Avoiding Collision

AtCoder

IDarc090_c

Time2000ms

Memory256MB

Difficulty—

We have a graph with $N$ vertices and $M$ edges, and there are two people on the graph: Takahashi and Aoki. The $i$\-th edge connects Vertex $U_i$ and Vertex $V_i$. The time it takes to traverse this edge is $D_i$ minutes, regardless of direction and who traverses the edge (Takahashi or Aoki). Takahashi departs Vertex $S$ and Aoki departs Vertex $T$ at the same time. Takahashi travels to Vertex $T$ and Aoki travels to Vertex $S$, both in the shortest time possible. Find the number of the pairs of ways for Takahashi and Aoki to choose their shortest paths such that they never meet (at a vertex or on an edge) during the travel, modulo $10^9 + 7$. ## Constraints * $1 \leq N \leq 100$ $000$ * $1 \leq M \leq 200$ $000$ * $1 \leq S, T \leq N$ * $S \neq T$ * $1 \leq U_i, V_i \leq N$ ($1 \leq i \leq M$) * $1 \leq D_i \leq 10^9$ ($1 \leq i \leq M$) * If $i \neq j$, then $(U_i, V_i) \neq (U_j, V_j)$ and $(U_i, V_i) \neq (V_j, U_j)$. * $U_i \neq V_i$ ($1 \leq i \leq M$) * $D_i$ are integers. * The given graph is connected. ## Input Input is given from Standard Input in the following format: $N$ $M$ $S$ $T$ $U_1$ $V_1$ $D_1$ $U_2$ $V_2$ $D_2$ $:$ $U_M$ $V_M$ $D_M$ [samples]

Samples

Input #1

Output #1

2

There are two ways to choose shortest paths that satisfies the condition:

*   Takahashi chooses the path $1 \rightarrow 2 \rightarrow 3$, and Aoki chooses the path $3 \rightarrow 4 \rightarrow 1$.
*   Takahashi chooses the path $1 \rightarrow 4 \rightarrow 3$, and Aoki chooses the path $3 \rightarrow 2 \rightarrow 1$.

Input #2

Output #2

Input #3

Output #3

Input #4

8 13
4 2
7 3 9
6 2 3
1 6 4
7 6 9
3 8 9
1 2 2
2 8 12
8 6 9
2 5 5
4 2 18
5 3 7
5 1 515371567
4 8 6

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Avoiding Collision",
    "description": {
      "content": "We have a graph with $N$ vertices and $M$ edges, and there are two people on the graph: Takahashi and Aoki. The $i$\\-th edge connects Vertex $U_i$ and Vertex $V_i$. The time it takes to traverse this ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc090_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a graph with $N$ vertices and $M$ edges, and there are two people on the graph: Takahashi and Aoki.\nThe $i$\\-th edge connects Vertex $U_i$ and Vertex $V_i$. The time it takes to traverse this ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments