All Pair Shortest Paths

AtCoder

IDarc158_e

Time4000ms

Memory256MB

Difficulty—

We have a grid with $2$ rows and $N$ columns. Let $(i,j)$ denote the square at the $i$\-th row from the top and $j$\-th column from the left. $(i,j)$ has a postive integer $x_{i,j}$ written on it. Two squares are said to be **adjacent** when they share a side. A **path** from square $X$ to $Y$ is a sequence $(P_1, \ldots, P_n)$ of distinct squares such that $P_1 = X$, $P_n = Y$, and $P_i$ and $P_{i+1}$ are adjacent for every $1\leq i \leq n-1$. Additionally, the **weight** of that path is the sum of integers written on $P_1, \ldots, P_n$. For two squares $X$ and $Y$, let $f(X, Y)$ denote the minimum weight of a path from $X$ to $Y$. Find the sum of $f(X, Y)$ over all pairs of squares $(X,Y)$, modulo $998244353$. ## Constraints * $1\leq N\leq 2\times 10^5$ * $1\leq x_{i,j} \leq 10^9$ ## Input The input is given from Standard Input in the following format: $N$ $x_{1,1}$ $\ldots$ $x_{1,N}$ $x_{2,1}$ $\ldots$ $x_{2,N}$ [samples]

Samples

Input #1

1
3
5

Output #1

24

You should find the sum of the following four values.

*   For $X = (1,1), Y = (1,1)$: $f(X, Y) = 3$.
*   For $X = (1,1), Y = (2,1)$: $f(X, Y) = 8$.
*   For $X = (2,1), Y = (1,1)$: $f(X, Y) = 8$.
*   For $X = (2,1), Y = (2,1)$: $f(X, Y) = 5$.

Input #2

2
1 2
3 4

Output #2

Input #3

5
1 1000000000 1 1 1
1 1 1 1000000000 1

Output #3

66714886

API Response (JSON)

{
  "problem": {
    "name": "All Pair Shortest Paths",
    "description": {
      "content": "We have a grid with $2$ rows and $N$ columns. Let $(i,j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left. $(i,j)$ has a postive integer $x_{i,j}$ written on it. Two",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 4000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc158_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a grid with $2$ rows and $N$ columns. Let $(i,j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column from the left. $(i,j)$ has a postive integer $x_{i,j}$ written on it.\nTwo...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments