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算法-动态连通性

动态连通性直接听起来会稍微绕口一点,简单的说就是输入一列整数对,其中每个整数都表示某种类型的对象,假设输入的的整数对是p和q,我们可以理解p和q是相连的,假设相连是一种等价关系,一般具有三种特性自反性,对称性,传递性,根据上面的特性,如果整数对不存在某种等价关系,那么直接输出,如果存在就不输出。简单的举几个例子,计算机网络中判断两个计算机是否需要建立新的连接通信,如果可以通过一个或某几个节点能通信,那么我们就不需要建立新的连接,数学中可以将p

和q看成集合,跑题了,看下如何实现的吧,四种方式依次渐进:

Quick-Find算法

p和q做网络上相连可以看成连接,单独的看p和q可以看成触点,可以判断是否存在p和q或者pq之间的等价连接:

@interface DynamicUnion : NSObject    @property (strong,nonatomic) NSMutableArray  *ids;//存储每个触点对应的值   @property (assign,nonatomic) NSInteger  count;//已经连通的连接的数量   -(BOOL)connected:(NSInteger)a secondNumber:(NSInteger)b;//是否已经存在连接或者等价的连接  -(NSInteger)find:(NSInteger)index;//取出触点的值  -(void)dynamicUnion:(NSInteger)a secondNumber:(NSInteger)b;//a,b之间建立一个连接  -(void)initData:(NSInteger)count;//初始化触点的数量   @end

具体实现代码:

-(void)initData:(NSInteger)count{  self.count=count;  self.ids=[[NSMutableArray alloc]initWithCapacity:count];  for (NSInteger i=0; i<count; i++) {   [self.ids addObject:[NSNumber numberWithInteger:i]];  } } //http://www.cnblogs.com/xiaofeixiang -(BOOL)connected:(NSInteger)a secondNumber:(NSInteger)b{  return [self find:a]==[self find:b]; } -(NSInteger)find:(NSInteger)index{  return [[self.ids objectAtIndex:index] integerValue]; } -(void)dynamicUnion:(NSInteger)a secondNumber:(NSInteger)b{  NSInteger aID=[self find:a];  NSInteger bID=[self find:b];  if (aID==bID) {   return;  }  for (NSInteger i=0;i<[self.ids count]; i++) {   if ([[self.ids objectAtIndex:i] integerValue]==aID) {    self.ids[i]=[NSNumber numberWithInteger:bID];   }  }  self.count=self.count-1; } 

Quick-Union算法

Quick-Find在Union的过程中,每次都会遍历数组一次,这样有损性能,还是用ids数组存每个触点的值,不过每个值存的意义不一样,我们可以通过ids中的值存储触点的父级,作为一棵树存在,我们就不用遍历ids数组,简单的讲就是4,3的时候ids[4]存值的时候存的是3,这样最后会只要判断根节点就可以。其他方法不用变,我们只需要改变find和dynamicUnion方法即可。

-(NSInteger)find:(NSInteger)index{  while (index!=[[self.ids objectAtIndex:index] integerValue]) {   index=[[self.ids objectAtIndex:index] integerValue];  }  return index; } //http://www.cnblogs.com/xiaofeixiang -(void)dynamicUnion:(NSInteger)a secondNumber:(NSInteger)b{  NSInteger aRoot=[self find:a];  NSInteger bRoot=[self find:b];  if (aRoot==bRoot) {   return;  }  self.ids[aRoot]=[NSNumber numberWithInteger:bRoot];  self.count--;  } 

加权Quick-Union算法

Quick-Union可能会出现一种情况,如果以树的形式去展示的话,最终可能会出现大树的父级别是小树的情况,因为我们需要通过一个权重值,避免出现这种情况,不过我们需要回顾一个树的基础概念。一棵树的大小是它的节点的数量,树中的一个节点的深度是它到根节点的路径上的链接数。输的高度是它的所有节点中的最大深度。加权能保每次find、connected和union都是lgn级别。

//http://www.cnblogs.com/xiaofeixiang @interface DynamicUnionWeight : NSObject   @property (strong,nonatomic) NSMutableArray  *ids;//存储每个触点对应的值 @property (strong,nonatomic) NSMutableArray  *weightArr;//各个根节点对ing的分量的大小   @property (assign,nonatomic) NSInteger  count;//已经连通的连接的数量   -(BOOL)connected:(NSInteger)a secondNumber:(NSInteger)b;//是否已经存在连接或者等价的连接  -(NSInteger)find:(NSInteger)index;//取出触点的值  -(void)dynamicUnion:(NSInteger)a secondNumber:(NSInteger)b;//a,b之间建立一个连接  -(void)initData:(NSInteger)count;//初始化触点的数量  @end 

具体实现:

-(void)initData:(NSInteger)count{  self.count=count;  self.ids=[[NSMutableArray alloc]initWithCapacity:count];  self.weightArr=[[NSMutableArray alloc]initWithCapacity:count];  for (NSInteger i=0; i<count; i++) {   [self.ids addObject:[NSNumber numberWithInteger:i]];   [self.weightArr addObject:[NSNumber numberWithInteger:1]];  } } //http://www.cnblogs.com/xiaofeixiang -(BOOL)connected:(NSInteger)a secondNumber:(NSInteger)b{  return [self find:a]==[self find:b]; } -(NSInteger)find:(NSInteger)index{  while (index!=[[self.ids objectAtIndex:index] integerValue]) {   index=[[self.ids objectAtIndex:index] integerValue];  }  return index; } -(void)dynamicUnion:(NSInteger)a secondNumber:(NSInteger)b{  NSInteger i=[self find:a];  NSInteger j=[self find:b];  if (i==j) {   return;  }  NSInteger  weightA=[[self.weightArr objectAtIndex:i] integerValue];  NSInteger  weightB=[[self.weightArr objectAtIndex:j] integerValue];  if (weightA<weightB) {   self.ids[i]=[NSNumber numberWithInteger:j];   self.weightArr[j]=[NSNumber numberWithInteger:weightA+weightB];  }else{   self.ids[j]=[NSNumber numberWithInteger:i];   self.weightArr[i]=[NSNumber numberWithInteger:weightA+weightB];  }  self.count--; } 

路径压缩算法

路径压缩会保证union都接近于1,这个实现只需要加权Quick-Union的算法稍微改动一下,改变一下find即可:

-(NSInteger)find:(NSInteger)index{  while (index!=[[self.ids objectAtIndex:index] integerValue]) {    self.ids[index]=self.ids[[self.ids[index] integerValue]];   index=[[self.ids objectAtIndex:index] integerValue];  }  return index; } //http://www.cnblogs.com/xiaofeixiang -(void)dynamicUnion:(NSInteger)a secondNumber:(NSInteger)b{  NSInteger i=[self find:a];  NSInteger j=[self find:b];  if (i==j) {   return;  }  NSInteger  weightA=[[self.weightArr objectAtIndex:i] integerValue];  NSInteger  weightB=[[self.weightArr objectAtIndex:j] integerValue];  if (weightA<weightB) {   self.ids[i]=[NSNumber numberWithInteger:j];   self.weightArr[j]=[NSNumber numberWithInteger:weightA+weightB];  }else{   self.ids[j]=[NSNumber numberWithInteger:i];   self.weightArr[i]=[NSNumber numberWithInteger:weightA+weightB];  }  self.count--; } 

模拟测试:

DynamicUnion  *dynamicUnion=[[DynamicUnion alloc]init]; [dynamicUnion initData:10]; NSMutableArray *dataSource=[[NSMutableArray alloc]initWithCapacity:10]; [dataSource addObject:@"4,3"]; [dataSource addObject:@"3,8"]; [dataSource addObject:@"6,5"]; [dataSource addObject:@"9,4"]; [dataSource addObject:@"2,1"]; [dataSource addObject:@"8,9"]; [dataSource addObject:@"5,0"]; [dataSource addObject:@"7,2"]; [dataSource addObject:@"6,1"]; [dataSource addObject:@"1,0"]; for (NSInteger i=0; i<[dataSource count]; i++) {  NSString *node=[dataSource objectAtIndex:i];  NSInteger a=[[node substringWithRange:NSMakeRange(0, 1)] integerValue];  NSInteger b=[[node substringWithRange:NSMakeRange(2, 1)] integerValue];  if ([dynamicUnion connected:a secondNumber:b]) {   continue;  }  [dynamicUnion dynamicUnion:a secondNumber:b];  NSLog(@"%ld---%ld",a,b); } NSLog(@"%ld已存在连接",dynamicUnion.count); NSLog(@"iOS技术交流群:228407086"); NSLog(@"原文地址:http://www.cnblogs.com/xiaofeixiang"); 

结果如下:

算法-动态连通性

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