算法-無向圖(連通分量，是否有環和二分圖)

前面的文章實現了無向圖深度優先搜索和廣度優先搜索解決了無向圖中的路徑尋找，不過無向圖中還有幾個比較常見的問題需要解決，判斷圖中的連通分量，在無向圖中，如果從頂點vi到頂點vj有路徑，則稱vi和vj連通。如果圖中任意兩個頂點之間都連通，則稱該圖為連通圖，否則，稱該圖為非連通圖，則其中的極大連通子圖稱為連通分量，這里所謂的極大是指子圖中包含的頂點個數極大。

連通分量

為了編程和理解，我們還是使用之前文章的非連通圖，如下圖所示，圖中有三個連通分量，看着這個圖對照上文中的概念比較好理解:

 

代碼中跟深度優先搜索有所改動，需要一個新的數組存儲對應的值，數組中0-6都屬於一個連通分量，那么我們可以將數組中索引在0-6之間的變量賦值為0，代碼如下:

@interface GraphCC : NSObject

//記錄頂點是否被標記
@property  (strong,nonatomic)  NSMutableArray  *marked;

@property (assign,nonatomic)  NSInteger count;//連通的分量

@property  (strong,nonatomic)  NSMutableArray  *ids;//頂點所在的連通分量的標識符

//連通分量遞歸初始化
-(instancetype)initWithGraph:(Graph *)graph;

-(void)depthSearch:(Graph *)graph  vertex:(NSInteger)vertex;


@end

實現代碼如下:

@implementation GraphCC

#pragma mark  getter and setter
-(NSMutableArray *)marked{
    if (!_marked) {
        _marked=[[NSMutableArray alloc]initWithCapacity:1];
    }
    return _marked;
}

-(NSMutableArray *)ids{
    if (!_ids) {
        _ids=[[NSMutableArray alloc]initWithCapacity:1];
    }
    return _ids;
}

-(instancetype)initWithGraph:(Graph *)graph{
    self=[super init];
    if (self) {
        for (NSInteger i=0; i<graph.vertexs;i++) {
            [self.marked addObject:[NSNull null]];
            [self.ids addObject:[NSNull null]];
        }
        //遍歷圖的頂點
        for (NSInteger j=0; j<graph.vertexs; j++) {
            if (![self isMarked:j]) {
                [self depthSearch:graph vertex:j];
                self.count++;
            }
        }
    }
    return self;
}
//博客園-FlyElephant:http://www.cnblogs.com/xiaofeixiang/
-(void)depthSearch:(Graph *)graph vertex:(NSInteger)vertex{
    self.marked[vertex]=[NSNumber numberWithBool:true];
    //同一分量中頂點的賦值
    self.ids[vertex]=[NSNumber numberWithInteger:self.count];
    for (NSInteger i=0; i<[graph.adjDataSource[vertex] count]; i++) {
        NSInteger temp=[[graph.adjDataSource[vertex] objectAtIndex:i] integerValue];
        if (![self isMarked:temp]) {
            [self depthSearch:graph vertex:temp];
        }
    }
}

-(Boolean)isMarked:(NSInteger)vertex{
    return self.marked[vertex]==[NSNull null]?false:[self.marked[vertex] boolValue];
}

@end

測試代碼:

        Graph  *graph=[[Graph alloc]initWithVertex:13];
        [graph addEdges:0 endVertex:1];
        [graph addEdges:0 endVertex:2];
        [graph addEdges:0 endVertex:5];
        [graph addEdges:0 endVertex:6];
        [graph addEdges:3 endVertex:4];
        [graph addEdges:3 endVertex:5];
        [graph addEdges:4 endVertex:5];
        [graph addEdges:4 endVertex:6];
        [graph addEdges:7 endVertex:8];
        [graph addEdges:9 endVertex:10];
        [graph addEdges:9 endVertex:11];
        [graph addEdges:9 endVertex:12];
        [graph addEdges:11 endVertex:12];
        GraphCC  *graphCC=[[GraphCC alloc]initWithGraph:graph];
        for (NSInteger i=0; i<graphCC.count; i++) {
            NSMutableArray  *dataSource=[[NSMutableArray alloc]initWithCapacity:1];
            for (NSInteger j=0; j<graph.vertexs; j++) {
                if ([graphCC.ids[j] integerValue]==i) {
                    [dataSource addObject:[NSNumber numberWithInteger:j]];
                }
            }
            NSLog(@"分量%ld:%@",i,[dataSource componentsJoinedByString:@"--"]);
        }

效果如下:

是否有環

環簡單就是幾個頂點之間是否存在閉合，從頂點1通過若干個頂點是否可以返回到頂點1，此次判斷通過之前文章的一個連通圖:

@interface GraphCycle : NSObject

//記錄頂點是否被標記
@property  (strong,nonatomic)  NSMutableArray  *marked;

@property (assign,nonatomic)  Boolean hasCycle;
//初始化
-(instancetype)initWithGraph:(Graph *)graph;

-(void)depthSearch:(Graph *)graph  vertex:(NSInteger)vertex nextVertex:(NSInteger)nextVertex;

@end

實現代碼:

@implementation GraphCycle
#pragma mark  getter and setter
-(NSMutableArray *)marked{
    if (!_marked) {
        _marked=[[NSMutableArray alloc]initWithCapacity:1];
    }
    return _marked;
}


-(instancetype)initWithGraph:(Graph *)graph{
    self=[super init];
    if (self) {
        for (NSInteger i=0; i<graph.vertexs;i++) {
            [self.marked addObject:[NSNull null]];
        }
        //遍歷圖的頂點
        for (NSInteger s=0; s<graph.vertexs; s++) {
            if (![self isMarked:s]) {
                [self depthSearch:graph vertex:s nextVertex:s];
            }
        }
    }
    return self;
}
//http://www.cnblogs.com/xiaofeixiang/
-(void)depthSearch:(Graph *)graph vertex:(NSInteger)vertex nextVertex:(NSInteger)nextVertex{
    self.marked[vertex]=[NSNumber numberWithBool:true];
    for (NSInteger i=0; i<[graph.adjDataSource[vertex] count]; i++) {
        NSInteger temp=[[graph.adjDataSource[vertex] objectAtIndex:i] integerValue];
        if (![self isMarked:temp]) {
            [self depthSearch:graph vertex:temp nextVertex:vertex];
        }
        //最開始檢測到的環是0-1-2
        else if (temp!=nextVertex) self.hasCycle=true;
    }
}

-(Boolean)isMarked:(NSInteger)vertex{
    return self.marked[vertex]==[NSNull null]?false:[self.marked[vertex] boolValue];
}

@end

測試代碼:

        Graph  *graph=[[Graph alloc]initWithVertex:6];
        [graph addEdges:0 endVertex:5];
        [graph addEdges:2 endVertex:4];
        [graph addEdges:2 endVertex:3];
        [graph addEdges:1 endVertex:2];
        [graph addEdges:0 endVertex:1];
        [graph addEdges:3 endVertex:4];
        [graph addEdges:5 endVertex:3];
        [graph addEdges:0 endVertex:2];
        GraphCycle *cycle=[[GraphCycle alloc]initWithGraph:graph];
        if ([cycle hasCycle]) {
            NSLog(@"圖中含有環");
        }else{
            NSLog(@"圖中不含有環");
        }
        NSLog(@"技術交流群:%@",@"228407086");
        NSLog(@"原文地址:http://www.cnblogs.com/xiaofeixiang");

測試效果:

二分圖

我們也許會看到過這么一個問題就是是否能夠用兩種顏色將圖中的所有頂點着色，，使得任意一條邊的兩個點都不相同，其實這個問題等價於圖是否是二分圖：

@interface GraphTwoColor : NSObject

//標記數組
@property  (strong,nonatomic)  NSMutableArray  *marked;

@property  (strong,nonatomic)  NSMutableArray  *color;

@property (assign,nonatomic)  Boolean isTwoColorable;
//初始化
-(instancetype)initWithGraph:(Graph *)graph;
//深度搜索
-(void)depthSearch:(Graph *)graph  vertex:(NSInteger)vertex;


@end

實現代碼:

@implementation GraphTwoColor
#pragma mark  getter and setter
-(NSMutableArray *)marked{
    if (!_marked) {
        _marked=[[NSMutableArray alloc]initWithCapacity:1];
    }
    return _marked;
}

-(NSMutableArray *)color{
    if (!_color) {
        _color=[[NSMutableArray alloc]initWithCapacity:1];
    }
    return _color;
}


-(instancetype)initWithGraph:(Graph *)graph{
    self=[super init];
    if (self) {
        for (NSInteger i=0; i<graph.vertexs;i++) {
            [self.marked addObject:[NSNull null]];
            [self.color addObject:[NSNull null]];
        }
        self.isTwoColorable=true;
        //遍歷圖的頂點
        for (NSInteger s=0; s<graph.vertexs; s++) {
            if (![self isMarked:s]) {
                [self depthSearch:graph vertex:s];
            }
        }
    }
    return self;
}
//http://www.cnblogs.com/xiaofeixiang/
-(void)depthSearch:(Graph *)graph vertex:(NSInteger)vertex{
    self.marked[vertex]=[NSNumber numberWithBool:true];
    
    for (NSInteger i=0; i<[graph.adjDataSource[vertex] count]; i++) {
        NSInteger temp=[[graph.adjDataSource[vertex] objectAtIndex:i] integerValue];
        if (![self isMarked:temp]) {
            //二分圖兩個相鄰的節點顏色不一樣
            self.color[temp]=[NSNumber numberWithBool:![self isColor:vertex]];
            [self depthSearch:graph vertex:temp];
        }
        //相鄰的節點顏色相同則不是二分圖
        else if ([self isColor:temp]==[self isColor:vertex]) self.isTwoColorable=false;
    }
}

-(Boolean)isMarked:(NSInteger)vertex{
    return self.marked[vertex]==[NSNull null]?false:[self.marked[vertex] boolValue];
}

-(Boolean)isColor:(NSInteger)vertex{
    return self.color[vertex]==[NSNull null]?false:[self.color[vertex] boolValue];
}

@end

測試代碼:

        Graph  *graph=[[Graph alloc]initWithVertex:6];
        [graph addEdges:0 endVertex:5];
        [graph addEdges:2 endVertex:4];
        [graph addEdges:2 endVertex:3];
        [graph addEdges:1 endVertex:2];
        [graph addEdges:0 endVertex:1];
        [graph addEdges:3 endVertex:4];
        [graph addEdges:5 endVertex:3];
        [graph addEdges:0 endVertex:2];
        GraphTwoColor  *graphColor=[[GraphTwoColor alloc]initWithGraph:graph];
        if ([graphColor isTwoColorable]) {
            NSLog(@"圖是一個二分圖");
        }else{
            NSLog(@"圖不是一個二分圖");
        }
        NSLog(@"技術交流群:%@",@"228407086");
        NSLog(@"原文地址:http://www.cnblogs.com/xiaofeixiang");

測試效果:

如果我們修改一下Graph，比如圖只有四個頂點，四條邊，肯定是一個二分圖:

                Graph  *graph=[[Graph alloc]initWithVertex:4];
                [graph addEdges:0 endVertex:1];
                [graph addEdges:0 endVertex:2];
                [graph addEdges:1 endVertex:3];
                [graph addEdges:2 endVertex:3];