Disjoint-set and Kruskal minimum spanning tree algorithm

First, what is the disjoint-set

In computer science, disjoint-set data structure of a tree for consolidation and deal with some problems disjoint sets of queries. There is a joint - lookup algorithm (union-find algorithm) defines two operation times for the data structures:

Find: determining which subset of elements belonging. It can be used to determine whether the two elements belong to the same subset.

Union: the two subsets and into a collection.

Second, the main operation

Initialization: initialize each point where the collection of its own.

for(int i=1;i<=n;i++)
    f[i]=i;

Find: Find a set of elements are located, that is the root node.

int find(int x)
{
    while(f[x]!=x)
        x=f[x];
    return x;
}

Merge: The two sets of elements where combined into one collection.

void Union(int x1,int x2)
{
    int t1=find(x1);
    int t2=find(x2);
    if(t1!=t2)
        f[t2]=t1;
}

Third, optimization

The above code seems simple, but the time complexity of each find operation is O (H), H is the height of the tree, because we did not do special treatment to the tree, so the tree continues to merge tree may cause a serious imbalance, the worst case each node has only one child node.

Therefore, the find function employed in path compression .

int find(int x)       //查找x元素所在的集合,回溯时压缩路径
{
    if (x != f[x])
    {
        f[x] = find(f[x]); 
        //从x结点搜索到祖先结点所经过的结点都指向该祖先结点
    }         
    return f[x];
}

Fourth, the template title

Luo Gu P3367 [template] disjoint-set

#include<bits/stdc++.h>
using namespace std;
int n,m;
int f[10002];
int find(int x)
{
    if(x!=f[x])
        f[x]=find(f[x]);
    return f[x];
}
void Union(int x1,int x2)
{
    int t1=find(x1);
    int t2=find(x2);
    if(t1!=t2) //祖先不一样 
        f[t2]=t1; //把t2的祖先变为x1的祖先t1 
}
int main()
{
    cin>>n>>m;
    for(int i=1;i<=n;i++)
        f[i]=i;
    for(int i=0;i<m;i++)
    {
        int z,x,y;
        cin>>z>>x>>y;
        if(z==1)
            Union(x,y); 
        else
        {
            if(find(x)!=find(y))cout<<"N"<<endl;
            else cout<<"Y"<<endl;
        }
    }
    return 0;
}

Fifth, the minimum spanning tree

A connected graph with n vertices of the original image is minimal spanning tree link subgraph, all of the n nodes comprising original image, and there is minimal communication while keeping FIG.

Minimum spanning tree is actually referred to as minimum weight spanning tree.

Kruskal's algorithm

The graph all the edges in accordance with weight in ascending order
Traversing all edges sorted, if constitute loop, then the edge is added to the collection
Until it finds the n-1 sides

Example: busy city

#include<bits/stdc++.h>
using namespace std;
int n,m;
int s,maxm;
int p[100002];
struct node{
    int u;
    int v;
    int c;
}info[100002];

bool cmp(node x1,node x2)
{
    if(x1.c!=x2.c)return x1.c<x2.c;
    else if(x1.u!=x2.u) return x1.u<x2.u;
    else return x1.v<x2.v;
}
int find(int x)       //查找x元素所在的集合,回溯时压缩路径
{
    if (x!=p[x])
    {
        p[x]=find(p[x]);  
    }         
    return p[x];
}
void bcj(int x1,int x2)//把x2并入x1的集合
{
    int t1,t2;//存储祖先节点
    t1=find(x1);
    t2=find(x2);
    if(t1!=t2)p[t2]=t1;
}
int main()
{
    cin>>n>>m;//n就是顶点数，m是边数
    for(int i=1;i<=n;i++)
    {
        p[i]=i;
    }
    for(int i=0;i<m;i++)
    {
        cin>>info[i].u>>info[i].v>>info[i].c;
    }
    sort(info,info+m,cmp);
    for(int i=0;i<m;i++)//遍历所有的边
    {
        if(find(info[i].u)!=find(info[i].v))
        {
            bcj(info[i].u,info[i].v);//把v并入u的集合
            maxm=max(maxm,info[i].c);
        }
    }
    cout<<n-1<<" "<<maxm;
    return 0;
}