K-Medians Clustering

AtCoder

ID1202Contest_f

Time3000ms

Memory256MB

Difficulty—

You are given positive integers $N, M, K$, and a sequence of $K$ positive integers $c=(c_1, c_2, \dots, c_K)$. A **multiset** $S$ consisting of $N$ positive integers less than or equal to $M$ is called _good_, if there exists at least one sequence of $K$ multisets $(S_1, S_2, \dots, S_K)$ satisfying the following conditions. * None of $S_1, S_2, \dots, S_K$ is empty. * The median of $S_i$ equals to $c_i$, for all $i=1, 2, \dots, K$. * $S_1, S_2, \dots, S_K$ have exactly $N$ elements in total. A multiset consisting of those $N$ elements coincides with $S$. Note that we define the median of a multiset $T$ with $n\ (\geq 1)$ elements as the $\lceil n / 2 \rceil$\-th element of $T$ in ascending order. For example, the median of $T=\lbrace 1, 2, 3, 4 \rbrace$ is $2$, and that of $T=\lbrace 1, 3, 5, 7, 7 \rbrace$ is $5$. Find the number of good multisets modulo $998244353$. ## Constraints * $1 \leq N, M \leq 10^7$ * $1 \leq K \leq \min(2 \times 10^5, N)$ * $1 \leq c_i \leq M$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N \ M \ K$ $c_1 \ c_2 \ \dots \ c_K$ [samples]

Samples

Input #1

8 5 3
4 1 5

Output #1

105

For example, $S=\lbrace 1,1,1,2,3,4,5,5 \rbrace$ is a good multiset because there exists $(S_1, S_2, S_3)$ satisfying the conditions as follows.

*   $S_1 = \lbrace 1, \mathbf{4}, 5 \rbrace$
*   $S_2 = \lbrace 1, \mathbf{1}, 2, 3 \rbrace$
*   $S_3 = \lbrace \mathbf{5} \rbrace$

Input #2

10000000 2 2
1 2

Output #2

Input #3

30 10 5
3 1 4 1 5

Output #3

38446044

API Response (JSON)

{
  "problem": {
    "name": "K-Medians Clustering",
    "description": {
      "content": "You are given positive integers $N, M, K$, and a sequence of $K$ positive integers $c=(c_1, c_2, \\dots, c_K)$. A **multiset** $S$ consisting of $N$ positive integers less than or equal to $M$ is calle",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "1202Contest_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N, M, K$, and a sequence of $K$ positive integers $c=(c_1, c_2, \\dots, c_K)$.\nA **multiset** $S$ consisting of $N$ positive integers less than or equal to $M$ is calle...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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