Problem:
Score : 400 points

Problem Statement

Let \mathrm{popcount}(n) be the number of 1s in the binary representation of n.
For example, \mathrm{popcount}(3) = 2, \mathrm{popcount}(7) = 3, and \mathrm{popcount}(0) = 0.

Let f(n) be the number of times the following operation will be done when we repeat it until n becomes 0: "replace n with the remainder when n is divided by \mathrm{popcount}(n)." (It can be proved that, under the constraints of this problem, n always becomes 0 after a finite number of operations.)

For example, when n=7, it becomes 0 after two operations, as follows:

\mathrm{popcount}(7)=3, so we divide 7 by 3 and replace it with the remainder, 1.

\mathrm{popcount}(1)=1, so we divide 1 by 1 and replace it with the remainder, 0.

You are given an integer X with N digits in binary.
For each integer i such that 1 \leq i \leq N, let X_i be what X becomes when the i-th bit from the top is inverted.
Find f(X_1), f(X_2), \ldots, f(X_N).

Constraints

1 \leq N \leq 2 \times 10^5

X is an integer with N digits in binary, possibly with leading zeros.

Input

Input is given from Standard Input in the following format:

N
X

Output

Print N lines. The i-th line should contain the value f(X_i).

Sample Input 1

3
011

Sample Output 1

2
1
1

X_1 = 7, which will change as follows: 7 \rightarrow 1 \rightarrow 0. Thus, f(7) = 2.

X_2 = 1, which will change as follows: 1 \rightarrow 0. Thus, f(1) = 1.

X_3 = 2, which will change as follows: 2 \rightarrow 0. Thus, f(2) = 1.

Sample Input 2

23
00110111001011011001110

Sample Output 2

2
1
2
2
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
3

Solution in Rust (1.42.0) (judge verdict: Time Limit Exceeded)

use std::fmt::Debug;
use std::str::FromStr;

pub struct TokenReader {
    reader: std::io::Stdin,
    tokens: Vec<String>,
    index: usize,
}

impl TokenReader {
    pub fn new() -> Self {
        Self {
            reader: std::io::stdin(),
            tokens: Vec::new(),
            index: 0,
        }
    }

    pub fn next<T>(&mut self) -> T
    where
        T: FromStr,
        T::Err: Debug,
    {
        if self.index >= self.tokens.len() {
            self.load_next_line();
        }
        self.index += 1;
        self.tokens[self.index - 1].parse().unwrap()
    }

    pub fn vector<T>(&mut self) -> Vec<T>
    where
        T: FromStr,
        T::Err: Debug,
    {
        if self.index >= self.tokens.len() {
            self.load_next_line();
        }
        self.index = self.tokens.len();
        self.tokens.iter().map(|tok| tok.parse().unwrap()).collect()
    }

    pub fn load_next_line(&mut self) {
        let mut line = String::new();
        self.reader.read_line(&mut line).unwrap();

        self.tokens = line
            .split_whitespace()
            .map(String::from)
            .collect();
        self.index = 0;
    }
}

const BASE_BITS: usize = 16;
const BASE: i32 = 1 << BASE_BITS;

#[derive(PartialEq, Eq, Clone, Hash)]
struct Num {
    digits: Vec<i32>,
}

impl Num {
    fn new(bits: &str) -> Self {
        let bytes = bits.as_bytes();
        let mut digits = vec![];

        for i in 0..bytes.len() {
            if i % BASE_BITS == 0 {
                digits.push(0);
            }
            if bytes[bytes.len() - 1 - i] == b'1' {
                let last = digits.len() - 1;
                digits[last] += 1 << (i % BASE_BITS);
            }
        }

        Self { digits }
    }

    fn popcount(&self) -> usize {
        self.digits.iter().map(|d| d.count_ones() as usize).sum()
    }

    fn rem(&self, value: i32) -> Self {
        let mut other = self.clone();
        while other.digits.len() > 1 || other.digits[0] >= value {
             other.sub(value);
        }
        other
    }

    fn sub(&mut self, value: i32) {
        let mut carry = value;
        for digit in &mut self.digits {
            *digit -= carry;
            if *digit < 0 {
                *digit += BASE;
                carry = 1;
            } else {
                carry = 0;
            }
        }

        while self.digits.len() > 1 && *self.digits.last().unwrap() == 0 {
            self.digits.pop();
        }
    }
}

fn calc_steps(limit: usize) -> Vec<i32> {
    let mut steps = vec![0; limit + 1];
    for i in 1..=limit {
        steps[i] = 1 + steps[i % i.count_ones() as usize];
    }
    steps
}

fn raise(base: i32, exp: i32, m: i32) -> i32 {
    if exp == 0 {
        return 1;
    } else if exp == 1 {
        return base % m;
    }

    if exp % 2 == 1 {
        let res = ((base % m) as i64) * (raise(base, exp - 1, m) as i64);
        return (res % (m as i64)) as i32
    }

    let root = raise(base, exp / 2, m) as i64;
    ((root * root) % m as i64) as i32
}

fn solve(bits: &String) {
    let bytes = bits.as_bytes();
    let n = bytes.len();

    let num = Num::new(&bits[..]);
    let popcount = num.popcount() as i32;

    let rem_plus = num.rem(popcount + 1).digits[0];
    let rem_minus = num.rem(popcount - 1).digits[0];

    const MAX_POPCOUNT: usize = 100_000;
    let steps = calc_steps(MAX_POPCOUNT);

    for i in 0..n {
        let res = if bytes[i] == b'0' {
            let m = popcount + 1;
            let v = (rem_plus % m + raise(2, (n - i - 1) as i32, m)) % m;
            1 + steps[v as usize]
        } else {
            let m = popcount - 1;
            let v = (rem_minus % m + m - raise(2, (n - i - 1) as i32, m)) % m;
            1 + steps[v as usize]
        };
        println!("{}", res);
    }
}

fn main() {
    let mut reader = TokenReader::new();
    let _ = reader.next::<usize>();
    let bits = reader.next();
    solve(&bits);
}