// 单点修改,路径查询,子树查询 package main import ( "bufio" "fmt" "os" ) func main() { // yosupoVertexAddPathSum() yuki1641() } func demo() { { // 0 // / \ // 1 2 // / \ // 3 4 tree := NewTree32(5) tree.AddEdge(0, 1, 0) tree.AddEdge(0, 2, 0) tree.AddEdge(2, 3, 0) tree.AddEdge(2, 4, 0) tree.Build(0) S := NewTreeAbleGroup(tree, false, true, true) S.Build(func(vidOrEid int32) E { return int(vidOrEid) }) fmt.Println(S.QuerySubtree(0)) // 7 fmt.Println(S.QuerySubtree(1)) // 1 fmt.Println(S.QuerySubtree(2)) // 9 fmt.Println(S.QuerySubtree(3)) // 3 fmt.Println(S.QuerySubtree(4)) // 4 fmt.Println(S.QueryPath(1, 3)) // 4 fmt.Println(S.QuerySubtreeRooted(0, 3)) // 4 S.Add(3, 10) fmt.Println(S.QuerySubtree(0)) // 20 fmt.Println(S.QuerySubtreeRooted(4, 3)) // 20 } } // https://judge.yosupo.jp/problem/vertex_add_path_sum func yosupoVertexAddPathSum() { in := bufio.NewReader(os.Stdin) out := bufio.NewWriter(os.Stdout) defer out.Flush() var n, q int32 fmt.Fscan(in, &n, &q) weights := make([]int, n) for i := 0; i < int(n); i++ { fmt.Fscan(in, &weights[i]) } tree := NewTree32(n) for i := 1; i < int(n); i++ { var u, v int32 fmt.Fscan(in, &u, &v) tree.AddEdge(u, v, 0) } tree.Build(0) S := NewTreeAbleGroup(tree, false, true, false) S.Build(func(vidOrEid int32) E { return weights[vidOrEid] }) for i := 0; i < int(q); i++ { var t int fmt.Fscan(in, &t) if t == 0 { var v, x int32 fmt.Fscan(in, &v, &x) S.Add(v, int(x)) } else { var u, v int32 fmt.Fscan(in, &u, &v) fmt.Fprintln(out, S.QueryPath(u, v)) } } } // https://yukicoder.me/problems/no/1641 func yuki1641() { in := bufio.NewReader(os.Stdin) out := bufio.NewWriter(os.Stdout) defer out.Flush() var n, q int32 fmt.Fscan(in, &n, &q) weights := make([]int, n) for i := 0; i < int(n); i++ { fmt.Fscan(in, &weights[i]) } tree := NewTree32(n) for i := 1; i < int(n); i++ { var u, v int32 fmt.Fscan(in, &u, &v) u, v = u-1, v-1 tree.AddEdge(u, v, 0) } tree.Build(0) S := NewTreeAbleGroup(tree, false, false, true) S.Build(func(vidOrEid int32) E { return weights[vidOrEid] }) for i := 0; i < int(q); i++ { var t int fmt.Fscan(in, &t) if t == 1 { var v, x int32 fmt.Fscan(in, &v, &x) v-- S.Add(v, int(x)) } else { var u int32 fmt.Fscan(in, &u) u-- fmt.Fprintln(out, S.QuerySubtree(u)) } } } type E = int func e() E { return 0 } func op(a, b E) E { return a ^ b } func inv(a E) E { return a } type TreeAbelGroup struct { pathQuery, subtreeQuery bool edge int32 n int32 tree *Tree32 bit, bitSubtree *bitGroup32 } func NewTreeAbleGroup(tree *Tree32, edge bool, pathQuery, subtreeQuery bool) *TreeAbelGroup { var edgeValue int32 if edge { edgeValue = 1 } return &TreeAbelGroup{pathQuery: pathQuery, subtreeQuery: subtreeQuery, edge: edgeValue, n: tree.n, tree: tree} } func (tag *TreeAbelGroup) Build(f func(vidOrEid int32) E) { bitRaw1 := make([]E, 2*tag.n) bitRaw2 := make([]E, tag.n) tree := tag.tree for v := int32(0); v < tag.n; v++ { var x E if tag.edge == 0 { x = f(v) } else { if v == 0 { x = e() } else { x = f(tree.vToE[v]) } } bitRaw1[tree.ELid(v)] = x bitRaw1[tree.ERid(v)] = inv(x) bitRaw2[tree.Lid[v]] = x } if tag.pathQuery { tag.bit = newBITGroup32From(2*tag.n, func(index int32) E { return bitRaw1[index] }) } if tag.subtreeQuery { tag.bitSubtree = newBITGroup32From(tag.n, func(index int32) E { return bitRaw2[index] }) } } func (tag *TreeAbelGroup) Add(i int32, x E) { v := i if tag.edge != 0 { v = tag.tree.EToV(i) } if tag.pathQuery { tag.bit.Update(tag.tree.ELid(v), x) tag.bit.Update(tag.tree.ERid(v), inv(x)) } if tag.subtreeQuery { tag.bitSubtree.Update(tag.tree.Lid[v], x) } } func (tag *TreeAbelGroup) QueryPath(from, to int32) E { if !tag.pathQuery { panic("path query not enabled") } lca := tag.tree.Lca(from, to) x1 := tag.bit.QueryRange(tag.tree.ELid(lca)+1, tag.tree.ELid(from)+1) x2 := tag.bit.QueryRange(tag.tree.ELid(lca)+tag.edge, tag.tree.ELid(to)+1) return op(x1, x2) } func (tag *TreeAbelGroup) QuerySubtree(u int32) E { return tag.QuerySubtreeRooted(u, -1) } func (tag *TreeAbelGroup) QuerySubtreeRooted(u, root int32) E { if !tag.subtreeQuery { panic("subtree query not enabled") } l, r := tag.tree.Lid[u], tag.tree.Rid[u] if root == -1 { return tag.bitSubtree.QueryRange(l+tag.edge, r) } if root == u { return tag.bitSubtree.QueryAll() } if tag.tree.InSubtree(u, root) { return tag.bitSubtree.QueryRange(l+tag.edge, r) } return op(tag.bitSubtree.QueryRange(0, l+1), tag.bitSubtree.QueryRange(r, tag.n)) } type bitGroup32 struct { n int32 data []E total E } func newBITGroup32(n int32) *bitGroup32 { data := make([]E, n) for i := range data { data[i] = e() } return &bitGroup32{n: n, data: data, total: e()} } func newBITGroup32From(n int32, f func(index int32) E) *bitGroup32 { total := e() data := make([]E, n) for i := range data { data[i] = f(int32(i)) total = op(total, data[i]) } for i := int32(1); i <= n; i++ { j := i + (i & -i) if j <= n { data[j-1] = op(data[j-1], data[i-1]) } } return &bitGroup32{n: n, data: data, total: total} } func (fw *bitGroup32) Update(i int32, x E) { fw.total = op(fw.total, x) for i++; i <= fw.n; i += i & -i { fw.data[i-1] = op(fw.data[i-1], x) } } func (fw *bitGroup32) QueryAll() E { return fw.total } // [0, end) func (fw *bitGroup32) QueryPrefix(end int32) E { if end > fw.n { end = fw.n } res := e() for end > 0 { res = op(res, fw.data[end-1]) end &= end - 1 } return res } // [start, end) func (fw *bitGroup32) QueryRange(start, end int32) E { if start < 0 { start = 0 } if end > fw.n { end = fw.n } if start == 0 { return fw.QueryPrefix(end) } if start > end { return e() } pos, neg := e(), e() for end > start { pos = op(pos, fw.data[end-1]) end &= end - 1 } for start > end { neg = op(neg, fw.data[start-1]) start &= start - 1 } return op(pos, inv(neg)) } type neighbor = struct { to int32 eid int32 cost int } type Tree32 struct { Lid, Rid []int32 IdToNode []int32 Depth []int32 DepthWeighted []int Parent []int32 Head []int32 // 重链头 Tree [][]neighbor Edges [][2]int32 vToE []int32 // 节点v的父边的id n int32 } func NewTree32(n int32) *Tree32 { res := &Tree32{Tree: make([][]neighbor, n), Edges: make([][2]int32, 0, n-1), n: n} return res } func (t *Tree32) AddEdge(u, v int32, w int) { eid := int32(len(t.Edges)) t.Tree[u] = append(t.Tree[u], neighbor{to: v, eid: eid, cost: w}) t.Tree[v] = append(t.Tree[v], neighbor{to: u, eid: eid, cost: w}) t.Edges = append(t.Edges, [2]int32{u, v}) } func (t *Tree32) AddDirectedEdge(from, to int32, cost int) { eid := int32(len(t.Edges)) t.Tree[from] = append(t.Tree[from], neighbor{to: to, eid: eid, cost: cost}) t.Edges = append(t.Edges, [2]int32{from, to}) } func (t *Tree32) Build(root int32) { if root != -1 && int32(len(t.Edges)) != t.n-1 { panic("edges count != n-1") } n := t.n t.Lid = make([]int32, n) t.Rid = make([]int32, n) t.IdToNode = make([]int32, n) t.Depth = make([]int32, n) t.DepthWeighted = make([]int, n) t.Parent = make([]int32, n) t.Head = make([]int32, n) t.vToE = make([]int32, n) for i := int32(0); i < n; i++ { t.Depth[i] = -1 t.Head[i] = root t.vToE[i] = -1 } if root != -1 { t._dfsSize(root, -1) time := int32(0) t._dfsHld(root, &time) } else { time := int32(0) for i := int32(0); i < n; i++ { if t.Depth[i] == -1 { t._dfsSize(i, -1) t._dfsHld(i, &time) } } } } // 从v开始沿着重链向下收集节点. func (t *Tree32) HeavyPathAt(v int32) []int32 { path := []int32{v} for { a := path[len(path)-1] for _, e := range t.Tree[a] { if e.to != t.Parent[a] && t.Head[e.to] == v { path = append(path, e.to) break } } if path[len(path)-1] == a { break } } return path } // 返回重儿子,如果没有返回 -1. func (t *Tree32) HeavyChild(v int32) int32 { k := t.Lid[v] + 1 if k == t.n { return -1 } w := t.IdToNode[k] if t.Parent[w] == v { return w } return -1 } // 从v开始向上走k步. func (t *Tree32) KthAncestor(v, k int32) int32 { if k > t.Depth[v] { return -1 } for { u := t.Head[v] if t.Lid[v]-k >= t.Lid[u] { return t.IdToNode[t.Lid[v]-k] } k -= t.Lid[v] - t.Lid[u] + 1 v = t.Parent[u] } } func (t *Tree32) Lca(u, v int32) int32 { for { if t.Lid[u] > t.Lid[v] { u, v = v, u } if t.Head[u] == t.Head[v] { return u } v = t.Parent[t.Head[v]] } } func (t *Tree32) LcaRooted(u, v, root int32) int32 { return t.Lca(u, v) ^ t.Lca(u, root) ^ t.Lca(v, root) } func (t *Tree32) Dist(a, b int32) int32 { c := t.Lca(a, b) return t.Depth[a] + t.Depth[b] - 2*t.Depth[c] } func (t *Tree32) DistWeighted(a, b int32) int { c := t.Lca(a, b) return t.DepthWeighted[a] + t.DepthWeighted[b] - 2*t.DepthWeighted[c] } // c 是否在 p 的子树中.c和p不能相等. func (t *Tree32) InSubtree(c, p int32) bool { return t.Lid[p] <= t.Lid[c] && t.Lid[c] < t.Rid[p] } // 从 a 开始走 k 步到 b. func (t *Tree32) Jump(a, b, k int32) int32 { if k == 1 { if a == b { return -1 } if t.InSubtree(b, a) { return t.KthAncestor(b, t.Depth[b]-t.Depth[a]-1) } return t.Parent[a] } c := t.Lca(a, b) dac := t.Depth[a] - t.Depth[c] dbc := t.Depth[b] - t.Depth[c] if k > dac+dbc { return -1 } if k <= dac { return t.KthAncestor(a, k) } return t.KthAncestor(b, dac+dbc-k) } func (t *Tree32) SubtreeSize(v int32) int32 { return t.Rid[v] - t.Lid[v] } func (t *Tree32) SubtreeSizeRooted(v, root int32) int32 { if v == root { return t.n } x := t.Jump(v, root, 1) if t.InSubtree(v, x) { return t.Rid[v] - t.Lid[v] } return t.n - t.Rid[x] + t.Lid[x] } func (t *Tree32) CollectChild(v int32) []int32 { var res []int32 for _, e := range t.Tree[v] { if e.to != t.Parent[v] { res = append(res, e.to) } } return res } // 收集与 v 相邻的轻边. func (t *Tree32) CollectLight(v int32) []int32 { var res []int32 skip := true for _, e := range t.Tree[v] { if e.to != t.Parent[v] { if !skip { res = append(res, e.to) } skip = false } } return res } func (tree *Tree32) RestorePath(from, to int32) []int32 { res := []int32{} composition := tree.GetPathDecomposition(from, to, 0) 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 } // 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 *Tree32) GetPathDecomposition(u, v int32, edge int32) [][2]int32 { up, down := [][2]int32{}, [][2]int32{} lid, head, parent := tree.Lid, tree.Head, tree.Parent for { if head[u] == head[v] { break } if lid[u] < lid[v] { down = append(down, [2]int32{lid[head[v]], lid[v]}) v = parent[head[v]] } else { up = append(up, [2]int32{lid[u], lid[head[u]]}) u = parent[head[u]] } } if lid[u] < lid[v] { down = append(down, [2]int32{lid[u] + edge, lid[v]}) } else if lid[v]+edge <= lid[u] { up = append(up, [2]int32{lid[u], lid[v] + edge}) } 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 *Tree32) EnumeratePathDecomposition(u, v int32, edge int32, f func(start, end int32)) { head, lid, parent := tree.Head, tree.Lid, tree.Parent for { if head[u] == head[v] { break } if lid[u] < lid[v] { a, b := lid[head[v]], lid[v] if a > b { a, b = b, a } f(a, b+1) v = parent[head[v]] } else { a, b := lid[u], lid[head[u]] if a > b { a, b = b, a } f(a, b+1) u = parent[head[u]] } } if lid[u] < lid[v] { a, b := lid[u]+edge, lid[v] if a > b { a, b = b, a } f(a, b+1) } else if lid[v]+edge <= lid[u] { a, b := lid[u], lid[v]+edge if a > b { a, b = b, a } f(a, b+1) } } // 返回 root 的欧拉序区间, 左闭右开, 0-indexed. func (tree *Tree32) 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 *Tree32) Eid(u, v int32) int32 { if tree.Lid[u] > tree.Lid[v] { return tree.Lid[u] } return tree.Lid[v] } // 点v对应的父边的边id.如果v是根节点则返回-1. func (tre *Tree32) VToE(v int32) int32 { return tre.vToE[v] } // 第i条边对应的深度更深的那个节点. func (tree *Tree32) EToV(i int32) int32 { u, v := tree.Edges[i][0], tree.Edges[i][1] if tree.Parent[u] == v { return u } return v } func (tree *Tree32) ELid(u int32) int32 { return 2*tree.Lid[u] - tree.Depth[u] } func (tree *Tree32) ERid(u int32) int32 { return 2*tree.Rid[u] - tree.Depth[u] - 1 } func (t *Tree32) _dfsSize(cur, pre int32) { size := t.Rid t.Parent[cur] = pre if pre != -1 { t.Depth[cur] = t.Depth[pre] + 1 } else { t.Depth[cur] = 0 } size[cur] = 1 nexts := t.Tree[cur] for i := int32(len(nexts)) - 2; i >= 0; i-- { e := nexts[i+1] if t.Depth[e.to] == -1 { nexts[i], nexts[i+1] = nexts[i+1], nexts[i] } } hldSize := int32(0) for i, e := range nexts { to := e.to if t.Depth[to] == -1 { t.DepthWeighted[to] = t.DepthWeighted[cur] + e.cost t.vToE[to] = e.eid t._dfsSize(to, cur) size[cur] += size[to] if size[to] > hldSize { hldSize = size[to] if i != 0 { nexts[0], nexts[i] = nexts[i], nexts[0] } } } } } func (t *Tree32) _dfsHld(cur int32, times *int32) { t.Lid[cur] = *times *times++ t.Rid[cur] += t.Lid[cur] t.IdToNode[t.Lid[cur]] = cur heavy := true for _, e := range t.Tree[cur] { to := e.to if t.Depth[to] > t.Depth[cur] { if heavy { t.Head[to] = t.Head[cur] } else { t.Head[to] = to } heavy = false t._dfsHld(to, times) } } } // 路径 [a,b] 与 [c,d] 的交集. // 如果为空则返回 {-1,-1},如果只有一个交点则返回 {x,x},如果有两个交点则返回 {x,y}. func (t *Tree32) PathIntersection(a, b, c, d int32) (int32, int32) { ab := t.Lca(a, b) ac := t.Lca(a, c) ad := t.Lca(a, d) bc := t.Lca(b, c) bd := t.Lca(b, d) cd := t.Lca(c, d) x := ab ^ ac ^ bc // meet(a,b,c) y := ab ^ ad ^ bd // meet(a,b,d) if x != y { return x, y } z := ac ^ ad ^ cd if x != z { x = -1 } return x, x } func max(a, b int) int { if a > b { return a } return b } 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 max32(a, b int32) int32 { if a > b { return a } return b } func abs(a int) int { if a < 0 { return -a } return a }