Serval Survival

AtCoder

IDarc178_e

Time2000ms

Memory256MB

Difficulty—

There are $N$ servals on a bridge of length $L$. The $i$\-th serval is located at position $A_{i}$ from the left end of the bridge. Here, $0 < A_{1} < A_{2} < \cdots < A_{N} < L$ holds. For each $i = 1, 2, \dots, N$, answer the following question: > The servals will perform the following three actions in order: > > * Action $1$: The $N - 1$ servals other than the $i$\-th serval face left or right. > * Action $2$: The $i$\-th serval faces left or right. > * Action $3$: All servals start moving simultaneously. All servals move at a constant speed of exactly $1$ unit distance per unit time. When a serval reaches the end of the bridge, it leaves the bridge. If two servals collide, they both reverse their direction and continue moving. > > The $i$\-th serval is smart and loves this bridge, so when choosing a direction in Action $2$, it will observe the directions of the other $N-1$ servals and choose the direction that allows it to stay on the bridge the longer during Action $3$. There are $2^{N-1}$ possible combinations of directions for the $N-1$ servals in Action $1$. Find the sum, modulo $998244353$, over all these combinations, of the durations the $i$\-th serval can stay on the bridge. It can be proved that the output value is an integer. ## Constraints * $1 \leq N \leq 10^{5}$ * $0 < A_{1} < A_{2} < \cdots < A_{N} < L \leq 10^{9}$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $L$ $A_{1}$ $A_{2}$ $\cdots$ $A_{N}$ [samples]

Samples

Input #1

2 167
9 24

Output #1

182
301

For $i = 1$, it is always optimal to face right.
For $i = 2$, it is optimal to face the opposite direction from the first serval.

Input #2

1 924
167

Output #2

Input #3

10 924924167
46001560 235529797 272749755 301863061 359726177 470023587 667800476 696193062 741860924 809211293

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Serval Survival",
    "description": {
      "content": "There are $N$ servals on a bridge of length $L$. The $i$\\-th serval is located at position $A_{i}$ from the left end of the bridge. Here, $0 < A_{1} < A_{2} < \\cdots < A_{N} < L$ holds. For each $i = ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc178_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ servals on a bridge of length $L$.\nThe $i$\\-th serval is located at position $A_{i}$ from the left end of the bridge.\nHere, $0 < A_{1} < A_{2} < \\cdots < A_{N} < L$ holds.\nFor each $i = ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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