Reachability in Functional Graph

AtCoder

IDabc357_e

Time2000ms

Memory256MB

Difficulty—

There is a directed graph with $N$ vertices numbered $1$ to $N$ and $N$ edges. The out-degree of every vertex is $1$, and the edge from vertex $i$ points to vertex $a_i$. Count the number of pairs of vertices $(u, v)$ such that vertex $v$ is reachable from vertex $u$. Here, vertex $v$ is reachable from vertex $u$ if there exists a sequence of vertices $w_0, w_1, \dots, w_K$ of length $K+1$ that satisfies the following conditions. In particular, if $u = v$, it is always reachable. * $w_0 = u$. * $w_K = v$. * For every $0 \leq i \lt K$, there is an edge from vertex $w_i$ to vertex $w_{i+1}$. ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq a_i \leq N$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $a_1$ $a_2$ $\dots$ $a_N$ [samples]

Samples

Input #1

4
2 1 1 4

Output #1

8

The vertices reachable from vertex $1$ are vertices $1, 2$.  
The vertices reachable from vertex $2$ are vertices $1, 2$.  
The vertices reachable from vertex $3$ are vertices $1, 2, 3$.  
The vertex reachable from vertex $4$ is vertex $4$.  
Therefore, the number of pairs of vertices $(u, v)$ such that vertex $v$ is reachable from vertex $u$ is $8$.  
Note that the edge from vertex $4$ is a self-loop, that is, it points to vertex $4$ itself.

Input #2

5
2 4 3 1 2

Output #2

Input #3

10
6 10 4 1 5 9 8 6 5 1

Output #3

API Response (JSON)

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      "content": "There is a directed graph with $N$ vertices numbered $1$ to $N$ and $N$ edges.   The out-degree of every vertex is $1$, and the edge from vertex $i$ points to vertex $a_i$.   Count the number of pairs",
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      "content": "There is a directed graph with $N$ vertices numbered $1$ to $N$ and $N$ edges.  \nThe out-degree of every vertex is $1$, and the edge from vertex $i$ points to vertex $a_i$.  \nCount the number of pairs...",
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