C++并查集算法简单详解

1、并查集的初始化并查集是用一个数组实现的。首先先定义一个数组1intfather[N];father[i]表示元素i的父亲结点。接下来进行初始化。一开始每个元素都分别是独立的一个集合父亲结点就是它自己所以初始化时将所有father[i]等于i123for(inti 1; i N; i){father[i] i;}这样就将father数组初始化完毕。2、并查集的查找操作由于规定同一个集合中只存在一个根结点因此查找操作就是查找给定结点的根结点的过程。可以通过递推或递归来实现思路都是一样的都是反复寻找父亲结点直到找到根结点为止。递推代码1234567//findFather函数返回元素x所在集合的根结点intfindFather(intx){while(x ! father[x]){//如果不是根结点继续循环x father[x];//获得自己的父亲结点}returnx;}上述代码中 while(x ! father[x])说明当x的父亲结点不等于本身时也就是x不是根结点时就继续循环因为父亲结点等于本身这个情况只有在根结点才会出现。递归代码1234intfindFather(intx){if(x father[x])returnx;//如找到根结点就返回根结点编号xelsereturnfindFather(father[x]);//否则递归判断x的父亲结点是否是根结点}3、并查集的合并操作合并就是把两个集合合并成一个集合。实现过程是先判断两个元素是否属于同一个集合不属于同一个集合就开始进行合并操作。判断两个元素是否属于同一个集合的具体思路就是调用上面的findFather函数分别查找两个元素所属集合的根结点根结点不同则两个元素不属于同一个集合。合并两个集合的具体思路就是将其中一个集合的根结点的父亲指向另外一个集合的根结点即可。合并操作的代码实现(假设有两个集合一个集合里有元素a一个集合有元素b)1234voidUnion(inta,intb){//让一个集合的根结点的父亲指向另一个集合的根结点father(findFather(a)) findFather(b);}注意合并操作之前最好先判断下待合并的两个元素是否位于同一个集合。4、为什么要路径压缩5、实现路径压缩由于findFather函数目的就是查找根结点所以我们在查找结点的路径上直接将所有结点的父亲都指向根结点查找的时候就不必一直回溯去寻找父亲了查询的复杂度可以降为O(1)。比如下面这张图观察图不难发现上图中father[1] 1father[2] 1father[3] 2father[4] 3。经过路径压缩就变成下面这幅图相当于将所有结点的父亲都直接指向根结点这就是路径压缩。如何用代码实现路径压缩呢以下是具体代码1234intfindFather(intx){if(father[x] ! x) father[x] findFather(father[x]);returnfather[x];}以上代码实现了在查询获取根结点的同时将路径进行压缩优化代码虽然很短但是很巧妙下面解释下上述代码if(father[x] ! x)当所查找的元素x的父亲结点不是自己也就是x不是根结点时findFather(father[x])就继续递归查找父结点直到找到根结点为止father[x] findFather(father[x])然后将找到的根结点直接赋给x的父亲结点。这样就实现了路径压缩即将结点的父亲直接指向根结点。return 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