200872: [AtCoder]ARC087 E - Prefix-free Game

Memory Limit:256 MB Time Limit:2 S
Judge Style:Text Compare Creator:
Submit:0 Solved:0

Description

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

Input

题意翻译

对于两个字符串 $s, t$ , 如果 $s$ 不是 $t$ 的前缀且 $t$ 不是 $s$ 的前缀,则称他们是 prefix-free 的。 给定一个正整数 $L$,如果一个字符串集合 $S$ 符合下列条件,则我们称他为 good string set: - $S$ 中的每个字符串的长度都在 $1$ 和 $L$ 之间(包括),并且之包含字符 $'0'$ 和 $'1'$ - $S$ 中的每两个的字符串都是 prefix-free 的 我们有一个 good string set $S = {s_1, s_2 ... , s_n}$,Alice 和 Bob 在玩一个游戏,他们轮流做下列操作: - 向 $S$ 中添加一个新字符串,并使 $S$ 仍是 good string set 无法进行这个操作的人输掉游戏。假设二人都按最优策略进行游戏,求谁会赢。 ## 数据范围 $1 \leq N \leq 10^5$ $1 \leq L \leq 10^{18}$ $s_1, s_2, ..., s_n$ 是不同的 ${s_1, s_2, ..., s_n}$ 是 good string set $|s_1| + |s_2| + ... + |s_n| \leq 10^5$ 翻译者:魔塔哈奇

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