Communicate Topological Order

AtCoder

IDagc074_a

Time2000ms

Memory256MB

Difficulty—

You are given a simple directed acyclic graph $G$ with $N$ vertices numbered $1$ to $N$. $G$ has $M$ edges, and the $i$\-th edge is directed from vertex $x_i$ to vertex $y_i$. A permutation $P=(P_1,P_2,\ldots,P_N)$ of $(1,2,\ldots,N)$ that satisfies the following condition is called a **special permutation**: * $P_{x_i} \lt P_{y_i}$ for all $i=1,2,\cdots,M$. You are given a special permutation $P$. Takahashi and Aoki will play a game using it. Takahashi knows the value of each term of $P$, but Aoki only knows that $P$ is a special permutation. Both of them know $G$. Takahashi selects some terms of $P$ and tells Aoki their positions and values. Find the minimum number of terms that Takahashi needs to tell Aoki so that Aoki can determine the values of all terms of $P$. Solve $T$ test cases per input. ## Constraints * $1 \leq T \leq 10^4$ * $1 \leq N \leq 2 \times 10^5$ * $0 \leq M \leq 2 \times 10^5$ * $1 \leq x_i,y_i \leq N$ * The given graph $G$ is a simple directed acyclic graph. * $1 \leq P_i \leq N$ * $P_1,P_2,\ldots,P_N$ are pairwise distinct. * $P_{x_i} \lt P_{y_i}$ * The sum of $N$ over all test cases is at most $2 \times 10^5$. * The sum of $M$ over all test cases is at most $2 \times 10^5$. * All input values are integers. ## Input The input is given from Standard Input in the following format: $T$ $case_1$ $case_2$ $\vdots$ $case_T$ Each case is given in the following format: $N$ $M$ $x_1$ $y_1$ $x_2$ $y_2$ $\vdots$ $x_M$ $y_M$ $P_1$ $P_2$ $\ldots$ $P_N$ [samples]

Samples

Input #1

3
5 4
3 4
4 1
3 1
5 2
5 4 1 2 3
4 0
4 2 1 3
10 15
6 2
8 5
4 9
7 10
3 7
5 6
8 9
3 5
5 2
8 2
3 9
5 9
10 2
3 2
7 4
8 9 2 6 4 5 3 1 10 7

Output #1

2
3
6

In the first case, if Takahashi tells that $P_1=5, \; P_4=2$, Aoki can determine all terms of $P$. Also, if Takahashi tells only one term of $P$, Aoki cannot determine all terms of $P$.

API Response (JSON)

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    "name": "Communicate Topological Order",
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      "content": "You are given a simple directed acyclic graph $G$ with $N$ vertices numbered $1$ to $N$. $G$ has $M$ edges, and the $i$\\-th edge is directed from vertex $x_i$ to vertex $y_i$. A permutation $P=(P_1,P_",
      "description_type": "Markdown"
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    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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      "statement_type": "Markdown",
      "content": "You are given a simple directed acyclic graph $G$ with $N$ vertices numbered $1$ to $N$. $G$ has $M$ edges, and the $i$\\-th edge is directed from vertex $x_i$ to vertex $y_i$.\nA permutation $P=(P_1,P_...",
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