图论练习4

内容：染色划分，带权并查集，扩展并查集

Arpa’s overnight party and Mehrdad’s silent entering

题目链接

题目大意

$2n$ 个点围成一圈，分为 $n$ 对，对内两点不同染色
同时，相邻3个点之间必须有两个点不同染色
问构造出一种染色方案

解题思路

将每对进行的连边看作一类边
将为满足相邻3个点必须有两个点不同染色的边，看作二类边
满足构造方案，即将 $2n$ 个点形成一个二分图，无奇环
对于只有一类边，形不成环，端点不同
对于二类边与一类边组合才能形成环
将二类边定义为 $2_i\leftrightarrow 2_i+1$
成环的情况只能，如下图
由于一类边不可能相连，中间必然是二类边，一类边交换，所以环上的点数为二类边的总点数，一定为偶数，只能形成偶环，构造成立
利用带权并查集实现


import java.io.*;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.BitSet;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.StringTokenizer;
import java.util.Vector;




public class Main{
	
	static int find(int x,int[] fa,int[] relation) {
		if(x==fa[x])return x;
		else {
			int f=fa[x];
			fa[x]=find(fa[x], fa, relation);
			relation[x]=(relation[x]+relation[f])%2;
			return fa[x];
		}
	}

	public static void main(String[] args) throws IOException{
		AReader input=new AReader();
	    PrintWriter out = new PrintWriter(new OutputStreamWriter(System.out));
	    int n=input.nextInt();
	    int[] fa=new int[2*n+1];
	    int[] relation=new int[2*n+1];
	    for(int i=1;i<=2*n;++i)fa[i]=i;
	    int[] a=new int[n+1];
	    int[] b=new int[n+1];
	    for(int i=1;i<=n;++i) {
	    	int x=input.nextInt();
	    	int y=input.nextInt();
	    	a[i]=x;b[i]=y;
	    	int fx=find(x, fa, relation);
	    	int fy=find(y, fa, relation);
	    	if(fx==fy)continue;
	    	fa[fx]=fy;
	    	relation[fx]=(relation[y]+1-relation[x]+2)%2;
	    }
	    for(int i=1;i<=n;++i) {//一圈
	    	int x=2*i;
	    	int y=i==n?1:2*i+1;
	    	int fx=find(x, fa, relation);
	    	int fy=find(y, fa, relation);
	    	if(fx==fy)continue;
	    	fa[fx]=fy;
	    	relation[fx]=(relation[y]+1-relation[x]+2)%2;
	    }
	    for(int i=1;i<=n;++i) {
	    	find(a[i], fa, relation);
	    	find(b[i], fa, relation);
	    	int x=relation[a[i]]+1;
	    	int y=relation[b[i]]+1;
	    	out.println(x+" "+y);
	    }
 	    out.flush();
	    out.close();
	}
	static
	class AReader{
	    BufferedReader bf;
	    StringTokenizer st;
	    BufferedWriter bw;

	    public AReader(){
	        bf=new BufferedReader(new InputStreamReader(System.in));
	        st=new StringTokenizer("");
	        bw=new BufferedWriter(new OutputStreamWriter(System.out));
	    }
	    public String nextLine() throws IOException{
	        return bf.readLine();
	    }
	    public String next() throws IOException{
	        while(!st.hasMoreTokens()){
	            st=new StringTokenizer(bf.readLine());
	        }
	        return st.nextToken();
	    }
	    public char nextChar() throws IOException{
	        //确定下一个token只有一个字符的时候再用
	        return next().charAt(0);
	    }
	    public int nextInt() throws IOException{
	        return Integer.parseInt(next());
	    }
	    public long nextLong() throws IOException{
	        return Long.parseLong(next());
	    }
	    public double nextDouble() throws IOException{
	        return Double.parseDouble(next());
	    }
	    public float nextFloat() throws IOException{
	        return Float.parseFloat(next());
	    }
	    public byte nextByte() throws IOException{
	        return Byte.parseByte(next());
	    }
	    public short nextShort() throws IOException{
	        return Short.parseShort(next());
	    }
	    public BigInteger nextBigInteger() throws IOException{
	        return new BigInteger(next());
	    }
	    public void println() throws IOException {
	        bw.newLine();
	    }
	    public void println(int[] arr) throws IOException{
	        for (int value : arr) {
	            bw.write(value + " ");
	        }
	        println();
	    }
	    public void println(int l, int r, int[] arr) throws IOException{
	        for (int i = l; i <= r; i ++) {
	            bw.write(arr[i] + " ");
	        }
	        println();
	    }
	    public void println(int a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(int a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(String a) throws IOException{
	        bw.write(a);
	        bw.newLine();
	    }
	    public void print(String a) throws IOException{
	        bw.write(a);
	    }
	    public void println(long a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(long a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(double a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(double a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void print(char a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(char a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	}
}

食物链

题目链接

题目大意

$n$ 个点， $k$ 对关系
每个点可能为 $A/B/C$ 类，它们之间满足 $A$ 吃 $B$ ， $B$ 吃 $C$ ， $C$ 吃 $A$
$k$ 对关系中， $1,x,y$ 表示 $x,y$ 同类， $2,x,y$ 表示 $x$ 吃 $y$
依次处理 $k$ 对关系，若当前关系与之前的关系矛盾，则视为错误
$x>n,y>n,(2,x,y)x=y$ （自己吃自己），均视为错误
问有多少对错误

解题思路

对 $n$ 个点进行 $3$ 种颜色的染色划分
类似黑白染色
利用带权并查集或扩展并查集实现
详见图论笔记

扩展并查集

 
import java.io.*;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.BitSet;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.StringTokenizer;
import java.util.Vector;
 
 
 
 
public class Main{
	
	static int find(int x,int[] fa) {
		if(x==fa[x])return x;
		else return fa[x]=find(fa[x], fa);
	}
	
	
	
	public static void main(String[] args) throws IOException{
		AReader input=new AReader();
	    PrintWriter out = new PrintWriter(new OutputStreamWriter(System.out));
	    int n=input.nextInt();
	    int k=input.nextInt();
	    int[] fa=new int[3*n+1];
	    for(int i=1;i<=3*n;++i)fa[i]=i;
	    int ans=0;
	    for(int i=1;i<=k;++i) {
	    	int op=input.nextInt();
	    	int x=input.nextInt();
	    	int y=input.nextInt();
	    	if(x>n||y>n) {
	    		ans++;
	    		continue;
	    	}
	    	if(op==1) {
	    		if(find(x, fa)==find(y+n, fa)||find(y, fa)==find(x+n, fa)) {
	    			//x吃y或y吃x
	    			//x吃y=>xA->yB,xB->yC,xC->yA
	    			//y吃x=>yA->xB,yB->xC->yC->xA
	    			ans++;
	    			continue;
	    		}
	    		fa[find(x, fa)]=find(y, fa);
	    		fa[find(x+n, fa)]=find(y+n, fa);
	    		fa[find(x+2*n, fa)]=find(y+2*n, fa);
	    	}else {
	    		if(x==y) {
	    			ans++;
	    			continue;
	    		}
	    		if(find(x, fa)==find(y, fa)||find(y, fa)==find(x+n, fa)) {
	    			//x和y是同类,y吃x
	    			ans++;
	    			continue;
	    		}
	    		fa[find(x, fa)]=find(y+n, fa);
	    		fa[find(x+n, fa)]=find(y+2*n, fa);
	    		fa[find(x+2*n, fa)]=find(y, fa);
	    	}
	    }
	    out.print(ans);
 	    out.flush();
	    out.close();
	}
	static
	class AReader{
	    BufferedReader bf;
	    StringTokenizer st;
	    BufferedWriter bw;
 
	    public AReader(){
	        bf=new BufferedReader(new InputStreamReader(System.in));
	        st=new StringTokenizer("");
	        bw=new BufferedWriter(new OutputStreamWriter(System.out));
	    }
	    public String nextLine() throws IOException{
	        return bf.readLine();
	    }
	    public String next() throws IOException{
	        while(!st.hasMoreTokens()){
	            st=new StringTokenizer(bf.readLine());
	        }
	        return st.nextToken();
	    }
	    public char nextChar() throws IOException{
	        //确定下一个token只有一个字符的时候再用
	        return next().charAt(0);
	    }
	    public int nextInt() throws IOException{
	        return Integer.parseInt(next());
	    }
	    public long nextLong() throws IOException{
	        return Long.parseLong(next());
	    }
	    public double nextDouble() throws IOException{
	        return Double.parseDouble(next());
	    }
	    public float nextFloat() throws IOException{
	        return Float.parseFloat(next());
	    }
	    public byte nextByte() throws IOException{
	        return Byte.parseByte(next());
	    }
	    public short nextShort() throws IOException{
	        return Short.parseShort(next());
	    }
	    public BigInteger nextBigInteger() throws IOException{
	        return new BigInteger(next());
	    }
	    public void println() throws IOException {
	        bw.newLine();
	    }
	    public void println(int[] arr) throws IOException{
	        for (int value : arr) {
	            bw.write(value + " ");
	        }
	        println();
	    }
	    public void println(int l, int r, int[] arr) throws IOException{
	        for (int i = l; i <= r; i ++) {
	            bw.write(arr[i] + " ");
	        }
	        println();
	    }
	    public void println(int a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(int a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(String a) throws IOException{
	        bw.write(a);
	        bw.newLine();
	    }
	    public void print(String a) throws IOException{
	        bw.write(a);
	    }
	    public void println(long a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(long a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(double a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(double a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void print(char a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(char a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	}
}

带权并查集

 
import java.io.*;
import java.math.BigInteger;
import java.util.Arrays;
import java.util.BitSet;
import java.util.HashMap;
import java.util.HashSet;
import java.util.Iterator;
import java.util.LinkedList;
import java.util.PriorityQueue;
import java.util.Queue;
import java.util.StringTokenizer;
import java.util.Vector;
 
 
 
 
public class Main{
	
	static int find(int x,int[] fa,int[] relation) {
		if(x==fa[x])return x;
		else {
			int f=fa[x];
			fa[x]=find(fa[x], fa, relation);
			relation[x]=(relation[x]+relation[f])%3;
			return fa[x];
		}
	}
	
	
	
	public static void main(String[] args) throws IOException{
		AReader input=new AReader();
	    PrintWriter out = new PrintWriter(new OutputStreamWriter(System.out));
	    int n=input.nextInt();
	    int k=input.nextInt();
	    int ans=0;
	    int[] fa=new int[n+1];
	    int[] relation=new int[n+1];
	    for(int i=1;i<=n;++i) fa[i]=i;
	    for(int i=1;i<=k;++i) {
	    	int op=input.nextInt();
	    	int x=input.nextInt();
	    	int y=input.nextInt();
	    	if(x>n||y>n) {
	    		ans++;
	    		continue;
	    	}
	    	if(op==1) {
	    		int fx=find(x, fa, relation);
	    		int fy=find(y, fa, relation);
	    		if(fx==fy) {
	    			if(relation[x]!=relation[y]) {
	    				ans++;
	    				continue;
	    			}
	    		}else {
	    			fa[fx]=fy;
	    			relation[fx]=(relation[y]+0-relation[x]+3)%3;
	    		}
	    	}else {
	    		if(x==y) {
	    			ans++;
	    			continue;
	    		}
	    		int fx=find(x, fa, relation);
	    		int fy=find(y, fa, relation);
	    		if(fx==fy) {//0吃1，1吃2，2吃0
	    			if(relation[x]==0&&relation[y]==1||
	    					relation[x]==1&&relation[y]==2||
	    					relation[x]==2&&relation[y]==0)continue;
	    			ans++;
	    			continue;
	    		}else {
	    			fa[fx]=fy;
	    			relation[fx]=(relation[y]+2-relation[x]+3)%3;
	    		}
	    	}
	    }
	    out.println(ans);
 	    out.flush();
	    out.close();
	}
	static
	class AReader{
	    BufferedReader bf;
	    StringTokenizer st;
	    BufferedWriter bw;
 
	    public AReader(){
	        bf=new BufferedReader(new InputStreamReader(System.in));
	        st=new StringTokenizer("");
	        bw=new BufferedWriter(new OutputStreamWriter(System.out));
	    }
	    public String nextLine() throws IOException{
	        return bf.readLine();
	    }
	    public String next() throws IOException{
	        while(!st.hasMoreTokens()){
	            st=new StringTokenizer(bf.readLine());
	        }
	        return st.nextToken();
	    }
	    public char nextChar() throws IOException{
	        //确定下一个token只有一个字符的时候再用
	        return next().charAt(0);
	    }
	    public int nextInt() throws IOException{
	        return Integer.parseInt(next());
	    }
	    public long nextLong() throws IOException{
	        return Long.parseLong(next());
	    }
	    public double nextDouble() throws IOException{
	        return Double.parseDouble(next());
	    }
	    public float nextFloat() throws IOException{
	        return Float.parseFloat(next());
	    }
	    public byte nextByte() throws IOException{
	        return Byte.parseByte(next());
	    }
	    public short nextShort() throws IOException{
	        return Short.parseShort(next());
	    }
	    public BigInteger nextBigInteger() throws IOException{
	        return new BigInteger(next());
	    }
	    public void println() throws IOException {
	        bw.newLine();
	    }
	    public void println(int[] arr) throws IOException{
	        for (int value : arr) {
	            bw.write(value + " ");
	        }
	        println();
	    }
	    public void println(int l, int r, int[] arr) throws IOException{
	        for (int i = l; i <= r; i ++) {
	            bw.write(arr[i] + " ");
	        }
	        println();
	    }
	    public void println(int a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(int a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(String a) throws IOException{
	        bw.write(a);
	        bw.newLine();
	    }
	    public void print(String a) throws IOException{
	        bw.write(a);
	    }
	    public void println(long a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(long a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(double a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	    public void print(double a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void print(char a) throws IOException{
	        bw.write(String.valueOf(a));
	    }
	    public void println(char a) throws IOException{
	        bw.write(String.valueOf(a));
	        bw.newLine();
	    }
	}
}