From 8508fa6ccee3ab48fdf142d25a60943b4e5d463d Mon Sep 17 00:00:00 2001
From: LogSingleDog <1034028483@qq.com>
Date: Thu, 25 Apr 2024 20:28:23 +0800
Subject: [PATCH] graph update

---
 "5-\345\233\276\350\256\272.md" | 346 +++++++++++++++-----------------
 1 file changed, 158 insertions(+), 188 deletions(-)

diff --git "a/5-\345\233\276\350\256\272.md" "b/5-\345\233\276\350\256\272.md"
index 07a001f..1b5397e 100644
--- "a/5-\345\233\276\350\256\272.md"
+++ "b/5-\345\233\276\350\256\272.md"
@@ -7,280 +7,250 @@
 ```cpp
 #include <bits/stdc++.h>
 using namespace std;
-#define int long long
-const int inf = 0x3f3f3f3f;
-typedef pair<int,int> PII;
-vector<PII> mp[100100];
-int n,m,s;
-int dis[100100];
-int vis[100100];
-priority_queue<PII,vector<PII>,greater<PII> > q;
+#define ll long long
+const ll maxn = 10+1e5;
+const ll inf = 0x3f3f3f3f;
+typedef pair<ll,ll> pll;
+    vector<pll> mp[100100];
+    ll n,m,s;
+    ll dis[100100];
+    ll vis[100100];
 
-void dj(int s)
-{
-    for(int i=1;i<=n;i++)dis[i]=inf;
-    dis[s]=0ll;
+void dij(ll s){
+    for(ll i=1;i<=n;i++){
+        dis[i]=inf;
+    }
+    dis[s]=0;
+    priority_queue<pll,vector<pll>,greater<pll> > q;
     q.push({dis[s],s});
-    
-    while(!q.empty())
-    {
-        int u=q.top().second;
+    while(!q.empty()){
+        ll u=q.top().second;
         q.pop();
-        if(vis[u])continue;
+        if(vis[u]) continue;
         vis[u]=1;
-        for(auto [w,v]:mp[u])
-        {
-            if(dis[u]+w<dis[v])
-            {
+        for(auto [w,v]:mp[u]){
+            if(dis[u]+w<dis[v]){
                 dis[v]=dis[u]+w;
                 q.push({dis[v],v});
             }
         }
     }
 }
-signed main() {
-cin>>n>>m>>s;
-for(int i=1;i<=m;i++)
-{
-    int u,v,w;
-    cin>>u>>v>>w;
-    mp[u].push_back({w,v});
-}
-dj(s);
-for(int i=1;i<=n;i++)cout<<dis[i]<<" ";
+int main() {
+	cin>>n>>m>>s;
+	for(ll i=1;i<=m;i++){
+    	ll u,v,w;
+    	cin>>u>>v>>w;
+    	mp[u].push_back({w,v});
+	}
+	dij(s);
+	for(ll i=1;i<=n;i++)
+        cout<<dis[i]<<" ";
     return 0;
 }
 ```
 ## 并查集
 
 ```cpp
-int fa[100100];
-void init(int n)
-{
-    for(int i=1;i<=n;i++)fa[i]=i;
+ll fa[maxn];
+void init(ll n){
+    for(ll i=1;i<=n;i++){
+        fa[i]=i;
+    }
 }
-int find(int x)
-{
+ll find(ll x){
     if(fa[x]==x)
         return x;
     fa[x]=find(fa[x]);
-    return find(fa[x]);
+    return fa[x];
 }
-void merge(int x,int y)
-{
+void merge(ll x,ll y){
+    if(find(x)==find(y)) return ;
     fa[find(x)]=find(y);
 }
 ```
 ## 最小生成树
 
-###   prim
+### prim O(n^2)
 
 ```cpp
-   #include <bits/stdc++.h>
-using namespace std;
-const int N = 5050;
-const int inf=0x3f3f;
-int g[N][N],dis[200200];
-bool vis[200200];
-int n,m,ans,u,v,w;
-void init(){
-    memset(g,inf,sizeof(g));
-    memset(dis,inf,sizeof(dis));
-}
-void addedge(int u,int v,int w)
-{
-    if (u != v && g[u][v] > w) g[u][v] = g[v][u] = w;
+    vector<pll> edge[maxn];
+    ll dis[maxn];
+    ll vis[maxn];
+void update(ll u){
+    for(auto [v,w]:edge[u]){
+        dis[v]=min(dis[v],w);
+    }
 }
-void prim(){
-    dis[1]=0;
-    for(int i=1;i<=n;i++){
-        int t=0;
-        for(int j=1;j<=n;j++){
-            if(!vis[j]&&dis[j]<dis[t])t=j;
-        }
-        vis[t]=1;
-        ans+=dis[t];
-        for(int j=1;j<=n;j++){
-            if(!vis[j]&&dis[j]>g[t][j]){
-                dis[j]=g[t][j];
+ll prim(ll n){
+    for(ll i=1;i<=n;i++){
+        vis[i]=0;
+        dis[i]=inf;
+    }
+    vis[1]=1;
+    update(1);
+    ll ans=0;
+    for(ll i=1;i<=n-1;i++){//连接n-1条边
+        ll pos,mi=inf;
+        for(ll j=1;j<=n;j++){
+            if(vis[j]) continue;
+            if(dis[j]<mi){
+                mi=dis[j];
+                pos=j;
             }
         }
+        ans+=mi;
+        vis[pos]=1;
+        update(pos);    
     }
+    return ans;
 }
+```
 
-signed main() {
-cin>>n>>m;
-init();
-for(int i=1;i<=m;i++){
-    cin>>u>>v>>w;
-    addedge(u,v,w);
-    addedge(u,v,w);
-}
-prim();
-for(int i=1;i<=n;i++){
-    if(!vis[i]) {
-        cout << "orz";
-        return 0;
+### prim O(mlogm) (小根堆优化)
+```cpp
+    vector<pll> edge[maxn];
+    ll dis[maxn];
+    ll vis[maxn];
+ll prim(ll n){
+    for(ll i=1;i<=n;i++){
+        vis[i]=0;
+        dis[i]=inf;
     }
-}
-cout<<ans;
-    return 0;
+    ll ans=0;
+    priority_queue<pll,vector<pll>,greater<pll>> pq;
+    pq.push({0,1});
+    while(!pq.empty()){
+        auto [d,u]=pq.top();
+        pq.pop();
+        if(vis[u]) continue;
+        vis[u]=1;
+        ans+=d;
+        for(auto [v,w]:edge[u]){
+            if(!vis[v]&&w<dis[v]){
+                dis[v]=w;
+                pq.push({dis[v],v});
+            }
+        }
+    }
+    return ans;
 }
 ```
 
 ### kruskal
 
 ```cpp
-#include<bits/stdc++.h>
-using namespace std;
-#define int long long 
-int fa[200100];
-struct edge{
-    int u,v,w;
-}e[200100],mst[200100];
-int n,m,k;
-int ans;
-bool cmp(edge a,edge b)
-{
+typedef struct{
+    ll u,v,w;
+}eg;
+    eg e[maxm];
+    vector<pll> edge[maxn];
+    ll fa[maxn];
+bool cmp(eg a,eg b){
     return a.w<b.w;
 }
-void init(int n)
-{
-	for(int i=1;i<=n;i++)
-	fa[i]=i;
+ll find(ll a){
+    return (fa[a]==a)?a:(fa[a]=find(fa[a]));
 }
-int find(int x) 
-{
-	if(fa[x]==x)
-		return x;
-	else{
-	fa[x]=find(fa[x]); 
-	return find(fa[x]);
-	}
-}
-void merge(int i,int j)
-{
-	fa[find(i)]=find(j);
+void merge(ll a,ll b){
+    fa[a]=b;
 }
-void kruskal(){
-    for(int i=1;i<=m;i++){
-        if(find(e[i].u)!=find(e[i].v)){
-            k++;
-            mst[i].u=e[i].u;
-            mst[i].v=e[i].v;
-            mst[i].w=e[i].w;
+ll kruscal(ll n,ll m){
+    for(ll i=1;i<=n;i++){
+        fa[i]=i;
+    }
+    sort(e+1,e+m+1,cmp);
+    ll ans=0;
+    for(ll i=1;i<=m;i++){
+        ll a=find(e[i].u);
+        ll b=find(e[i].v);
+        if(a!=b){
+            merge(a,b);
             ans+=e[i].w;
-            merge(e[i].u,e[i].v);
+            n--;
         }
     }
+    if(n==1){
+        return ans;
+    }else{
+        return inf;
+    }
 }
-signed main()
-{
- std::ios::sync_with_stdio(false);
-    std::cin.tie(0);
-cin>>n>>m;
-init(n);
-for(int i=1;i<=m;i++)cin>>e[i].u>>e[i].v>>e[i].w;
-sort(e+1,e+m+1,cmp);
-kruskal();
-if(k==n-1)cout<<ans;
-else cout<<"orz";
-	return 0;
+for(ll i=1;i<=m;i++){
+    ll u,v,w;
+    cin>>u>>v>>w;
+    edge[u].push_back({v,w});
+    edge[v].push_back({u,w});
+    e[i].u=u;e[i].v=v;e[i].w=w;
 }
 ```
 
 ## LCA
 
 ```cpp
-   #include <bits/stdc++.h>
-
+#include <bits/stdc++.h>
 using namespace std;
-#define IOS ios::sync_with_stdio(false), cin.tie(nullptr), cout.tie(nullptr);
-#define int long long
-#define ull unsigned long long
+#define IOS ios::sync_with_stdio(false),cin.tie(nullptr), cout.tie(nullptr);
+#define ll long long
+#define ull unint long long
 #define lowbit(i) ((i) & (-i))
 #define ls(p) (p << 1)
 #define rs(p) (p << 1 | 1)
-#define rep(i, a, b) for (int i = a; i <= b; i++)
-#define per(i, a, b) for (int i = a; i >= b; i--)
+#define rep(i, a, b) for (ll i = a; i <= b; i++)
+#define per(i, a, b) for (ll i = a; i >= b; i--)
 
-typedef pair<int, int> PII;
-const int mod = 1e9 + 7;
-const int inf = 0x3f3f3f3f;
-const int N = 5e5 + 200;
+typedef pair<ll, ll> pll;
+const ll mod = 1e9 + 7;
+const ll inf = 0x3f3f3f3f;
+const ll maxn = 5e5 + 200;
 
-int qpow(int a, int n)
-{
-    int ans = 1;
-    while (n)
-    {
-        if (n & 1)
-        {
-            ans = ans * a % mod;
-        }
-        a = a * a % mod;
-        n >>= 1;
-    }
-    return ans;
-}
-
-int n, q, root; 
-vector<int> mp[N];
-int lg2[N];
-int dep[N];
-int f[N][20];
-int vis[N];
-void dfs(int u, int fa = 0)
-{
-    if (vis[u])
-        return;
+    ll n, q, root; 
+    vector<ll> mp[N];
+    ll lg2[maxn];
+    ll dep[maxn];
+    ll f[maxn][20];
+    ll vis[maxn];
+void dfs(ll u, ll fa = 0){
+    if (vis[u]) return;
     vis[u] = 1;
     dep[u] = dep[fa] + 1;
     f[u][0] = fa;
-    for (int i = 1; i <= lg2[dep[u]]; i++)
-    {
+    for (ll i = 1; i <= lg2[dep[u]]; i++){
         f[u][i] = f[f[u][i - 1]][i - 1];
     }
-    for (auto v : mp[u])
-    {
+    for (auto v : mp[u]){
         dfs(v, u);
     }
 }
-int lca(int a, int b)
-{
+ll lca(ll a, ll b){
     if (dep[a] > dep[b])
         swap(a, b);
     while (dep[a] != dep[b])
         b = f[b][lg2[dep[b] - dep[a]]];
     if (a == b)
         return a;
-    for (int k = lg2[dep[a]]; k >= 0; k--)
-    {
-        if (f[a][k] != f[b][k])
-        {
+    for (ll k = lg2[dep[a]]; k >= 0; k--){
+        if (f[a][k] != f[b][k]){
             a = f[a][k], b = f[b][k];
         }
     }
     return f[a][0];
 }
-signed main()
-{
-    IOS cin >> n >> q >> root;
-    for (int i = 1; i < n; i++)
-    {
-        int u, v;
+int main(){
+    IOS 
+    cin >> n >> q >> root;
+    for (ll i = 1; i < n; i++){
+        ll u, v;
         cin >> u >> v;
         mp[u].push_back(v);
         mp[v].push_back(u);
     }
-    for (int i = 2; i <= n; i++)
-    {
+    for (ll i = 2; i <= n; i++){
         lg2[i] = lg2[i / 2] + 1;
-    } 
+    }
     dfs(root);
-    while (q--)
-    {
-        int u, v;
+    while (q--){
+        ll u, v;
         cin >> u >> v;
         cout << lca(u, v) << endl;
     }