a, ll b, ll &x, ll &y)
{
if(b == 0)
{
x = 1;
y = 0;
return;
}
ll p;
exgcd(b, a % b, x, y);
p = x;
x = y;
y = p - y * (a / b);
return;
}
int t;
ll a, b, c, d;
int main()
{
ll t = read();
while(t --)
{
a = read(), b = read(), c = read();
ll x = 0, y = 0, dx, dy, l, r, s;
d = gcd(a, b);
if(c % d != 0)
{
printf("-1
");
continue;
}
a /= d, b /= d, c /= d;
exgcd(a, b, x, y);
x *= c, y *= c;
dx = b, dy = a;
//printf("%lld %lld
", dx, dy);
l = ll(double(ceil(1.0 * (-x + 1) / (1.0 * dx))));
r = ll(double(floor(1.0 * (y - 1) / (1.0 * dy))));
//l = ll((-x + 1) / dx);
//r = ll((y - 1) / dy);
if(l > r)
{
printf("%lld %lld
", x + l * dx, y - r * dy);
}
else
{
printf("%lld %lld %lld %lld %lld
", r - l + 1, x + l * dx, y - r * dy, x + r * dx, y - l * dy);
}
}
}
乘法逆元（递推）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 3000005;
ll n, p;
ll inv[N];
int main()
{
scanf("%lld%lld", &n, &p);
inv[1] = 1;
for(int i = 2; i <= n; i ++)
inv[i] = ((-(p / i) * inv[p % i]) % p + p) % p;
for(int i = 1; i <= n; i ++) printf("%lld
", inv[i]);
}
乘法逆元（(a^{p-2})）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 3000005;
ll n, p;
ll qpow(ll a, ll b)
{
ll ans = 1;
while(b)
{
if(b & 1) ans = ans * a % p;
a = a * a % p;
b >>= 1;
}
return ans;
}
ll inv[N];
int main()
{
scanf("%lld%lld", &n, &p);
printf("%lld", qpow(n, p - 2));
}
裴蜀定理
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
int n, ans;
int gcd(int a, int b)
{
if(b == 0) return a;
return gcd(b, a % b);
}
int main()
{
scanf("%d", &n);
scanf("%d", &ans);
for(int i = 1, x; i < n; i ++)
{
scanf("%d", &x);
ans = gcd(ans, abs(x));
}
printf("%d", ans);
}
矩阵快速幂
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 105;
const ll mod = 1e9 + 7;
ll n, k;
struct martix
{
ll a[N][N];
martix(){memset(a, 0, sizeof(a));}
void clear()
{
for(int i = 1; i <= n; i ++) a[i][i] = 1;
}
}m;
martix cheng(martix a, martix b)
{
martix c;
for(int k = 1; k <= n; k ++)
for(int i = 1; i <= n; i ++)
for(int j = 1; j <= n; j ++)
c.a[i][j] = (c.a[i][j] + 1ll * a.a[i][k] * b.a[k][j] + mod % mod) % mod;
return c;
}
martix qpow(martix a, ll b)
{
martix ans;
ans.clear();
while(b)
{
if(b & 1) ans = cheng(ans, a);
a = cheng(a, a);
b >>= 1;
}
return ans;
}
int main()
{
scanf("%lld%lld", &n, &k);
for(int i = 1; i <= n; i ++)
for(int j = 1; j <= n; j ++)
scanf("%lld", &m.a[i][j]);
martix ans = qpow(m, k);
for(int i = 1; i <= n; i ++)
{
for(int j = 1; j <= n; j ++)
printf("%lld ", ans.a[i][j]);
printf("
");
}
}
矩阵加速（数列）
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int N = 105;
const ll mod = 1e9 + 7;
ll n, k;
struct martix
{
ll a[N][N];
martix(){memset(a, 0, sizeof(a));}
void clear()
{
for(int i = 1; i <= 3; i ++) a[i][i] = 1;
}
}m, t;
martix cheng(martix a, martix b)
{
martix c;
for(int k = 1; k <= 3; k ++)
for(int i = 1; i <= 3; i ++)
for(int j = 1; j <= 3; j ++)
c.a[i][j] = (c.a[i][j] + 1ll * a.a[i][k] * b.a[k][j] + mod % mod) % mod;
return c;
}
martix qpow(martix a, ll b)
{
martix ans;
ans.clear();
while(b)
{
if(b & 1) ans = cheng(ans, a);
a = cheng(a, a);
b >>= 1;
}
return ans;
}
int T;
int main()
{
scanf("%d", &T);
while(T --)
{
scanf("%lld", &n);
m.a[1][1] = m.a[1][2] = m.a[1][3] = 1;
t.a[1][1] = t.a[1][2] = t.a[2][3] = t.a[3][1] = 1;
t.a[1][3] = t.a[2][1] = t.a[2][2] = t.a[3][2] = t.a[3][3]= 0;
martix ans = qpow(t, n - 1);
martix qwq;
for(int k = 1; k <= 3; k ++)
for(int i = 1; i <= 1; i ++)
for(int j = 1; j <= 3; j ++)
qwq.a[i][j] = (qwq.a[i][j] + (1ll * m.a[i][k] * ans.a[k][j] + mod) % mod) % mod;
printf("%lld
", qwq.a[1][3]);
}
}