// FunctionalGraph-函数图(每个顶点出度为1的有向图,有向的NamoriGraph) // 定义:Directed graphs in which every vertex has exactly one outgoing edge. // !每个点的出度为1(如果顶点没有出边,那么它的出边指向自己) // 连通分量个数=环的个数 // 1. 如果竞赛图无环,那么竞赛图的拓扑序是唯一确定的 // 2. 竞赛图的强连通分量缩点后呈链状 // 3. 如果竞赛图是强连通的,则一定存在一条哈密顿回路 // 4. 竞赛图存在一条哈密顿路径 // 5. 大小为n的竞赛图如果强连通,则恰好有长度为3,4,…,n的简单环。 // // 例如:下图是一个竞赛图。 // // 0 // ↓ // 1 // ↙ ↖ // 2 3 ← 4 ← 5 // ↘ ↗ // 6 ← 7 // // root: 6 // bottom: 3 // to: [1 2 6 1 3 4 3 6] // directedGraph: // // 8 (虚拟节点) // / // 6(分量root) // / \ // 2 7 // / // 1 // / \ // 0 3 (bottom) // \ // 4 // \ // 5 // // !1.n作为树的虚拟根节点,联通各个分量的起点. // !2.bottom的祖先节点都在环中. // !3.点u在所在子树的根节点(在环上)为lca(u, bottom). package main import ( "bufio" "fmt" "os" ) func main() { yuki1242() // abc296_e() // demo() } func demo() { edges := [][]int{{0, 1, 1}, {1, 2, 1}, {3, 1, 1}, {2, 6, 1}, {6, 3, 9}, {4, 3, 1}, {5, 4, 1}, {7, 6, 1}} n := int32(8) F := NewFunctionalGraph(n) for _, e := range edges { F.AddDirectedEdge(int32(e[0]), int32(e[1]), e[2]) } F.Build() fmt.Println(F.Dist(7, 1, true)) // 11 fmt.Println(F.Dist(7, 1, false)) // 3 fmt.Println(F.Root, F.graph, F.To) fmt.Println(F.Jump(7, 12)) // 2 fmt.Println(F.JumpAll(100)) for i := int32(0); i < n; i++ { fmt.Println(F.InCycle(i)) } for i := int32(0); i < n; i++ { if F.Root[i] == i { fmt.Println(F.CollectCycle(i)) } } } // No.1242 高橋君とすごろく // https://yukicoder.me/problems/no/1242 func yuki1242() { in := bufio.NewReader(os.Stdin) out := bufio.NewWriter(os.Stdout) defer out.Flush() var n int var k int32 fmt.Fscan(in, &n, &k) nums := make([]int, k) for i := int32(0); i < k; i++ { fmt.Fscan(in, &nums[i]) } G := NewFunctionalGraph(64) for s := int32(0); s < 64; s++ { t := (2 * s) & 63 ok := true if s&1 == 0 && s&32 == 0 { ok = false } if s&2 == 0 && s&16 == 0 { ok = false } if s&4 == 0 && s&8 == 0 { ok = false } if ok { t |= 1 } G.AddDirectedEdge(s, t, 1) } G.Build() x := n s := int32(63) for i := k - 1; i >= 0; i-- { y := nums[i] s = G.Jump(s, x-y) s &= 62 x = y } s = G.Jump(s, x-1) if s&1 == 1 { fmt.Fprintln(out, "Yes") } else { fmt.Fprintln(out, "No") } } func abc296_e() { // https://atcoder.jp/contests/abc296/tasks/abc296_e // 给定一个竞赛图,求多少个点在环中 in := bufio.NewReader(os.Stdin) out := bufio.NewWriter(os.Stdout) defer out.Flush() var n int32 fmt.Fscan(in, &n) nums := make([]int32, n) for i := int32(0); i < n; i++ { fmt.Fscan(in, &nums[i]) } F := NewFunctionalGraph(n) for i := int32(0); i < n; i++ { F.AddDirectedEdge(i, nums[i]-1, 1) } F.Build() res := 0 for i := int32(0); i < n; i++ { if F.InCycle(i) { res++ } } fmt.Fprintln(out, res) } // 给定一个竞赛图,求最长的环的长度,如果没有环,返回-1. func longestCycle(edges []int) int { n := int32(len(edges)) F := NewFunctionalGraph(n) for i := int32(0); i < n; i++ { if edges[i] != -1 { F.AddDirectedEdge(i, int32(edges[i]), 1) } else { F.AddDirectedEdge(i, i, 1) // !如果顶点没有出边,那么它的出边指向自己. } } F.Build() res := -1 for i := int32(0); i < n; i++ { if F.Root[i] == i { cycle := F.CollectCycle(int32(i)) if len(cycle) > 1 { res = max(res, len(cycle)) } } } return res } type neighbor struct { to int32 weight int } type FunctionalGraph struct { To []int32 // 每个顶点的出边指向的顶点 Weight []int // 每个顶点的出边的权值 Root []int32 // 每个联通分量的起点 n, m int32 graph [][]neighbor tree *Tree } func NewFunctionalGraph(n int32) *FunctionalGraph { to, weight, root := make([]int32, n), make([]int, n), make([]int32, n) for i := int32(0); i < n; i++ { to[i] = -1 root[i] = -1 } return &FunctionalGraph{n: n, To: to, Weight: weight, Root: root} } func (f *FunctionalGraph) AddDirectedEdge(from, to int32, weight int) { if f.To[from] != -1 { panic("FunctionalGraph: each vertex must have exactly one outgoing edge") } f.m++ f.To[from] = to f.Weight[from] = weight } func (f *FunctionalGraph) Build() ([][]neighbor, *Tree) { if f.n != f.m { panic("FunctionalGraph: vertex count must be equal to edge count") } n := f.n uf := newUnionFindArraySimple32(n) for v := int32(0); v < n; v++ { if !uf.Union(v, f.To[v]) { f.Root[v] = v } } for v := int32(0); v < n; v++ { if f.Root[v] == v { f.Root[uf.Find(v)] = v } } for v := int32(0); v < n; v++ { f.Root[v] = f.Root[uf.Find(v)] } graph := make([][]neighbor, n+1) for v := int32(0); v < n; v++ { if f.Root[v] == v { graph[n] = append(graph[n], neighbor{to: v, weight: f.Weight[v]}) } else { graph[f.To[v]] = append(graph[f.To[v]], neighbor{to: v, weight: f.Weight[v]}) } } f.graph = graph tree := NewTree(graph) tree.Build(n) f.tree = tree return graph, tree } // 从from到to的距离,不可达返回-1. func (f *FunctionalGraph) Dist(from, to int32, weighted bool) int { if weighted { if f.tree.IsInSubtree(from, to) { return f.tree.DepthWeighted[from] - f.tree.DepthWeighted[to] } root := f.Root[from] bottom := f.To[root] // from -> root -> bottom -> to if f.tree.IsInSubtree(bottom, to) { x := f.tree.DepthWeighted[from] - f.tree.DepthWeighted[root] x += f.Weight[root] x += f.tree.DepthWeighted[bottom] - f.tree.DepthWeighted[to] return x } return -1 } else { if f.tree.IsInSubtree(from, to) { return int(f.tree.Depth[from] - f.tree.Depth[to]) } root := f.Root[from] bottom := f.To[root] // from -> root -> bottom -> to if f.tree.IsInSubtree(bottom, to) { x := f.tree.Depth[from] - f.tree.Depth[root] x++ x += f.tree.Depth[bottom] - f.tree.Depth[to] return int(x) } return -1 } } // 从v向前跳step步,返回跳到的节点,不可达返回-1. func (f *FunctionalGraph) Jump(v int32, step int) int32 { d := f.tree.Depth[v] if step <= int(d-1) { return f.tree.Jump(v, f.n, int32(step)) } v = f.Root[v] step -= int(d - 1) bottom := f.To[v] c := f.tree.Depth[bottom] step %= int(c) if step == 0 { return v } return f.tree.Jump(bottom, f.n, int32(step-1)) } func (f *FunctionalGraph) JumpAll(step int) []int32 { n := f.n res := make([]int32, n) for v := int32(0); v < n; v++ { res[v] = -1 } query := make([][][2]int32, n) for v := int32(0); v < n; v++ { d := int(f.tree.Depth[v]) r := f.Root[v] if d-1 > step { query[v] = append(query[v], [2]int32{v, int32(step)}) } if d-1 <= step { k := step - (d - 1) bottom := f.To[r] c := int(f.tree.Depth[bottom]) k %= c if k == 0 { res[v] = r continue } query[bottom] = append(query[bottom], [2]int32{v, int32(k - 1)}) } } path := []int32{} var dfs func(int32) dfs = func(v int32) { path = append(path, v) for _, e := range query[v] { w, k := e[0], e[1] res[w] = path[int32(len(path))-1-k] } for _, e := range f.graph[v] { dfs(e.to) } path = path[:len(path)-1] } for _, e := range f.graph[n] { dfs(e.to) } return res } func (f *FunctionalGraph) InCycle(v int32) bool { root := f.Root[v] bottom := f.To[root] return f.tree.IsInSubtree(bottom, v) } func (f *FunctionalGraph) CollectCycle(r int32) []int32 { if r != f.Root[r] { panic("FunctionalGraph: r must be root") } cycle := []int32{f.To[r]} for last := cycle[len(cycle)-1]; last != r; last = cycle[len(cycle)-1] { cycle = append(cycle, f.To[last]) } return cycle } type Tree struct { Tree [][]neighbor // (next, weight) Depth []int32 DepthWeighted []int Parent []int32 LID, RID []int32 // 欧拉序[in,out) IdToNode []int32 top, heavySon []int32 timer int32 } func NewTree(graph [][]neighbor) *Tree { n := int32(len(graph)) tree := graph lid := make([]int32, n) rid := make([]int32, n) IdToNode := make([]int32, n) top := make([]int32, n) // 所处轻/重链的顶点(深度最小),轻链的顶点为自身 depth := make([]int32, n) // 深度 depthWeighted := make([]int, n) parent := make([]int32, n) // 父结点 heavySon := make([]int32, n) // 重儿子 for i := range parent { parent[i] = -1 } return &Tree{ Tree: tree, Depth: depth, DepthWeighted: depthWeighted, Parent: parent, LID: lid, RID: rid, IdToNode: IdToNode, top: top, heavySon: heavySon, } } // root:0-based // // 当root设为-1时,会从0开始遍历未访问过的连通分量 func (tree *Tree) Build(root int32) { if root != -1 { tree.build(root, -1, 0, 0) tree.markTop(root, root) } else { for i := int32(0); i < int32(len(tree.Tree)); i++ { if tree.Parent[i] == -1 { tree.build(i, -1, 0, 0) tree.markTop(i, i) } } } } // 返回 root 的欧拉序区间, 左闭右开, 0-indexed. func (tree *Tree) Id(root int32) (int32, int32) { return tree.LID[root], tree.RID[root] } // 返回返回边 u-v 对应的 欧拉序起点编号, 1 <= eid <= n-1., 0-indexed. func (tree *Tree) Eid(u, v int32) int32 { if tree.LID[u] > tree.LID[v] { return tree.LID[u] } return tree.LID[v] } func (tree *Tree) LCA(u, v int32) int32 { for { if tree.LID[u] > tree.LID[v] { u, v = v, u } if tree.top[u] == tree.top[v] { return u } v = tree.Parent[tree.top[v]] } } func (tree *Tree) RootedLCA(u, v int32, root int32) int32 { return tree.LCA(u, v) ^ tree.LCA(u, root) ^ tree.LCA(v, root) } func (tree *Tree) RootedParent(u int32, root int32) int32 { return tree.Jump(u, root, 1) } func (tree *Tree) Dist(u, v int32, weighted bool) int { if weighted { return tree.DepthWeighted[u] + tree.DepthWeighted[v] - 2*tree.DepthWeighted[tree.LCA(u, v)] } return int(tree.Depth[u] + tree.Depth[v] - 2*tree.Depth[tree.LCA(u, v)]) } // k: 0-based // // 如果不存在第k个祖先,返回-1 // kthAncestor(root,0) == root func (tree *Tree) KthAncestor(root, k int32) int32 { if k > tree.Depth[root] { return -1 } for { u := tree.top[root] if tree.LID[root]-k >= tree.LID[u] { return tree.IdToNode[tree.LID[root]-k] } k -= tree.LID[root] - tree.LID[u] + 1 root = tree.Parent[u] } } // 从 from 节点跳向 to 节点,跳过 step 个节点(0-indexed) // // 返回跳到的节点,如果不存在这样的节点,返回-1 func (tree *Tree) Jump(from, to, step int32) int32 { if step == 1 { if from == to { return -1 } if tree.IsInSubtree(to, from) { return tree.KthAncestor(to, tree.Depth[to]-tree.Depth[from]-1) } return tree.Parent[from] } c := tree.LCA(from, to) dac := tree.Depth[from] - tree.Depth[c] dbc := tree.Depth[to] - tree.Depth[c] if step > dac+dbc { return -1 } if step <= dac { return tree.KthAncestor(from, step) } return tree.KthAncestor(to, dac+dbc-step) } func (tree *Tree) CollectChild(root int32) []int32 { res := []int32{} for _, e := range tree.Tree[root] { next := e.to if next != tree.Parent[root] { res = append(res, next) } } return res } // 返回沿着`路径顺序`的 [起点,终点] 的 欧拉序 `左闭右闭` 数组. // // !eg:[[2 0] [4 4]] 沿着路径顺序但不一定沿着欧拉序. func (tree *Tree) GetPathDecomposition(u, v int32, vertex bool) [][2]int32 { up, down := [][2]int32{}, [][2]int32{} for { if tree.top[u] == tree.top[v] { break } if tree.LID[u] < tree.LID[v] { down = append(down, [2]int32{tree.LID[tree.top[v]], tree.LID[v]}) v = tree.Parent[tree.top[v]] } else { up = append(up, [2]int32{tree.LID[u], tree.LID[tree.top[u]]}) u = tree.Parent[tree.top[u]] } } edgeInt := int32(1) if vertex { edgeInt = 0 } if tree.LID[u] < tree.LID[v] { down = append(down, [2]int32{tree.LID[u] + edgeInt, tree.LID[v]}) } else if tree.LID[v]+edgeInt <= tree.LID[u] { up = append(up, [2]int32{tree.LID[u], tree.LID[v] + edgeInt}) } for i := 0; i < len(down)/2; i++ { down[i], down[len(down)-1-i] = down[len(down)-1-i], down[i] } return append(up, down...) } // 遍历路径上的 `[起点,终点)` 欧拉序 `左闭右开` 区间. func (tree *Tree) EnumeratePathDecomposition(u, v int32, vertex bool, f func(start, end int32)) { for { if tree.top[u] == tree.top[v] { break } if tree.LID[u] < tree.LID[v] { a, b := tree.LID[tree.top[v]], tree.LID[v] if a > b { a, b = b, a } f(a, b+1) v = tree.Parent[tree.top[v]] } else { a, b := tree.LID[u], tree.LID[tree.top[u]] if a > b { a, b = b, a } f(a, b+1) u = tree.Parent[tree.top[u]] } } edgeInt := int32(1) if vertex { edgeInt = 0 } if tree.LID[u] < tree.LID[v] { a, b := tree.LID[u]+edgeInt, tree.LID[v] if a > b { a, b = b, a } f(a, b+1) } else if tree.LID[v]+edgeInt <= tree.LID[u] { a, b := tree.LID[u], tree.LID[v]+edgeInt if a > b { a, b = b, a } f(a, b+1) } } func (tree *Tree) GetPath(u, v int32) []int32 { res := []int32{} composition := tree.GetPathDecomposition(u, v, true) for _, e := range composition { a, b := e[0], e[1] if a <= b { for i := a; i <= b; i++ { res = append(res, tree.IdToNode[i]) } } else { for i := a; i >= b; i-- { res = append(res, tree.IdToNode[i]) } } } return res } // 以root为根时,结点v的子树大小. func (tree *Tree) SubSize(v, root int32) int32 { if root == -1 { return tree.RID[v] - tree.LID[v] } if v == root { return int32(len(tree.Tree)) } x := tree.Jump(v, root, 1) if tree.IsInSubtree(v, x) { return tree.RID[v] - tree.LID[v] } return int32(len(tree.Tree)) - tree.RID[x] + tree.LID[x] } // child 是否在 root 的子树中 (child和root不能相等) func (tree *Tree) IsInSubtree(child, root int32) bool { return tree.LID[root] <= tree.LID[child] && tree.LID[child] < tree.RID[root] } // 寻找以 start 为 top 的重链 ,heavyPath[-1] 即为重链底端节点. func (tree *Tree) GetHeavyPath(start int32) []int32 { heavyPath := []int32{start} cur := start for tree.heavySon[cur] != -1 { cur = tree.heavySon[cur] heavyPath = append(heavyPath, cur) } return heavyPath } // 结点v的重儿子.如果没有重儿子,返回-1. func (tree *Tree) GetHeavyChild(v int32) int32 { k := tree.LID[v] + 1 if k == int32(len(tree.Tree)) { return -1 } w := tree.IdToNode[k] if tree.Parent[w] == v { return w } return -1 } func (tree *Tree) ELID(u int32) int32 { return 2*tree.LID[u] - tree.Depth[u] } func (tree *Tree) ERID(u int32) int32 { return 2*tree.RID[u] - tree.Depth[u] - 1 } func (tree *Tree) build(cur, pre, dep int32, dist int) int32 { subSize, heavySize, heavySon := int32(1), int32(0), int32(-1) for _, e := range tree.Tree[cur] { next, weight := e.to, e.weight if next != pre { nextSize := tree.build(next, cur, dep+1, dist+int(weight)) subSize += nextSize if nextSize > heavySize { heavySize, heavySon = nextSize, next } } } tree.Depth[cur] = dep tree.DepthWeighted[cur] = dist tree.heavySon[cur] = heavySon tree.Parent[cur] = pre return subSize } func (tree *Tree) markTop(cur, top int32) { tree.top[cur] = top tree.LID[cur] = tree.timer tree.IdToNode[tree.timer] = cur tree.timer++ heavySon := tree.heavySon[cur] if heavySon != -1 { tree.markTop(heavySon, top) for _, e := range tree.Tree[cur] { next := e.to if next != heavySon && next != tree.Parent[cur] { tree.markTop(next, next) } } } tree.RID[cur] = tree.timer } type unionFindArraySimple32 struct { Part int32 n int32 data []int32 } func newUnionFindArraySimple32(n int32) *unionFindArraySimple32 { data := make([]int32, n) for i := int32(0); i < n; i++ { data[i] = -1 } return &unionFindArraySimple32{Part: n, n: n, data: data} } func (u *unionFindArraySimple32) Union(key1, key2 int32) bool { root1, root2 := u.Find(key1), u.Find(key2) if root1 == root2 { return false } if u.data[root1] > u.data[root2] { root1, root2 = root2, root1 } u.data[root1] += u.data[root2] u.data[root2] = int32(root1) u.Part-- return true } func (u *unionFindArraySimple32) Find(key int32) int32 { if u.data[key] < 0 { return key } u.data[key] = u.Find(u.data[key]) return u.data[key] } func (u *unionFindArraySimple32) GetSize(key int32) int32 { return -u.data[u.Find(key)] } func min(a, b int) int { if a < b { return a } return b } func min32(a, b int32) int32 { if a < b { return a } return b } func max(a, b int) int { if a > b { return a } return b } func max32(a, b int32) int32 { if a > b { return a } return b }