[WIP] Matroid Intersection code

content/graph/MatroidIntersection.h

@@ -0,0 +1,81 @@
+/**
+ * Author: tfg, Aeren, pajenegod, chilli
+ * Date: 2020-04-15
+ * License: CC0
+ * Source: https://codeforces.com/blog/entry/69287
+ * Description: Given two matroids, finds the largest common independent set.
+ * For the color and graph matroids, this would be the largest forest where no
+ * two edges are the same color. A matroid has 3 functions
+ *  - check(int x): returns if current matroid can add x without becoming dependent
+ *  - add(int x): adds an element to the matroid (guaranteed to never make it dependent)
+ *  - clear(): sets the matroid to the empty matroid
+ * The matroid is given an int representing the element, and is expected to
+ * convert it (e.g: the color or the endpoints)
+ * Pass the matroid with more expensive add/clear operations to M1.
+ * Time: R^2N(M2.add + M1.check + M2.check) + R^3 M1.add + R^2 M1.clear + RN M2.clear
+ * Status: Tested on SWERC 2011D, Pick Your Own Nim
+ * Usage:
+ * Details:
+ */
+#include "../data-structures/UnionFind.h"
+
+struct ColorMat {
+	vi cnt, clr;
+	ColorMat(int n, vector<int> clr) : cnt(n), clr(clr) {}
+	bool check(int x) { return !cnt[clr[x]]; }
+	void add(int x) { cnt[clr[x]]++; }
+	void clear() { fill(all(cnt), 0); }
+};
+struct GraphMat {
+	UF uf;
+	vector<array<int, 2>> e;
+	GraphMat(int n, vector<array<int, 2>> e) : uf(n), e(e) {}
+	bool check(int x) { return !uf.sameSet(e[x][0], e[x][1]); }
+	void add(int x) { uf.join(e[x][0], e[x][1]); }
+	void clear() { uf = UF(sz(uf.e)); }
+};
+template <class M1, class M2> struct MatroidIsect {
+	int n;
+	vector<char> iset;
+	M1 m1; M2 m2;
+	MatroidIsect(M1 m1, M2 m2, int n) : n(n), iset(n + 1), m1(m1), m2(m2) {}
+	vi solve() {
+		rep(i,0,n) if (m1.check(i) && m2.check(i))
+			iset[i] = true, m1.add(i), m2.add(i);
+		while (augment());
+		vi ans;
+		rep(i,0,n) if (iset[i]) ans.push_back(i);
+		return ans;
+	}
+	bool augment() {
+		vector<int> frm(n, -1);
+		queue<int> q({n}); // starts at dummy node
+		auto fwdE = [&](int a) {
+			vi ans;
+			m1.clear();
+			rep(v, 0, n) if (iset[v] && v != a) m1.add(v);
+			rep(b, 0, n) if (!iset[b] && frm[b] == -1 && m1.check(b))
+				ans.push_back(b), frm[b] = a;
+			return ans;
+		};
+		auto backE = [&](int b) {
+			m2.clear();
+			rep(cas, 0, 2) rep(v, 0, n)
+				if ((v == b || iset[v]) && (frm[v] == -1) == cas) {
+					if (!m2.check(v))
+						return cas ? q.push(v), frm[v] = b, v : -1;
+					m2.add(v);
+				}
+			return n;
+		};
+		while (!q.empty()) {
+			int a = q.front(), c; q.pop();
+			for (int b : fwdE(a))
+				while((c = backE(b)) >= 0) if (c == n) {
+					while (b != n) iset[b] ^= 1, b = frm[b];
+					return true;
+				}
+		}
+		return false;
+	}
+};

content/graph/chapter.tex

@@ -16,6 +16,9 @@ \section{Network flow}
 	\kactlimport{MinCut.h}
 	\kactlimport{GlobalMinCut.h}
 
+\section{Matroids}
+	\kactlimport{MatroidIntersection.h}
+
 \section{Matching}
 	\kactlimport{hopcroftKarp.h}
 	\kactlimport{DFSMatching.h}