Problem:
Score : 700 points

Problem Statement

For strings s and t, we will say that s and t are prefix-free when neither is a prefix of the other.

Let L be a positive integer. A set of strings S is a good string set when the following conditions hold true:

Each string in S has a length between 1 and L (inclusive) and consists of the characters 0 and 1.

Any two distinct strings in S are prefix-free.

We have a good string set S = \{ s_1, s_2, ..., s_N \}. Alice and Bob will play a game against each other. They will alternately perform the following operation, starting from Alice:

Add a new string to S. After addition, S must still be a good string set.

The first player who becomes unable to perform the operation loses the game. Determine the winner of the game when both players play optimally.

Constraints

1 \leq N \leq 10^5

1 \leq L \leq 10^{18}

s_1, s_2, ..., s_N are all distinct.

{ s_1, s_2, ..., s_N } is a good string set.

|s_1| + |s_2| + ... + |s_N| \leq 10^5

Input

Input is given from Standard Input in the following format:

N L
s_1
s_2
:
s_N

Output

If Alice will win, print Alice; if Bob will win, print Bob.

Sample Input 1

2 2
00
01

Sample Output 1

Alice

If Alice adds 1, Bob will be unable to add a new string.

Sample Input 2

2 2
00
11

Sample Output 2

Bob

There are two strings that Alice can add on the first turn: 01 and 10.
In case she adds 01, if Bob add 10, she will be unable to add a new string.
Also, in case she adds 10, if Bob add 01, she will be unable to add a new string.

Sample Input 3

3 3
0
10
110

Sample Output 3

Alice

If Alice adds 111, Bob will be unable to add a new string.

Sample Input 4

2 1
0
1

Sample Output 4

Bob

Alice is unable to add a new string on the first turn.

Sample Input 5

1 2
11

Sample Output 5

Alice

Sample Input 6

2 3
101
11

Sample Output 6

Bob

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::collections::{BTreeMap, BTreeSet};

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };
    let n: usize = sc.read();
    let l: u64 = sc.read();
    let mut trie = Trie {
        nodes: vec![[0; 2]],
    };
    for _ in 0..n {
        let s: Vec<usize> = sc
            .chars()
            .into_iter()
            .map(|c| c as usize - '0' as usize)
            .collect();
        trie.add(&s, 0, 0);
    }

    let mut results = vec![];
    trie.traverse(0, &mut results, l);
    println!(
        "{}",
        if results
            .into_iter()
            .map(|h| 1 << h.trailing_zeros())
            .fold(0, |xor, x| xor ^ x)
            == 0
        {
            "Bob"
        } else {
            "Alice"
        }
    );
}

struct Trie {
    nodes: Vec<[usize; 2]>,
}

impl Trie {
    fn add(&mut self, s: &[usize], pos: usize, cur: usize) {
        if pos == s.len() {
            return;
        }
        if self.nodes[cur][s[pos]] == 0 {
            self.nodes[cur][s[pos]] = self.nodes.len();
            self.nodes.push([0; 2]);
        }
        let cur = self.nodes[cur][s[pos]];
        self.add(s, pos + 1, cur);
    }

    fn traverse(&self, cur: usize, results: &mut Vec<u64>, height: u64) {
        let (a, b) = (self.nodes[cur][0], self.nodes[cur][1]);
        if a == 0 && b == 0 {
            return;
        }
        if a == 0 || b == 0 {
            results.push(height);
        }
        if a != 0 {
            self.traverse(a, results, height - 1);
        }
        if b != 0 {
            self.traverse(b, results, height - 1);
        }
    }
}

fn grundy(l: usize, map: &mut BTreeMap<usize, usize>) -> usize {
    match map.get(&l) {
        Some(&x) => x,
        None => {
            let mut set = BTreeSet::new();
            for i in 1..(l + 1) {
                let mut g = 0;
                for j in 1..i {
                    g ^= grundy(l - j, map);
                }
                set.insert(g);
            }
            let g = (0..).find(|x| !set.contains(x)).unwrap();
            map.insert(l, g);
            g
        }
    }
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}