128 lines
3.6 KiB
Text
128 lines
3.6 KiB
Text
---
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id: BCC-2CC
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title: "BCCs and 2CCs"
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author: Benjamin Qi
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prerequisites:
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- Silver - DFS
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- Platinum - Binary Jumping
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description: "Precomputing parents of nodes in powers of two to find the xth parent of a node efficently."
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frequency: 1
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---
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import { Problem } from "../models";
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export const metadata = {
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problems: {
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sam2: [
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new Problem("YS", "Two-Edge-Connected Components", "two_edge_connected_components", "Easy", false, [], ""),
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],
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disrupt: [
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new Problem("Plat", "Disruption", "842", "Normal", false, ["Merging"]),
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],
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probs2: [
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new Problem("CSA", "One-Way Streets", "one-way-streets", "Easy", false, [], ""),
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],
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bccSam: [
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new Problem("CSES", "Forbidden Cities", "1705", "Normal", false, [], ""),
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],
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gen: [
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new Problem("POI", "Blockade", "https://szkopul.edu.pl/problemset/problem/eDt8w290owtatmCjad0O0ywk/site/?key=statement", "Normal", false, [], ""),
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new Problem("ojuz", "APIO - Duathlon", "APIO18_duathlon", "Normal", false, [], ""),
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new Problem("Plat", "Push a Box", "769", "Very Hard", false, [], ""),
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new Problem("DMOJ", "Investment", "tle17c1p6", "Hard", false, [], ""),
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new Problem("ojuz", "CEOI - Pipes", "CEOI15_pipes", "Hard", false, [], ""),
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],
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}
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};
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<resources>
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<resource source="CF" title="DFS Tree + Bridges" url="blog/entry/68138"> </resource>
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</resources>
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## 2-Edge-Connected Components
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<problems-list problems={metadata.problems.sam2} />
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(implementation)
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### With DSU
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<problems-list problems={metadata.problems.disrupt} />
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The analysis for the above problem mentions an $O(m\alpha(n))$ solution. Although this is not a two-connected component problem, we can in fact use DSU to generate two-connected components.
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(explanation?)
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<spoiler title="Code">
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The DSU operations take $O(\log n)$ rather than $O(\alpha(n))$ because the DSU does not use union by size, but it's easy to change this.
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```cpp
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struct TwoEdgeCC {
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struct {
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vi e; void init(int n) { e = vi(n,-1); }
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int get(int x) { return e[x] < 0 ? x : e[x] = get(e[x]); }
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bool unite(int x, int y) { // set par[y] = x
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x = get(x), y = get(y); if (x == y) return 0;
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e[x] += e[y]; e[y] = x; return 1;
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}
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} DSU;
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int N; vector<vi> adj; vi depth, par;
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vpi extra;
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void init(int _N) {
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N = _N; DSU.init(N);
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adj.rsz(N), depth.rsz(N), par = vi(N,-1);
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}
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void dfs(int x) {
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trav(t,adj[x]) if (t != par[x])
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par[t] = x, depth[t] = depth[x]+1, dfs(t);
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}
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void ae(int a, int b) {
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if (DSU.unite(a,b)) adj[a].pb(b), adj[b].pb(a); // edge of forest
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else extra.pb({a,b}); // extra edge
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}
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void ad(int a, int b) {
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while (1) {
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a = DSU.get(a), b = DSU.get(b);
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if (a == b) return;
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if (depth[a] < depth[b]) swap(a,b);
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assert(par[a] != -1 && DSU.unite(par[a],a));
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}
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}
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void gen() {
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F0R(i,N) if (par[i] == -1) dfs(i); // independently for each connected component
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DSU.init(N); trav(t,extra) ad(t.f,t.s); // add non-spanning edges
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}
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};
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```
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</spoiler>
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### Problems
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<problems-list problems={metadata.problems.probs2} />
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- SRM 787 1000
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## [Biconnected Components](https://en.wikipedia.org/wiki/Biconnected_component)
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<problems-list problems={metadata.problems.bccSam} />
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note that BCCs contain EDGES not VERTICES
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Related topics include
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- Articulation Points
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- Bridges
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- Block-Cut Tree
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### Tutorial
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<resources>
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<resource source="GFG" title="Articulation Points (aka Cut Vertices)" url="articulation-points-or-cut-vertices-in-a-graph">maybe not completely correct</resource>
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</resources>
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(implementation)
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### Problems
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<problems-list problems={metadata.problems.gen} /> |