From 66a22f8547b2d9c5ad95287ac7938d2a0f5bc9fe Mon Sep 17 00:00:00 2001 From: CyC2018 Date: Fri, 30 Oct 2026 20:19:18 +0800 Subject: [PATCH] =?UTF-8?q?Update=20=E7=AE=97=E6=B3=95=20-=20=E5=B9=B6?= =?UTF-8?q?=E6=9F=A5=E9=9B=86.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 重新绘制 Quick Union 树结构配图;调整 Quick Union 与加权合并的原理说明,补充路径压缩实现代码及复杂度分析。 --- notes/pics/uf-quick-union-tree.png | Bin 87376 -> 87789 bytes notes/算法 - 并查集.md | 30 +++++++++++++++++++++++------ 2 files changed, 24 insertions(+), 6 deletions(-) diff --git a/notes/pics/uf-quick-union-tree.png b/notes/pics/uf-quick-union-tree.png index 5007daa6a859fc6b091140c6d82a144a9ab6bad9..e9df9dd6eb1c559528607593664626e09e844dfa 100644 GIT binary patch literal 87789 zcmX6_Wmr~S)4k~~=}rOZ?v{`TDV6T-?rxMW=>`R)L%JL3?nXemyS~l)ogW4loco-; zXV$D)Yc^p@3NmQOM92^b1WopX{*XR=C6tuR3RxAT)#~r(fruEG^$QPttTboF%K2q%!XG{5qN9uh!Gwi!3-N=76Ym%zeV5EMgcs3k 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@@ -94,7 +94,7 @@ public class QuickFindUF extends UF { 可以快速进行 union 操作,只需要修改一个节点的 id 值即可。 -但是 find 操作开销很大,因为同一个连通分量的节点 id 值不同,id 值只是用来指向另一个节点。因此需要一直向上查找操作,直到找到最上层的节点。 +这种方法把每个连通分量看成一棵树:id[p] 保存的是 p 的父节点,根节点的 id 等于它自身。同一个连通分量的节点 id 值不一定相同,但根节点相同,因此 find 需要沿着 id 一直向上查找,直到找到根节点,开销较大。

@@ -133,16 +133,16 @@ public class QuickUnionUF extends UF { ## 加权 Quick Union -为了解决 quick-union 的树通常会很高的问题,加权 quick-union 在 union 操作时会让较小的树连接到较大的树上面。 +为了解决 Quick Union 的树通常会很高的问题,加权 Quick Union 在 union 操作时会让节点较少的树连接到节点较多的树上面。 -理论研究证明,加权 quick-union 算法构造的树深度最多不超过 logN。 +这样构造的树深度最多不超过 logN。这是因为一个节点的深度只在它所在的树连接到另一棵树上时加 1,而那棵树的节点数不少于它,合并后的树至少是原来的两倍大;N 个节点的树最多只能翻倍 logN 次。

```java public class WeightedQuickUnionUF extends UF { - // 保存节点的数量信息 + // 以该节点为根的树的节点数,只有根节点的值有意义 private int[] sz; @@ -189,6 +189,24 @@ public class WeightedQuickUnionUF extends UF {

+```java +@Override +public int find(int p) { + int root = p; + while (root != id[root]) { + root = id[root]; + } + while (p != root) { // 把路径上的节点直接链接到根节点 + int next = id[p]; + id[p] = root; + p = next; + } + return root; +} +``` + +这样处理后,一组 union 和 find 操作的均摊代价非常接近 1(与反阿克曼函数 α(N) 成正比,实际中 α(N) 不超过 4),但单次操作在最坏情况下仍与 logN 成正比。 + ## 比较 | 算法 | union | find | @@ -196,7 +214,7 @@ public class WeightedQuickUnionUF extends UF { | Quick Find | N | 1 | | Quick Union | 树高 | 树高 | | 加权 Quick Union | logN | logN | -| 路径压缩的加权 Quick Union | 非常接近 1 | 非常接近 1 | +| 路径压缩的加权 Quick Union | 均摊非常接近 1 | 均摊非常接近 1 | ## 参考资料