Prefix-free Game

AtCoder

IDarc087_c

Time2000ms

Memory256MB

Difficulty—

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: $N$ $L$ $s_1$ $s_2$ $:$ $s_N$ [samples]

Samples

Input #1

2 2
00
01

Output #1

Alice

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

Input #2

2 2
00
11

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.

Input #3

Output #3

Alice

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

Input #4

2 1
0
1

Output #4

Bob

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

Input #5

1 2
11

Output #5

Alice

Input #6

2 3
101
11

Output #6

Bob

API Response (JSON)

{
  "problem": {
    "name": "Prefix-free Game",
    "description": {
      "content": "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 follo",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc087_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For strings $s$ and $t$, we will say that $s$ and $t$ are _prefix-free_ when neither is a prefix of the other.\nLet $L$ be a positive integer. A set of strings $S$ is a _good string set_ when the follo...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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