Counting Shortest Paths

AtCoder

IDabc319_g

Time2000ms

Memory256MB

Difficulty—

We will perform the following operation on a complete undirected graph $G$ with $N$ vertices. > For each $i = 1, 2, \ldots, M$, delete the undirected edge connecting vertices $u_i$ and $v_i$. Determine if there is a path from vertex $1$ to vertex $N$ in $G$ after the operation. If there is, find the number, modulo $998244353$, of shortest paths from vertex $1$ to vertex $N$. Here, a shortest path from vertex $1$ to vertex $N$ is a path from vertex $1$ to vertex $N$ that contains the minimum number of edges. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $0 \leq M \leq \min\lbrace 2 \times 10^5, N(N-1)/2 \rbrace$ * $1 \leq u_i, v_i \leq N$ * $u_i \neq v_i$ * $i \neq j \implies \lbrace u_i, v_i \rbrace \neq \lbrace u_j, v_j \rbrace$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$ [samples]

Samples

Input #1

Output #1

3

In $G$ after the operation, the shortest paths from vertex $1$ to vertex $N$ are the following three, each containing three edges.

*   vertex $1$ $\rightarrow$ vertex $2$ $\rightarrow$ vertex $3$ $\rightarrow$ vertex $6$
*   vertex $1$ $\rightarrow$ vertex $2$ $\rightarrow$ vertex $5$ $\rightarrow$ vertex $6$
*   vertex $1$ $\rightarrow$ vertex $4$ $\rightarrow$ vertex $5$ $\rightarrow$ vertex $6$

Input #2

Output #2

\-1

$G$ has no edges after the operation. There is no path from vertex $1$ to vertex $N$, so print `-1`.

API Response (JSON)

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    "name": "Counting Shortest Paths",
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      "content": "We will perform the following operation on a complete undirected graph $G$ with $N$ vertices. > For each $i = 1, 2, \\ldots, M$, delete the undirected edge connecting vertices $u_i$ and $v_i$. Determ",
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "sync_url": null,
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We will perform the following operation on a complete undirected graph $G$ with $N$ vertices.\n\n> For each $i = 1, 2, \\ldots, M$, delete the undirected edge connecting vertices $u_i$ and $v_i$.\n\nDeterm...",
      "is_translate": false,
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