[leetcode] 785. Is Graph Bipartite?

Description

Given an undirected graph, return true if and only if it is bipartite.

Recall that a graph is bipartite if we can split it’s set of nodes into two independent subsets A and B such that every edge in the graph has one node in A and another node in B.

The graph is given in the following form: graph[i] is a list of indexes j for which the edge between nodes i and j exists. Each node is an integer between 0 and graph.length - 1. There are no self edges or parallel edges: graph[i] does not contain i, and it doesn’t contain any element twice.

Example 1:

Input: [[1,3], [0,2], [1,3], [0,2]]
Output: true
Explanation: 
The graph looks like this:
0----1
|    |
|    |
3----2
We can divide the vertices into two groups: {0, 2} and {1, 3}.      

Example 2:

Input: [[1,2,3], [0,2], [0,1,3], [0,2]]
Output: false
Explanation: 
The graph looks like this:
0----1
| \  |
|  \ |
3----2
We cannot find a way to divide the set of nodes into two independent subsets.      

Note:

• graph will have length in range [1, 100].
• graph[i] will contain integers in range [0, graph.length - 1].
• graph[i] will not contain i or duplicate values.
• The graph is undirected: if any element j is in graph[i], then i will be in graph[j].

分析

題目的意思是：判斷一個無向圖是否是一個二分圖。

• 我們采用一種很機智的染色法，大體上的思路是要将相連的兩個頂點染成不同的顔色，一旦在染的過程中發現有相連的兩個頂點已經被染成相同的顔色，說明不是二分圖。

代碼

class Solution {
public:
    bool isBipartite(vector<vector<int>>& graph) {
        vector<int> colors(graph.size());
        for(int i=0;i<graph.size();i++){
            if(colors[i]==0&&!isValid(graph,i,1,colors)){
                return false;
            }
        }
        return true;
    }
    bool isValid(vector<vector<int>>& graph,int cur,int color,vector<int>& colors){
        if(colors[cur]!=0) return colors[cur]==color;
        colors[cur]=color;
        for(int i:graph[cur]){
            if(!isValid(graph,i,-1*color,colors)){
                return false;
            }
        }
        return true;
    }
};      

代碼二 （python）

class Solution:
    def isValid(self,graph,pos):
        for i in graph[pos]:
            if(i in self.colors):
                if(self.colors[i]==self.colors[pos]):
                    return False
            else:
                self.colors[i]=1-self.colors[pos]
                if(not self.isValid(graph,i)):
                    return False
        return True
            
        
    def isBipartite(self, graph: List[List[int]]) -> bool:
        self.colors={}
        for i in range(len(graph)):
            if(i not in self.colors):
                self.colors[i]=0
                if(not self.isValid(graph,i)):
                    return False
        return True      

參考文獻