Shortest Path Queries 2

AtCoder

IDabc208_d

Time3000ms

Memory256MB

Difficulty—

There are $N$ cities and $M$ roads in the Kingdom of Takahashi. The cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ is a **one-way** road that leads from City $A_i$ to City $B_i$, and it takes $C_i$ minutes to go through it. Let us define $f(s, t, k)$ as the answer to the following query. * Compute the minimum time needed to get from City $s$ to City $t$. Here, other than the Cities $s$ and $t$, it is only allowed to pass through Cities $1$ through $k$. If City $t$ is unreachable or $s = t$, the answer should be $0$. Compute $f(s,t,k)$ for all triples $s,t,k$ and print the sum of them. More formally, print $\displaystyle \sum_{s = 1}^N \sum_{t = 1}^N \sum_{k = 1}^N f(s, t, k)$. ## Constraints * $1 \leq N \leq 400$ * $0 \leq M \leq N(N-1)$ * $1 \leq A_i \leq N$ $(1 \leq i \leq M)$ * $1 \leq B_i \leq N$ $(1 \leq i \leq M)$ * $A_i \neq B_i$ $(1 \leq i \leq M)$ * $1 \leq C_i \leq 10^6$ $(1 \leq i \leq M)$ * $A_i \neq A_j$ or $B_i \neq B_j$, if $i \neq j$. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ $A_1$ $B_1$ $C_1$ $\vdots$ $A_M$ $B_M$ $C_M$ [samples]

Samples

Input #1

3 2
1 2 3
2 3 2

Output #1

25

The triples $s,t,k$ such that $f(s,t,k) \neq 0$ are as follows.

*   For $k = 1$: $f(1,2,1) = 3, f(2,3,1) = 2$.
*   For $k = 2$: $f(1,2,2) = 3, f(2,3,2) = 2, f(1,3,2) = 5$.
*   For $k = 3$: $f(1,2,3) = 3, f(2,3,3) = 2, f(1,3,3) = 5$.

Input #2

3 0

Output #2

0

We have $f(s,t,k) = 0$ for all $s,t,k$.

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Shortest Path Queries 2",
    "description": {
      "content": "There are $N$ cities and $M$ roads in the Kingdom of Takahashi. The cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ is a **one-way** road that leads from City",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc208_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There are $N$ cities and $M$ roads in the Kingdom of Takahashi.\nThe cities are numbered $1$ through $N$, and the roads are numbered $1$ through $M$. Road $i$ is a **one-way** road that leads from City...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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