// Add // QueryPath // QuerySubtree package main import ( "bufio" "fmt" "os" ) func main2() { tree := NewTree(5) tree.AddEdge(0, 1, 1) tree.AddEdge(0, 2, 1) tree.AddEdge(1, 3, 1) tree.AddEdge(1, 4, 1) tree.Build(0) A := NewTreeAbelGroup(tree, []Abel{1, 2, 3, 4, 5}, true, true, true) fmt.Println(A.QueryPath(0, 4)) } func main() { // https://yukicoder.me/problems/no/1641 in := bufio.NewReader(os.Stdin) out := bufio.NewWriter(os.Stdout) defer out.Flush() var n, q int fmt.Fscan(in, &n, &q) A := make([]Abel, n) for i := 0; i < n; i++ { fmt.Fscan(in, &A[i]) } tree := NewTree(n) for i := 0; i < n-1; i++ { var a, b int fmt.Fscan(in, &a, &b) tree.AddEdge(a-1, b-1, 1) } tree.Build(0) TA := NewTreeAbelGroup(tree, A, true, false, true) for i := 0; i < q; i++ { var t, x, y int fmt.Fscan(in, &t, &x, &y) x-- if t == 1 { TA.Add(x, Abel(y)) } if t == 2 { fmt.Fprintln(out, TA.QuerySubtree(x)) } } } type Abel = int func e() Abel { return 0 } func op(e1, e2 Abel) Abel { return e1 ^ e2 } func inv(e Abel) Abel { return e } // 如果只查询前缀, 可以不需要是群 type TreeAbelGroup struct { tree *Tree n int isVertex bool pathQuery bool subtreeQuery bool bit *_FTree bitSubtree *_FTree unit Abel } // 静态树的路径查询,维护的量需要满足阿贝尔群的性质. // data: 顶点的值, 或者边的值.(边的编号为两个定点中较深的那个点的编号) // isVertex: data是否为顶点的值以及查询的时候是否是顶点权值. func NewTreeAbelGroup(tree *Tree, data []Abel, isVertex, pathQuery, subtreeQuery bool) *TreeAbelGroup { res := &TreeAbelGroup{ tree: tree, n: len(tree.Tree), isVertex: isVertex, pathQuery: pathQuery, subtreeQuery: subtreeQuery, unit: e(), } if pathQuery { bitRaw := make([]Abel, 2*res.n) if isVertex { for v := 0; v < res.n; v++ { bitRaw[tree.ELID(v)] = data[v] bitRaw[tree.ERID(v)] = inv(data[v]) } } else { for i := range bitRaw { bitRaw[i] = res.unit } for e := 0; e < res.n-1; e++ { v := tree.EidtoV(e) bitRaw[tree.ELID(v)] = data[e] bitRaw[tree.ERID(v)] = inv(data[e]) } } res.bit = _NewFTree(len(bitRaw), bitRaw...) } if subtreeQuery { bitRaw := make([]Abel, res.n) if isVertex { for v := 0; v < res.n; v++ { bitRaw[tree.LID[v]] = data[v] } } else { for i := range bitRaw { bitRaw[i] = res.unit } for e := 0; e < res.n-1; e++ { v := tree.EidtoV(e) bitRaw[tree.LID[v]] = data[e] } } res.bitSubtree = _NewFTree(len(bitRaw), bitRaw...) } return res } func (ta *TreeAbelGroup) Add(i int, x Abel) { v := i if !ta.isVertex { v = ta.tree.EidtoV(i) } if ta.pathQuery { inv_x := inv(x) ta.bit.Update(ta.tree.ELID(v), x) ta.bit.Update(ta.tree.ERID(v), inv_x) } if ta.subtreeQuery { ta.bitSubtree.Update(ta.tree.LID[v], x) } } func (ta *TreeAbelGroup) QueryPath(from, to int) Abel { lca := ta.tree.LCA(from, to) x1 := ta.bit.QueryRange(ta.tree.ELID(lca)+1, ta.tree.ELID(from)+1) offset := 1 if ta.isVertex { offset = 0 } x2 := ta.bit.QueryRange(ta.tree.ELID(lca)+offset, ta.tree.ELID(to)+1) return op(x1, x2) } func (ta *TreeAbelGroup) QuerySubtree(root int) Abel { l, r := ta.tree.LID[root], ta.tree.RID[root] offset := 1 if ta.isVertex { offset = 0 } return ta.bitSubtree.QueryRange(l+offset, r) } type Tree struct { Tree [][][2]int // (next, weight) Edges [][3]int // (u, v, w) Depth, DepthWeighted []int Parent []int LID, RID []int // 欧拉序[in,out) idToNode []int top, heavySon []int timer int } func NewTree(n int) *Tree { tree := make([][][2]int, n) lid := make([]int, n) rid := make([]int, n) idToNode := make([]int, n) top := make([]int, n) // 所处轻/重链的顶点(深度最小),轻链的顶点为自身 depth := make([]int, n) // 深度 depthWeighted := make([]int, n) parent := make([]int, n) // 父结点 heavySon := make([]int, n) // 重儿子 edges := make([][3]int, 0, n-1) 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, Edges: edges, } } // 添加无向边 u-v, 边权为w. func (tree *Tree) AddEdge(u, v, w int) { tree.Tree[u] = append(tree.Tree[u], [2]int{v, w}) tree.Tree[v] = append(tree.Tree[v], [2]int{u, w}) tree.Edges = append(tree.Edges, [3]int{u, v, w}) } // 添加有向边 u->v, 边权为w. func (tree *Tree) AddDirectedEdge(u, v, w int) { tree.Tree[u] = append(tree.Tree[u], [2]int{v, w}) tree.Edges = append(tree.Edges, [3]int{u, v, w}) } // 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 int) { if root != -1 { tree.build(root, -1, 0, 0) tree.markTop(root, root) } else { for i := 0; i < 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 int) (int, int) { return tree.LID[root], tree.RID[root] } // 返回边 u-v 对应的 欧拉序起点编号, 0-indexed. func (tree *Tree) Eid(u, v int) int { if tree.LID[u] > tree.LID[v] { return tree.LID[u] } return tree.LID[v] } // 较深的那个点作为边的编号. func (tree *Tree) EidtoV(eid int) int { e := tree.Edges[eid] u, v := e[0], e[1] if tree.Parent[u] == v { return u } return v } func (tree *Tree) LCA(u, v int) int { 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) Dist(u, v int, weighted bool) int { if weighted { return tree.DepthWeighted[u] + tree.DepthWeighted[v] - 2*tree.DepthWeighted[tree.LCA(u, v)] } return 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 func (tree *Tree) KthAncestor(root, k int) int { 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 int) int { 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 int) []int { res := []int{} for _, e := range tree.Tree[root] { next := e[0] 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 int, vertex bool) [][2]int { up, down := [][2]int{}, [][2]int{} for { if tree.top[u] == tree.top[v] { break } if tree.LID[u] < tree.LID[v] { down = append(down, [2]int{tree.LID[tree.top[v]], tree.LID[v]}) v = tree.Parent[tree.top[v]] } else { up = append(up, [2]int{tree.LID[u], tree.LID[tree.top[u]]}) u = tree.Parent[tree.top[u]] } } edgeInt := 1 if vertex { edgeInt = 0 } if tree.LID[u] < tree.LID[v] { down = append(down, [2]int{tree.LID[u] + edgeInt, tree.LID[v]}) } else if tree.LID[v]+edgeInt <= tree.LID[u] { up = append(up, [2]int{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...) } func (tree *Tree) GetPath(u, v int) []int { res := []int{} 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 } func (tree *Tree) SubtreeSize(u int) int { return tree.RID[u] - tree.LID[u] } // child 是否在 root 的子树中 (child和root不能相等) func (tree *Tree) IsInSubtree(child, root int) bool { return tree.LID[root] <= tree.LID[child] && tree.LID[child] < tree.RID[root] } func (tree *Tree) ELID(u int) int { return 2*tree.LID[u] - tree.Depth[u] } func (tree *Tree) ERID(u int) int { return 2*tree.RID[u] - tree.Depth[u] - 1 } func (tree *Tree) build(cur, pre, dep, dist int) int { subSize, heavySize, heavySon := 1, 0, -1 for _, e := range tree.Tree[cur] { next, weight := e[0], e[1] if next != pre { nextSize := tree.build(next, cur, dep+1, dist+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 int) { tree.top[cur] = top tree.LID[cur] = tree.timer tree.idToNode[tree.timer] = cur tree.timer++ if tree.heavySon[cur] != -1 { tree.markTop(tree.heavySon[cur], top) for _, e := range tree.Tree[cur] { next := e[0] if next != tree.heavySon[cur] && next != tree.Parent[cur] { tree.markTop(next, next) } } } tree.RID[cur] = tree.timer } type _FTree struct { n int data []Abel unit Abel } func _NewFTree(n int, nums ...Abel) *_FTree { fw := &_FTree{n: n, unit: e()} if len(nums) == 0 { nums = make([]Abel, n) for i := range nums { nums[i] = fw.unit } } fw.build(nums) return fw } // [0, right) func (fw *_FTree) QueryPrefix(right int) Abel { if right > fw.n { right = fw.n } res := fw.unit for right > 0 { res = op(res, fw.data[right-1]) right &= right - 1 } return res } // [left, right) func (fw *_FTree) QueryRange(left, right int) Abel { if left < 0 { left = 0 } if right > fw.n { right = fw.n } if left == 0 { return fw.QueryPrefix(right) } if left > right { return fw.unit } pos, neg := fw.unit, fw.unit for right > left { pos = op(pos, fw.data[right-1]) right &= right - 1 } for left > right { neg = op(neg, fw.data[left-1]) left &= left - 1 } return op(pos, inv(neg)) } func (fw *_FTree) Update(i int, x Abel) { for i++; i <= fw.n; i += i & -i { fw.data[i-1] = op(fw.data[i-1], x) } } func (fw *_FTree) build(nums []Abel) { n := fw.n fw.data = append(fw.data, nums...) for i := 1; i <= n; i++ { j := i + (i & -i) if j <= n { fw.data[j-1] = op(fw.data[i-1], fw.data[j-1]) } } }