Problem:
Score : 400 points

Problem Statement

We have an undirected graph G with N vertices numbered 1 to N and N edges as follows:

For each i=1,2,...,N-1, there is an edge between Vertex i and Vertex i+1.

There is an edge between Vertex X and Vertex Y.

For each k=1,2,...,N-1, solve the problem below:

Find the number of pairs of integers (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j in G is k.

Constraints

3 \leq N \leq 2 \times 10^3

1 \leq X,Y \leq N

X+1 < Y

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X Y

Output

For each k=1, 2, ..., N-1 in this order, print a line containing the answer to the problem.

Sample Input 1

5 2 4

Sample Output 1

5
4
1
0

The graph in this input is as follows:

There are five pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 1: (1,2)\,,(2,3)\,,(2,4)\,,(3,4)\,,(4,5).

There are four pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 2: (1,3)\,,(1,4)\,,(2,5)\,,(3,5).

There is one pair (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 3: (1,5).

There are no pairs (i,j) (1 \leq i < j \leq N) such that the shortest distance between Vertex i and Vertex j is 4.

Sample Input 2

3 1 3

Sample Output 2

3
0

The graph in this input is as follows:

Sample Input 3

7 3 7

Sample Output 3

7
8
4
2
0
0

Sample Input 4

10 4 8

Sample Output 4

10
12
10
8
4
1
0
0
0

Solution in Rust (1.15.1) (judge verdict: Time Limit Exceeded)

use std::io::Read;
use std::collections::{VecDeque, HashMap};

fn main() {
    let mut buf = String::new();
    std::io::stdin().read_to_string(&mut buf).unwrap();

    let answer = solve(&buf);

    println!("{}", answer);
}

fn solve(input: &str) -> String {
    let mut iterator = input.split_whitespace();

    let n: usize = iterator.next().unwrap().parse().unwrap();
    let x: usize = iterator.next().unwrap().parse().unwrap();
    let y: usize = iterator.next().unwrap().parse().unwrap();

    let mut map: Vec<Vec<bool>> = vec![vec![false; n]; n];
    for i in 0..(n - 1) {
        map[i][i + 1] = true;
        map[i + 1][i] = true;
    }
    map[x - 1][y - 1] = true;
    map[y - 1][x - 1] = true;

    let mut counts: HashMap<usize, usize> = HashMap::with_capacity(n);
    for i in 0..n {
        let distance = bfs(&map, i);
        for j in i..n {
            *counts.entry(distance[j]).or_insert(0) += 1;
        }
    }

    let mut ans = String::new();
    for k in 1..n {
        let default = 0;
        let count = counts.get(&k).unwrap_or(&default);
        ans.push_str(&format!("{}\n", count));
    }

    return ans.trim().to_string();
}

fn bfs(map: &Vec<Vec<bool>>, s: usize) -> Vec<usize> {
    let n = map.len();
    let mut color: Vec<Color> = vec![Color::WHITE; n];
    let mut d: Vec<usize> = vec![std::usize::MAX; n];

    let mut q: VecDeque<usize> = VecDeque::new();
    color[s] = Color::GRAY;
    d[s] = 0;
    q.push_back(s);

    while !q.is_empty() {
        let u = q.pop_front().unwrap();
        for v in 0..n {
            if map[u][v] && color[v] == Color::WHITE {
                color[v] = Color::GRAY;
                d[v] = d[u] + 1;
                q.push_back(v);
            }
        }
        color[u] = Color::BLACK;
    }
    d
}

#[derive(Clone, PartialEq)]
enum Color {
    WHITE, GRAY, BLACK,
}