算法笔记--lca倍增算法

算法笔记

模板：

vector<int>g[N];
vector<int>edge[N];
int anc[20][N];
int deep[N];
int h[N];
void dfs(int o,int u,int w)
{
    if(u!=o)deep[u]=deep[o]+1,h[u]=h[o]+w;
    for(int j=0;j<g[u].size();j++)
    {
        if(g[u][j]!=o)
        {
            anc[0][g[u][j]]=u;
            for(int i=1;i<20;i++)anc[i][g[u][j]]=anc[i-1][anc[i-1][g[u][j]]];
            dfs(u,g[u][j],edge[u][j]);
        }
    }
}
int lca(int u,int v)
{
    if(deep[u]<deep[v])swap(u,v);
    for(int i=19;i>=0;i--)if(deep[anc[i][u]]>=deep[v])u=anc[i][u];
    if(u==v)return u;
    for(int i=19;i>=0;i--)if(anc[i][u]!=anc[i][v])u=anc[i][u],v=anc[i][v];
    return anc[0][u];
}
int dis(int u,int v)
{
    int l=lca(u,v);
    return h[u]+h[v]-2*h[l];
}

例题：

1.CODEVS 2370 小机房的树

带权lca

代码：

#include<bits/stdc++.h>
using namespace std;
#define ll long long
const int INF=0x3f3f3f3f;
const int N=5e4+5;
vector<int>g[N];
vector<int>edge[N];
int anc[20][N];
int deep[N];
int h[N];
void dfs(int o,int u,int w)
{
    if(u!=o)deep[u]=deep[o]+1,h[u]=h[o]+w;
    for(int j=0;j<g[u].size();j++)
    {
        if(g[u][j]!=o)
        {
            anc[0][g[u][j]]=u;
            for(int i=1;i<20;i++)anc[i][g[u][j]]=anc[i-1][anc[i-1][g[u][j]]];
            dfs(u,g[u][j],edge[u][j]);
        }
    }
}
int lca(int u,int v)
{
    if(deep[u]<deep[v])swap(u,v);
    for(int i=19;i>=0;i--)if(deep[anc[i][u]]>=deep[v])u=anc[i][u];
    if(u==v)return u;
    for(int i=19;i>=0;i--)if(anc[i][u]!=anc[i][v])u=anc[i][u],v=anc[i][v];
    return anc[0][u];
}
int dis(int u,int v)
{
    int l=lca(u,v);
    return h[u]+h[v]-2*h[l];
}
int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,u,v,c,m;
    cin>>n;
    for(int i=0;i<n-1;i++)
    {
        cin>>u>>v>>c;
        g[u].push_back(v);
        g[v].push_back(u);
        edge[u].push_back(c);
        edge[v].push_back(c);
    }
    cin>>m;
    for(int i=0;i<20;i++)anc[i][0]=0;
    dfs(0,0,0);
    while(m--)
    {
        cin>>u>>v;
        cout<<dis(u,v)<<endl;
    }
    return 0;
}

2.CODEVS 1036 商务旅行

普通lca

代码：

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
const int INF=0x3f3f3f3f;
const int N=1e5+5;
vector<int>g[N];
int anc[20][N];
int deep[N];
void dfs(int o,int u)
{
    if(o!=u)deep[u]=deep[o]+1;
    for(int j=0;j<g[u].size();j++)
    {
        if(o!=g[u][j])
        {
            anc[0][g[u][j]]=u;
            for(int i=1;i<20;i++)anc[i][g[u][j]]=anc[i-1][anc[i-1][g[u][j]]];
            dfs(u,g[u][j]);
        }
    }
}
int lca(int u,int v)
{
    if(deep[u]<deep[v])swap(u,v);
    for(int i=19;i>=0;i--)if(deep[anc[i][u]]>=deep[v])u=anc[i][u];
    if(u==v)return u;
    for(int i=19;i>=0;i--)if(anc[i][u]!=anc[i][v])u=anc[i][u],v=anc[i][v];
    return anc[0][u];
}
int dis(int u,int v)
{
    return deep[u]+deep[v]-2*deep[lca(u,v)];
}
void init()
{
    for(int i=0;i<20;i++)anc[0][1]=1;
    dfs(0,1);
}
int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m;
    cin>>n;
    for(int i=1;i<n;i++)
    {
        int a,b;
        cin>>a>>b;
        g[a].push_back(b);
        g[b].push_back(a);
    }
    init();
    cin>>m;
    int a,b,ans=0;
    cin>>a;
    for(int i=1;i<m;i++)
    {
        cin>>b;
        ans+=dis(a,b);
        a=b;
    }
    cout<<ans<<endl;
    return 0;
}

3.METO CODE P223 拉力赛

带权lca,当lca(a,b)==a时，韵韵才能参加。

代码：

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define pb push_back
const int INF=0x3f3f3f3f;
const int N=1e4+5;
vector<int>g[N];
vector<ll>edge[N];
int anc[20][N];
int deep[N];
ll h[N];
bool vis[N]={false};
int s;
void dfs(int o,int u,ll w)
{
    if(o!=u)deep[u]=deep[o]+1,h[u]=h[o]+w;
    for(int j=0;j<g[u].size();j++)
    {
        if(g[u][j]!=o)
        {
            anc[0][g[u][j]]=u;
            for(int i=1;i<20;i++)anc[i][g[u][j]]=anc[i-1][anc[i-1][g[u][j]]];
            dfs(u,g[u][j],edge[u][j]);
        }
    }
}
int lca(int u,int v)
{
    if(deep[u]<deep[v])swap(u,v);
    for(int i=19;i>=0;i--)if(deep[anc[i][u]]>=deep[v])u=anc[i][u];
    if(u==v)return u;
    for(int i=19;i>=0;i--)if(anc[i][u]!=anc[i][v])u=anc[i][u],v=anc[i][v];
    return anc[0][u];
}
ll dis(int u,int v)
{
    return h[u]+h[v]-2*h[lca(u,v)];
}
void init()
{
    for(int i=0;i<20;i++)anc[i][1]=1;
    dfs(0,1,1);
}
int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m,a,b;
    ll t;
    cin>>n>>m;
    for(int i=1;i<n;i++)
    {
        cin>>a>>b>>t;
        g[a].pb(b);
        g[b].pb(a);
        edge[a].pb(t);
        edge[b].pb(t);
    }
    init();
    int cnt=0;
    ll ans=0;
    for(int i=0;i<m;i++)
    {
        int a,b;
        cin>>a>>b;
        if(lca(a,b)==a)
        {
            cnt++;
            ans+=h[b]-h[a];
        }
        }
    cout<<cnt<<endl;
    cout<<ans<<endl;
    return 0;
}

4.HDU 2586 How far way?

带权lca

代码：

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define pb push_back
const int INF=0x3f3f3f3f;
const int N=4e4+5;
vector<int>g[N];
vector<int>edge[N];
int deep[N];
int h[N];
int anc[20][N];
bool vis[N]={false};
int s=1;
void dfs(int o,int u,int w)
{
    deep[u]=deep[o]+1;
    h[u]=h[o]+w;
    for(int j=0;j<g[u].size();j++)
    {
        if(g[u][j]!=o)
        {
            anc[0][g[u][j]]=u;
            for(int i=1;i<20;i++)anc[i][g[u][j]]=anc[i-1][anc[i-1][g[u][j]]];
            dfs(u,g[u][j],edge[u][j]);
        }
    }
}
int lca(int u,int v)
{
    if(deep[u]<deep[v])swap(u,v);
    for(int i=19;i>=0;i--)if(deep[anc[i][u]]>=deep[v])u=anc[i][u];
    if(u==v)return u;
    for(int i=19;i>=0;i--)if(anc[i][u]!=anc[i][v])u=anc[i][u],v=anc[i][v];
    return anc[0][u];
}
ll dis(int u,int v)
{
    return h[u]+h[v]-2*h[lca(u,v)];
}
void init()
{
    for(int i=0;i<20;i++)anc[i][1]=1;
    dfs(0,1,1);
}
int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    int t;
    cin>>t;
    while(t--)
    {
        int n,m,a,b,k;
        cin>>n>>m;
        for(int i=1;i<n;i++)
        {
            cin>>a>>b>>k;
            g[a].pb(b);
            g[b].pb(a);
            edge[a].pb(k);
            edge[b].pb(k);
            vis[b]=true;
        }
        init();
        for(int i=0;i<m;i++)
        {
            cin>>a>>b;
            cout<<dis(a,b)<<endl;
        }
        //cout<<endl;
    }
    return 0;
}

5.ZOJ P3195 Design the city

带权lca，三点之间的最短路径公式h[a]+h[b]+h[c]-h[lca(a,b)]-h[lca(a,c)]-h[lca(b,c)]。

代码：

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define pb push_back
const int INF=0x3f3f3f3f;
const int N=5e4+5;
vector<int>g[N];
vector<int>edge[N];
int deep[N];
int h[N];
int anc[20][N];
void dfs(int o,int u,int w)
{
    if(u!=o)deep[u]=deep[o]+1,h[u]=h[o]+w;
    for(int j=0;j<g[u].size();j++)
    {
        if(g[u][j]!=o)
        {
            anc[0][g[u][j]]=u;
            for(int i=1;i<20;i++)anc[i][g[u][j]]=anc[i-1][anc[i-1][g[u][j]]];
            dfs(u,g[u][j],edge[u][j]);
        }
    }
}
int lca(int u,int v)
{
    if(deep[u]<deep[v])swap(u,v);
    for(int i=19;i>=0;i--)if(deep[anc[i][u]]>=deep[v])u=anc[i][u];
    if(u==v)return u;
    for(int i=19;i>=0;i--)if(anc[i][u]!=anc[i][v])u=anc[i][u],v=anc[i][v];
    return anc[0][u];
}
int dis(int u,int v)
{
    return h[u]+h[v]-2*h[lca(u,v)];
}
void init()
{
    for(int i=0;i<20;i++)anc[i][0]=0;
    dfs(0,0,0);
}
int main()
{
    int n,q,a,b,c;
    bool flag=true;
    while(~scanf("%d",&n)&&n)
    {
        if(flag) flag=false;
        else printf("\n");
        for(int i=0;i<n;i++)g[i].clear(),edge[i].clear();
        for(int i=1;i<n;i++)
        {
            scanf("%d%d%d",&a,&b,&c);
            g[a].pb(b);
            g[b].pb(a);
            edge[a].pb(c);
            edge[b].pb(c);
        }
        init();
        scanf("%d",&q);
        while(q--)
        {
            scanf("%d%d%d",&a,&b,&c);
            printf("%d\n",h[a]+h[b]+h[c]-h[lca(a,b)]-h[lca(a,c)]-h[lca(b,c)]);
        }
    }
    return 0;
}