Social Distance

AtCoder

IDarc113_f

Time4000ms

Memory256MB

Difficulty—

Given is an integer sequence of length $N+1$: $X_0,X_1,\ldots,X_N$, where $0=X_0 < X_1 < \ldots < X_N$ holds. Now, $N$ people numbered $1$ through $N$ will appear on a number line. Person $i$ will appear at a **real** coordinate chosen uniformly at random from the interval $[X_{i-1},X_i]$. Find the expected value of the smallest distance between two people, modulo $998244353$. Definition of the expected value modulo $998244353$We can prove that the expected value in question is always a rational number. We can also prove that, under the constraints of this problem, if we express the expected value as an irreducible fraction $\frac{P}{Q}$, we have $Q \neq 0 \pmod{998244353}$. Thus, there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Report this $R$. ## Constraints * $2 \leq N \leq 20$ * $0=X_0 < X_1 < \cdots < X_N \leq 10^6$ ## Input Input is given from Standard Input in the following format: $N$ $X_0$ $X_1$ $\ldots$ $X_N$ [samples]

Samples

Input #1

2
0 1 3

Output #1

499122178

There are just two people, so the expected value of the smallest distance between two people is just the expected value of the distance between Person $1$ and Person $2$. The answer is $3/2$.

Input #2

5
0 3 4 8 9 14

Output #2

324469854

The answer is $196249/172800$.

Input #3

20
0 38927 83112 125409 165053 204085 246405 285073 325658 364254 406395 446145 485206 525532 563762 605769 644863 683453 722061 760345 798556

Output #3

29493181

API Response (JSON)

{
  "problem": {
    "name": "Social Distance",
    "description": {
      "content": "Given is an integer sequence of length $N+1$: $X_0,X_1,\\ldots,X_N$, where $0=X_0 < X_1 < \\ldots < X_N$ holds. Now, $N$ people numbered $1$ through $N$ will appear on a number line. Person $i$ will app",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc113_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is an integer sequence of length $N+1$: $X_0,X_1,\\ldots,X_N$, where $0=X_0 < X_1 < \\ldots < X_N$ holds.\nNow, $N$ people numbered $1$ through $N$ will appear on a number line. Person $i$ will app...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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