Simple APSP Problem

AtCoder

IDapc001_i

Time3000ms

Memory256MB

Difficulty—

You are given an $H \times W$ grid. The square at the top-left corner will be represented by $(0, 0)$, and the square at the bottom-right corner will be represented by $(H-1, W-1)$. Of those squares, $N$ squares $(x_1, y_1), (x_2, y_2), ..., (x_N, y_N)$ are painted black, and the other squares are painted white. Let the shortest distance between white squares $A$ and $B$ be the minimum number of moves required to reach $B$ from $A$ **visiting only white squares**, where one can travel to an adjacent square sharing a side (up, down, left or right) in one move. Since there are $H × W - N$ white squares in total, there are $_{(H×W-N)}C_2$ ways to choose two of the white squares. For each of these $_{(H×W-N)}C_2$ ways, find the shortest distance between the chosen squares, then find the sum of all those distances, modulo $1$ $000$ $000$ $007=10^9+7$. ## Constraints * $1 \leq H, W \leq 10^6$ * $1 \leq N \leq 30$ * $0 \leq x_i \leq H-1$ * $0 \leq y_i \leq W-1$ * If $i \neq j$, then either $x_i \neq x_j$ or $y_i \neq y_j$. * There is at least one white square. * For every pair of white squares $A$ and $B$, it is possible to reach $B$ from $A$ visiting only white squares. ## Input Input is given from Standard Input in the following format: $H$ $W$ $N$ $x_1$ $y_1$ $x_2$ $y_2$ $:$ $x_N$ $y_N$ [samples]

Samples

Input #1

2 3
1
1 1

Output #1

20

We have the following grid (`.`: a white square, `#`: a black square).

...
.#.

Let us assign a letter to each white square as follows:

ABC
D#E

Then, we have:

*   dist(A, B) = $1$
*   dist(A, C) = $2$
*   dist(A, D) = $1$
*   dist(A, E) = $3$
*   dist(B, C) = $1$
*   dist(B, D) = $2$
*   dist(B, E) = $2$
*   dist(C, D) = $3$
*   dist(C, E) = $1$
*   dist(D, E) = $4$

where dist(A, B) is the shortest distance between A and B. The sum of these distances is $20$.

Input #2

2 3
1
1 2

Output #2

16

Let us assign a letter to each white square as follows:

ABC
DE#

Then, we have:

*   dist(A, B) = $1$
*   dist(A, C) = $2$
*   dist(A, D) = $1$
*   dist(A, E) = $2$
*   dist(B, C) = $1$
*   dist(B, D) = $2$
*   dist(B, E) = $1$
*   dist(C, D) = $3$
*   dist(C, E) = $2$
*   dist(D, E) = $1$

The sum of these distances is $16$.

Input #3

3 3
1
1 1

Output #3

Input #4

Output #4

Input #5

1000000 1000000
1
0 0

Output #5

333211937

API Response (JSON)

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    "name": "Simple APSP Problem",
    "description": {
      "content": "You are given an $H \\times W$ grid. The square at the top-left corner will be represented by $(0, 0)$, and the square at the bottom-right corner will be represented by $(H-1, W-1)$. Of those squares, ",
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      "time_limit": 3000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "apc001_i"
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    {
      "statement_type": "Markdown",
      "content": "You are given an $H \\times W$ grid. The square at the top-left corner will be represented by $(0, 0)$, and the square at the bottom-right corner will be represented by $(H-1, W-1)$.\nOf those squares, ...",
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      "language": "English"
    }
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}

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