Merge pull request #4347 from danielzsh/patch8

content/5_Plat/Centroid.mdx

@@ -30,10 +30,6 @@ or equal to $\frac{N}{2}$, then it is a centroid. Otherwise, move to the child
 with a subtree size that is more than $\frac{N}{2}$ and repeat until you find a
 centroid.
 
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 ### Implementation
 
 <LanguageSection>
@@ -198,6 +194,28 @@ subgraphs.
 The implementation by
 [Carpanese](https://medium.com/carpanese/an-illustrated-introduction-to-centroid-decomposition-8c1989d53308#d10f)
 (ll. 20-21) leads to a segmentation fault due to an invalidated iterator in the range-based for loop after erasing the element.
+Instead, you can store an `is_removed` array and check it before visiting children (this approach is presented below).
+Alternatively, the following code in the Carpanese article (line 3 is the problematic one):
+
+```cpp
+for (auto v : tree[centroid])
+	tree[centroid].erase(v),      // remove the edge to disconnect
+	    tree[v].erase(centroid),  // the component from the tree
+	    build(v, centroid);
+```
+
+Can be rewritten like so:
+
+```cpp
+for (auto v : tree[centroid]) {
+	tree[v].erase(centroid);
+	build(v, centroid);
+}
+tree[centroid].clear();
+```
+
+The article also mentions an update complexity of $\mathcal{O}(\log^2n)$ because it assumes an additional factor of $\mathcal{O}(\log n)$ to calculate the distance between a node and a given centroid ancestor.
+However, we can precompute these (as demonstrated in the code below) to reduce update complexity to $\mathcal{O}(\log n)$ and overall complexity to $\mathcal{O}((N + Q)\log n)$.
 
 </Warning>