use proconio::input; use union_find::UnionFind; fn main() { input! { (n, m): (usize, usize), mut ijw: [(usize, usize, usize); m], t: usize, rr: [usize; t], } ijw.sort_unstable(); let mut uf = UnionFind::new(n); let mut ir = rr.iter().cloned().enumerate().collect::>(); ir.sort_unstable_by_key(|&(_, r)| r); let mut num_components_vec = vec![0_usize; t]; let mut progress = 0; for &(i, r) in &ir { while progress < m && ijw[progress].2 <= r { let (u, v, _) = ijw[progress]; uf.merge(u, v); progress += 1; } num_components_vec[i] = uf.group_num(); } for &num_components in &num_components_vec { println!("{}", num_components); } } pub mod union_find { //! Union-Find processes the following queries on undirected graphs. //! * Merge two connected components. //! * Determine whether two given nodes are in the same connected component. //! //! To seed up processing, merge optimization using the number of nodes //! of the connected components and path compression are performed. //! //! The time complexity of each query is `O(A(n))`. //! where `n` is the number of nodes in the graph and //! `A(n)` is the inverse of the Ackermann function. /// This is the value that will be associated with each nodes of the graph. #[derive(Debug, Clone, Copy)] enum ParentOrSize { /// It is used for non-representative nodes and stores the parent node. Parent(usize), /// It is used for the representative node and /// stores the number of nodes of the connected component. Size(usize), } /// Union-Find processes the following queries on undirected graphs. /// * Merge two connected components. /// * Determine whether two given nodes are in the same connected component. /// /// To seed up processing, merge optimization using the number of nodes /// of the connected components and path compression are performed. /// /// The time complexity of each query is `O(A(n))`. /// where `n` is the number of nodes in the graph and /// `A(n)` is the inverse of the Ackermann function. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(0, 1); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(1, 2); /// assert_eq!(uf.same(0, 2), true); /// ``` #[derive(Debug, Default, Clone)] pub struct UnionFind { /// For each node, one of the following is stored. /// * The number of nodes of the connected component to which it belongs. /// (If it is a representative node of the connected component.) /// * Index of the parent node. (Otherwise.) parent_or_size: Vec, /// Number of connected components. group_num: usize, } impl UnionFind { /// Create an undirected graph with `n` nodes and `0` edges. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(0, 1); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(2, 1); /// assert_eq!(uf.same(0, 2), true); /// ``` pub fn new(n: usize) -> Self { UnionFind { parent_or_size: vec![ParentOrSize::Size(1); n], group_num: n, } } /// Return the representative node of the connected component containing node `a`. /// /// At that time, perform path compression on the nodes on the path from node `a` to the representative node. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// uf.merge(1, 2); /// assert_eq!(uf.leader(0), 0); /// assert_eq!(uf.leader(1), uf.leader(2)); /// ``` pub fn leader(&mut self, a: usize) -> usize { // If node `a` is a representative node of the connected component, return `a`. if let ParentOrSize::Size(_) = self.parent_or_size[a] { return a; } // Path from node `a` to the representative node. let mut path = vec![]; // Current node. let mut current = a; // Record the path to the representative node. while let ParentOrSize::Parent(parent) = self.parent_or_size[current] { // Add current node to the path. path.push(current); // Move to the parent node. current = parent; } // The representative node of the connected component. let leader = current; // Set nodes on the path as direct children of the representative node. path.iter().for_each(|&node| { self.parent_or_size[node] = ParentOrSize::Parent(leader); }); // Return the representative node of the connected component. leader } /// Return whether two nodes `a` and `b` are in the same connected component. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(0, 1); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(2, 1); /// assert_eq!(uf.same(0, 2), true); /// ``` pub fn same(&mut self, a: usize, b: usize) -> bool { self.leader(a) == self.leader(b) } /// Merge each connected component containing nodes `a` and `b`. /// /// Return `true` if different connected components are newly merged. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(0, 1); /// assert_eq!(uf.same(0, 2), false); /// uf.merge(2, 1); /// assert_eq!(uf.same(0, 2), true); /// ``` pub fn merge(&mut self, a: usize, b: usize) -> bool { // Representative node of the connected component that contains the node `a`. let leader_a = self.leader(a); // Representative node of the connected component that contains the node `b`. let leader_b = self.leader(b); // If nodes `a` and `b` are in the same connected component, return `false` without processing. if leader_a == leader_b { return false; } // Number of nodes of the component containing node `a`. let component_size_a = self.size(leader_a); // Number of nodes of the component containing node `b`. let component_size_b = self.size(leader_b); // Number of nodes of the merged component. let merged_component_size = component_size_a + component_size_b; // Set the parent of the representative node of the smaller sized connected component // to be the parent of the other connected component. if component_size_a <= component_size_b { self.parent_or_size[leader_a] = ParentOrSize::Parent(leader_b); self.parent_or_size[leader_b] = ParentOrSize::Size(merged_component_size); } else { self.parent_or_size[leader_b] = ParentOrSize::Parent(leader_a); self.parent_or_size[leader_a] = ParentOrSize::Size(merged_component_size); } // Decrease the number of connected components by one. self.group_num -= 1; // Return `true` because different connected components are newly combined. true } /// Return a list of connected components. /// /// Each connected component consists of indexes of nodes. /// The indexes of the nodes in each connected component are arranged in ascending order. /// The list of connected components is sorted in ascending order /// with respect to the smallest index of the included nodes. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(5); /// uf.merge(1, 2); /// uf.merge(2, 3); /// assert_eq!(uf.groups(), vec![vec![0], vec![1, 2, 3], vec![4]]); /// ``` pub fn groups(&mut self) -> Vec> { // Number of nodes in graph. let element_num = self.parent_or_size.len(); // List of connected components. let mut groups: Vec> = vec![]; // Correspondence between the representative node and group index. let mut leader_to_idx: Vec> = vec![None; element_num]; // Assign each node in the graph to a group. for node in 0..element_num { // Representative node of the connected component to which the `node` belongs. let leader = self.leader(node); if let Some(group_idx) = leader_to_idx[leader] { // Assign to an existing group. groups[group_idx].push(node); } else { // Adding a new group. leader_to_idx[leader] = Some(groups.len()); groups.push(vec![node]); } } // Return a list of groups. groups } /// Return the number of nodes in the connected component to which node `a` belongs. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// assert_eq!(uf.size(0), 1); /// uf.merge(0, 1); /// assert_eq!(uf.size(0), 2); /// uf.merge(2, 1); /// assert_eq!(uf.size(0), 3); /// ``` pub fn size(&mut self, a: usize) -> usize { let leader = self.leader(a); match self.parent_or_size[leader] { ParentOrSize::Parent(_) => panic!(), ParentOrSize::Size(size) => size, } } /// Add a new node with degree `0`. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(4); /// uf.merge(1, 2); /// uf.merge(2, 3); /// assert_eq!(uf.groups(), vec![vec![0], vec![1, 2, 3]]); /// uf.add(); /// assert_eq!(uf.groups(), vec![vec![0], vec![1, 2, 3], vec![4]]); /// ``` pub fn add(&mut self) { self.parent_or_size.push(ParentOrSize::Size(1)); self.group_num += 1; } /// Return the number of connected components. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(3); /// assert_eq!(uf.group_num(), 3); /// uf.merge(0, 1); /// assert_eq!(uf.group_num(), 2); /// uf.merge(2, 1); /// assert_eq!(uf.group_num(), 1); /// ``` pub fn group_num(&self) -> usize { self.group_num } /// Return the number of nodes in the graph. /// /// # Examples /// /// ``` /// use atcoder8_library::union_find::UnionFind; /// /// let mut uf = UnionFind::new(5); /// assert_eq!(uf.elem_num(), 5); /// ``` pub fn elem_num(&self) -> usize { self.parent_or_size.len() } } }