算法-有向图及可达性

图是由顶点和边连接而成，如果边是没有方向的是就是之前文章中说的无向图，关于无向图可以参考本人之前的文章，如果边是有方向的，则称之为有向图。从顶点A→B，我们可以理解为A到B可达，有向图和无向图一样通过邻接表保存每一条边，由于边是有方向的，因此在添加边的过程中只需要添加一条边即可。关于可达性一个节点数组的可达性，采用的方法是之前的深度优先搜索一样的代码，通过递归将标记位Bool标记位判断数组中每个顶点的可达性。为了测试，选择下面一张有向图:

有向图基础

通过图片我们可以发现图中有13个节点，22条边，顶点0指出的节点有1，5，指入的节点有2，6，我们先实现所有顶点指出的节点，之后可以通过反转判断所有节点的指入节点：

@interface Digraph : NSObject

//顶点的总数
@property  (assign,nonatomic) NSInteger  vertexs;
//边的数总数
@property (assign,nonatomic) NSInteger  edges;
//连接点的边
@property (strong,nonatomic)  NSMutableArray  *adjDataSource;

-(instancetype)initWithVertex:(NSInteger)vertexs;
//添加一条有向边 startVertex→endVertex
-(void)addEdges:(NSInteger)startVertex  endVertex:(NSInteger)endVertex;

-(Digraph *)reverse;//该图的反向图

@end

实现代码:

@implementation Digraph

-(instancetype)initWithVertex:(NSInteger)vertexs{
self=[super init];
if (self) {
self.vertexs=vertexs;
for (NSInteger i=0; i<vertexs; i++) {
NSMutableArray  *neighbourVertex=[[NSMutableArray alloc]initWithCapacity:1];
[self.adjDataSource addObject:neighbourVertex];//创建邻接表,将所有链表初始化为空
}
}
return self;
}
//http://www.cnblogs.com/xiaofeixiang
-(void)addEdges:(NSInteger)startVertex endVertex:(NSInteger)endVertex{
//将endVertex添加到startVertex的链表中
[self.adjDataSource[startVertex] insertObject:[NSNumber numberWithInteger:endVertex] atIndex:0];
self.edges=self.edges+1;
}

-(Digraph *)reverse{
Digraph  *digraph=[[Digraph alloc]initWithVertex:self.vertexs];
for (NSInteger i=0; i<self.vertexs; i++) {
NSMutableArray  *tempArr=self.adjDataSource[i];
for (NSInteger j=0; j<[tempArr count]; j++) {
[digraph addEdges:[tempArr[j] integerValue] endVertex:i];
}
}
return digraph;
}

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

@end

测试代码:

Digraph  *graph=[[Digraph alloc]initWithVertex:13];
[graph addEdges:4 endVertex:2];
[graph addEdges:2 endVertex:3];
[graph addEdges:3 endVertex:2];
[graph addEdges:6 endVertex:0];
[graph addEdges:0 endVertex:1];
[graph addEdges:2 endVertex:0];
[graph addEdges:11 endVertex:12];
[graph addEdges:12 endVertex:9];
[graph addEdges:9 endVertex:10];
[graph addEdges:9 endVertex:11];
[graph addEdges:8 endVertex:9];
[graph addEdges:10 endVertex:12];
[graph addEdges:11 endVertex:4];
[graph addEdges:4 endVertex:3];
[graph addEdges:3 endVertex:5];
[graph addEdges:7 endVertex:8];
[graph addEdges:8 endVertex:7];
[graph addEdges:5 endVertex:4];
[graph addEdges:0 endVertex:5];
[graph addEdges:6 endVertex:4];
[graph addEdges:6 endVertex:9];
[graph addEdges:7 endVertex:6];
for (NSInteger i=0; i<[graph.adjDataSource count]; i++) {
NSLog(@"节点%ld指出→的节点:%@",i,[graph.adjDataSource[i] componentsJoinedByString:@"--"]);
}
NSLog(@"技术交流群:%@",@"228407086");
NSLog(@"原文地址:http://www.cnblogs.com/xiaofeixiang");

测试效果:



现在可以判断出顶点的指出节点，实现文件中有一个reverse方法将图反转，求出顶点的转入节点:

Digraph  *digraph=[graph reverse];
for (NSInteger i=0; i<[digraph.adjDataSource count]; i++) {
NSLog(@"指入%ld⬅️的节点:%@",i,[digraph.adjDataSource[i] componentsJoinedByString:@"--"]);
}
NSLog(@"技术交流群:%@",@"228407086");
NSLog(@"原文地址:http://www.cnblogs.com/xiaofeixiang");

测试效果:

可达性

可达性的判断和之前的深度优先搜索基本没变化，先来看一下需要实现的方法:

@interface DirectedDFS : NSObject

//标记数组
@property  (strong,nonatomic)  NSMutableArray  *marked;

//找到arr中顶点可达的所有顶点
-(instancetype)initDirectedDFSWithVertex:(Digraph *)graph  vertexArr:(NSArray *)arr;
//在graph中找到从vertex可达的所有顶点
-(void)directedDFS:(Digraph *)graph  vertex:(NSInteger)vertex;

//vertex是否可达
-(Boolean)isMarked:(NSInteger)vertex;

@end

实现代码:

@implementation DirectedDFS

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

-(instancetype)initDirectedDFSWithVertex:(Digraph *)graph vertexArr:(NSArray *)arr{
self=[super init];
if (self) {
for (NSInteger i=0; i<graph.vertexs;i++) {
[self.marked addObject:[NSNull null]];
}
//遍历有向图中的顶点
for (NSInteger j=0; j<[arr count]; j++) {
if (![self isMarked:[arr[j] integerValue]]) {
[self directedDFS:graph vertex:[arr[j] integerValue]];
}
}
}
return self;
}
//博客园-FlyElephant:http://www.cnblogs.com/xiaofeixiang/
-(void)directedDFS:(Digraph *)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]) {
[self directedDFS:graph vertex:temp];
}
}
}

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

@end

测试代码:

NSArray  *sources=[NSArray arrayWithObjects:@"2", nil];
DirectedDFS  *directedDFS=[[DirectedDFS alloc]initDirectedDFSWithVertex:graph vertexArr:sources];
NSMutableArray  *reachableArr=[[NSMutableArray alloc]initWithCapacity:1];
for (NSInteger i=0; i<graph.vertexs; i++) {

if (directedDFS.marked[i]&&directedDFS.marked[i]!=[NSNull null]) {
[reachableArr addObject:[NSNumber numberWithInteger:i]];
}
}
NSLog(@"可达的节点:%@",[reachableArr componentsJoinedByString:@"--"]);
NSLog(@"技术交流群:%@",@"228407086");
NSLog(@"博客园-FlyElephant:http://www.cnblogs.com/xiaofeixiang");

测试效果: