Best Concatenation

AtCoder

IDabc268_f

Time2000ms

Memory256MB

Difficulty—

You are given $N$ strings $S_1, S_2, \ldots, S_N$ consisting of digits from `1` through `9` and the character `X`. We will choose a permutation $P = (P_1, P_2, \ldots, P_N)$ of $(1, 2, \ldots, N)$ to construct a string $T = S_{P_1} + S_{P_2} + \cdots + S_{P_N}$, where $+$ denotes a concatenation of strings. Then, we will calculate the "score" of the string $T = T_1T_2\ldots T_{|T|}$ (where $|T|$ denotes the length of $T$). The score is calculated by the following $9$ steps, starting from the initial score $0$: * Add $1$ point to the score as many times as the number of integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `1`. * Add $2$ points to the score as many times as the number of integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `2`. * Add $3$ points to the score as many times as the number of integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `3`. * $\cdots$ * Add $9$ points to the score as many times as the number of integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `9`. Find the maximum possible score of $T$ when $P$ can be chosen arbitrarily. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $N$ is an integer. * $S_i$ is a string of length at least $1$ consisting of digits from `1` through `9` and the character `X`. * The sum of lengths of $S_1, S_2, \ldots, S_N$ is at most $2 \times 10^5$. ## Input Input is given from Standard Input in the following format: $N$ $S_1$ $S_2$ $\vdots$ $S_N$ [samples]

Samples

Input #1

3
1X3
59
XXX

Output #1

71

When $P = (3, 1, 2)$, we have $T = S_3 + S_1 + S_2 = $ `XXX1X359`. Then, the score of $T$ is calculated as follows:

*   there are $3$ integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `1`;
*   there are $4$ integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `3`;
*   there are $4$ integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `5`;
*   there are $4$ integer pairs $(i, j)$ such that $1 \leq i \lt j \leq |T|$, $T_i = $ `X`, and $T_j = $ `9`.

Therefore, the score of $T$ is $1 \times 3 + 3 \times 4 + 5 \times 4 + 9 \times 4 = 71$, which is the maximum possible value.

Input #2

10
X63X395XX
X2XX3X22X
13
3716XXX6
45X
X6XX
9238
281X92
1XX4X4XX6
54X9X711X1

Output #2

API Response (JSON)

{
  "problem": {
    "name": "Best Concatenation",
    "description": {
      "content": "You are given $N$ strings $S_1, S_2, \\ldots, S_N$ consisting of digits from `1` through `9` and the character `X`. We will choose a permutation $P = (P_1, P_2, \\ldots, P_N)$ of $(1, 2, \\ldots, N)$ to ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc268_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given $N$ strings $S_1, S_2, \\ldots, S_N$ consisting of digits from `1` through `9` and the character `X`.\nWe will choose a permutation $P = (P_1, P_2, \\ldots, P_N)$ of $(1, 2, \\ldots, N)$ to ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments