結果
| 問題 |
No.3079 Unite Japanese Prefectures
|
| ユーザー |
 atcoder8
|
| 提出日時 | 2025-03-31 16:34:22 |
| 言語 | Rust (1.83.0 + proconio) |
| 結果 |
AC
|
| 実行時間 | 417 ms / 4,000 ms |
| コード長 | 13,281 bytes |
| コンパイル時間 | 11,148 ms |
| コンパイル使用メモリ | 401,384 KB |
| 実行使用メモリ | 46,848 KB |
| 最終ジャッジ日時 | 2025-03-31 16:34:55 |
| 合計ジャッジ時間 | 15,437 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge4 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 27 |
ソースコード
use std::collections::BTreeMap;
use proconio::{input, marker::Usize1};
use union_find::UnionFind;
const NUM_FACES: usize = 6;
fn main() {
input! {
(n, m): (usize, usize),
mut uvc: [(Usize1, Usize1, Usize1); m],
}
uvc.sort_unstable_by_key(|edge| edge.2);
let mut uf = UnionFind::new(n);
let mut edges_by_cost = [const { vec![] }; NUM_FACES];
for &(u, v, cost) in &uvc {
if uf.merge(u, v) {
edges_by_cost[cost].push((u, v));
}
}
let ans = dfs(&edges_by_cost, [0; NUM_FACES], &mut BTreeMap::new());
println!("{}", ans);
}
fn dfs(
edges_by_cost: &[Vec<(usize, usize)>; NUM_FACES],
progresses: [usize; NUM_FACES],
memo: &mut BTreeMap<[usize; NUM_FACES], f64>,
) -> f64 {
if edges_by_cost
.iter()
.zip(&progresses)
.all(|(edge, &progress)| progress == edge.len())
{
return 0.0;
}
if let Some(&expected_value) = memo.get(&progresses) {
return expected_value;
}
let mut num_rolls_without_action = 0_usize;
let mut sum_next_expected_value = 0.0;
for roll in 0..NUM_FACES {
let Some(selected_cost) = (0..=roll)
.rev()
.find(|&cost| progresses[cost] < edges_by_cost[cost].len())
else {
num_rolls_without_action += 1;
continue;
};
let mut next_progresses = progresses.clone();
next_progresses[selected_cost] += 1;
sum_next_expected_value += dfs(edges_by_cost, next_progresses, memo);
}
let num_faces = NUM_FACES as f64;
let expected_value = num_faces / (num_faces - num_rolls_without_action as f64)
* (1.0 + sum_next_expected_value / num_faces);
memo.insert(progresses, expected_value);
expected_value
}
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<ParentOrSize>,
/// 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<Vec<usize>> {
// Number of nodes in graph.
let element_num = self.parent_or_size.len();
// List of connected components.
let mut groups: Vec<Vec<usize>> = vec![];
// Correspondence between the representative node and group index.
let mut leader_to_idx: Vec<Option<usize>> = 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()
}
}
}