add 次小生成树

Author: EndlessCheng

README.md

@@ -187,6 +187,7 @@
     - Kruskal
     - Prim
   - 单度限制最小生成树
+  - 次小生成树
   - 最小差值生成树
   - 最小树形图
     - 朱刘算法

copypasta/graph.go

@@ -1497,8 +1497,8 @@ func (*graph) mstKruskal(in io.Reader, n, m int) int64 {
 	cntE := 0
 	for _, e := range edges {
 		if fv, fw := find(e.v), find(e.w); fv != fw {
-			sum += int64(e.wt)
 			fa[fv] = fw
+			sum += int64(e.wt)
 			cntE++
 		}
 	}
@@ -1701,14 +1701,128 @@ func (*graph) limitDegreeMST(dis [][]int, root, lim int) int {
 	return mstSum
 }
 
-// 次小生成树 Second best Minimum Spanning Tree
-// todo 非严格/严格
-// Using Kruskal and Lowest Common Ancestor
-// https://oi-wiki.org/graph/mst/#_9
+// 严格次小生成树 Second best Minimum Spanning Tree
+// https://oi-wiki.org/graph/mst/#_13
 // https://cp-algorithms.com/graph/second_best_mst.html
-// todo 模板题 https://www.luogu.com.cn/problem/P4180
-func (*graph) secondMST(n, m int) (sum int64) {
-	return
+// 模板题（严格）https://www.luogu.com.cn/problem/P4180 https://www.acwing.com/problem/content/358/
+// 注：非严格次小生成树
+//     做法更加简单，维护路径最大值即可，见 https://oi-wiki.org/graph/mst/#_10
+func (*graph) strictlySecondMST(n int, edges []struct{ v, w, wt int }, min, max func(int, int) int) int {
+	sort.Slice(edges, func(i, j int) bool { return edges[i].wt < edges[j].wt })
+
+	fa := make([]int, n)
+	for i := range fa {
+		fa[i] = i
+	}
+	var find func(int) int
+	find = func(x int) int {
+		if fa[x] != x {
+			fa[x] = find(fa[x])
+		}
+		return fa[x]
+	}
+
+	mstSum := 0 // int64
+	inMST := make([]bool, len(edges))
+	type nb struct{ to, wt int }
+	g := make([][]nb, n)
+	for i, e := range edges {
+		v, w, wt := e.v, e.w, e.wt
+		if fv, fw := find(v), find(w); fv != fw {
+			fa[fv] = fw
+			mstSum += wt
+			inMST[i] = true
+			g[v] = append(g[v], nb{w, wt}) // MST
+			g[w] = append(g[w], nb{v, wt})
+		}
+	}
+
+	const mx = 17
+	type pair struct{ p, fi, se int }
+	pa := make([][mx]pair, n)
+	dep := make([]int, n)
+	var build func(v, p, d int)
+	build = func(v, p, d int) {
+		pa[v][0].p = p
+		dep[v] = d
+		for _, e := range g[v] {
+			if w := e.to; w != p {
+				pa[w][0].fi = e.wt
+				build(w, v, d+1)
+			}
+		}
+	}
+	build(0, -1, 0)
+
+	merge := func(xFi, xSe, yFi, ySe int) (int, int) {
+		fi, se := max(xFi, yFi), 0
+		if xFi == yFi {
+			se = max(xSe, ySe)
+		} else if xFi > yFi {
+			se = max(xSe, yFi)
+		} else {
+			se = max(xFi, ySe)
+		}
+		return fi, se
+	}
+	for i := 0; i+1 < mx; i++ {
+		for v := range pa {
+			if p := pa[v][i]; p.p != -1 {
+				pp := pa[p.p][i]
+				fi, se := merge(p.fi, p.se, pp.fi, pp.se)
+				pa[v][i+1] = pair{pp.p, fi, se}
+			} else {
+				pa[v][i+1].p = -1
+			}
+		}
+	}
+
+	// 返回路径最大边权和严格次大边权
+	queryPath := func(v, w int) (fi, se int) {
+		if dep[v] > dep[w] {
+			v, w = w, v
+		}
+		for i := 0; i < mx; i++ {
+			if (dep[w]-dep[v])>>i&1 > 0 {
+				p := pa[w][i]
+				fi, se = merge(fi, se, p.fi, p.se)
+				w = p.p
+			}
+		}
+		if w != v {
+			for i := mx - 1; i >= 0; i-- {
+				if pv, pw := pa[v][i], pa[w][i]; pv.p != pw.p {
+					fi, se = merge(fi, se, pv.fi, pv.se)
+					fi, se = merge(fi, se, pw.fi, pw.se)
+					v, w = pv.p, pw.p
+				}
+			}
+			fi, se = merge(fi, se, pa[v][0].fi, pa[v][0].se)
+			fi, se = merge(fi, se, pa[w][0].fi, pa[w][0].se)
+		}
+		return
+	}
+
+	const inf int = 1e9 // 1e18
+	delta := inf
+	for i, e := range edges {
+		v, w, wt := e.v, e.w, e.wt
+		if inMST[i] || v == w { // 注意跳过自环
+			continue
+		}
+		fi, se := queryPath(v, w)
+		if wt > fi {
+			delta = min(delta, wt-fi) // 替换从而得到更大的 MST，取最小的替换差值
+		} else if se > 0 { // 此时必然有 wt == fi
+			delta = min(delta, wt-se) // wt = fi > se，同样可以替换
+		}
+	}
+	if delta == inf {
+		return -1
+	}
+	mstSum += delta
+
+	return mstSum
 }
 
 // Kruskal 重构树

copypasta/graph_tree.go

@@ -647,10 +647,11 @@ func (*tree) numPairsWithDistanceLimit(in io.Reader, n, root int, upperDis int64
 // https://atcoder.jp/contests/arc060/tasks/arc060_c
 // 路径点权乘积 https://ac.nowcoder.com/acm/contest/6913/C
 // 树上倍增应用（静态路径查询）：代码见下面的 EXTRA 部分
-//    最大值（与 MST 结合）https://codeforces.com/problemset/problem/609/E
+//    维护最大值（与 MST 结合）https://codeforces.com/problemset/problem/609/E
 //       变体 https://codeforces.com/problemset/problem/733/F
-//    最大值（与 MST 结合）LC1697 的在线做法 https://leetcode-cn.com/problems/checking-existence-of-edge-length-limited-paths/
-//    前十大（点权）https://codeforces.com/problemset/problem/587/C
+//    维护最大值（与 MST 结合）LC1697 的在线做法 https://leetcode-cn.com/problems/checking-existence-of-edge-length-limited-paths/
+//    维护最大值和严格次大值（严格次小 MST）：见 graph.go 中的 strictlySecondMST
+//    维护前十大（点权）https://codeforces.com/problemset/problem/587/C
 // 树上倍增-查询深度最小的未被标记的点 https://codeforces.com/problemset/problem/980/E
 // 题目推荐 https://cp-algorithms.com/graph/lca.html#toc-tgt-2
 // todo poj2763 poj1986 poj3728
@@ -766,18 +767,19 @@ func (*tree) lcaBinarySearch(n, root int, g [][]int) {
 		type pair struct{ p, maxWt int }
 		pa := make([][mx]pair, n)
 		dep := make([]int, n)
-		var f func(v, p, d int)
-		f = func(v, p, d int) {
+		var build func(v, p, d int)
+		build = func(v, p, d int) {
 			pa[v][0].p = p
 			dep[v] = d
 			for _, e := range g[v] {
 				if w := e.to; w != p {
 					pa[w][0].maxWt = e.wt
-					f(w, v, d+1)
+					build(w, v, d+1)
 				}
 			}
 		}
-		f(0, -1, 0)
+		build(0, -1, 0)
+
 		for i := 0; i+1 < mx; i++ {
 			for v := range pa {
 				if p := pa[v][i]; p.p != -1 {
@@ -788,6 +790,7 @@ func (*tree) lcaBinarySearch(n, root int, g [][]int) {
 				}
 			}
 		}
+
 		// 求 LCA(v,w) 的同时，顺带求出 v-w 上的边权最值
 		_lca := func(v, w int) (lca, maxWt int) {
 			if dep[v] > dep[w] {
@@ -811,8 +814,10 @@ func (*tree) lcaBinarySearch(n, root int, g [][]int) {
 				v = pa[v][0].p
 			}
 			// 如果是点权的话这里加上 maxWt = max(maxWt, pa[v][0].maxWt)
-			return v, maxWt
+			lca = v
+			return
 		}
+
 		_ = _lca
 	}