Nowcoder problem: Game on Tree

Правка en6, от Aveiro_quanyue, 2023-05-15 14:06:45

Alice and Bob are playing a game on a tree $$$T$$$ whose root is $$$1$$$. $$$T$$$ has $$$n$$$ vertices and $$$n-1$$$ edges, each vertex has an attribute $$$a \in \{1,2,3\}$$$. If $$$a=1$$$, a person on this vertex $$$v$$$ must jump to the parent of $$$v$$$, i.e., v->parent. If $$$a=2$$$, a person on this vertex $$$v$$$ must jump to $$$v$$$'s grandparent, i.e., v->parent->parent. If $$$a=3$$$, a person on this vertex can either jump to v->parent or v->parent->parent. Alice starts the first and she can choose which vertex to start. If somebody cannot make a move, she/he loses the game. For example, if Alice stands on a child of the root and this child has attribute $$$2$$$, then Alice loses as this child does not have a grandparent. Similarly if Bob stands on the root, he will lose as he cannot move no matter what the attribute of the root is.

Input contains $$$n+1$$$ ($$$2 \leq n \leq 10^5$$$) lines.

The first line contains a single integer $$$n$$$, which is the number of vertices.

The second line contains $$$n$$$ integers $$$a_i$$$ ($$$1 \leq a_i \leq 3$$$) representing the attribute of each vertex $$$i$$$.

The following $$$n-1$$$ lines representing edges, each line contains two integers $$$u_i$$$ and $$$v_i$$$. $$$u_i$$$-$$$v_i$$$ is an undirected edge of the tree $$$T$$$.

Output should contain $$$n-1$$$ lines, each line is a query. The $$$i$$$-th ($$$1 \leq i \leq n-1$$$) line represents the winner if Alice and Bob plays optimally when the $$$i$$$-th edge is deleted from $$$T$$$. The deletion operations are independent along queries, i.e., the $$$i$$$-th edge is recovered from deletion when you process the $$$i+1$$$-th query.

Test case 1:

Input:

3

1 2 1

1 2

1 3

Output:

Alice

Bob

In the first line, the first edge, i.e., $$$1-2$$$ is destroyed. In this case, Alice can choose to start from vertex $$$3$$$, and jump to vertex $$$1$$$. Then, Bob will stuck at vertex $$$1$$$. Alice wins.

In the second line, the first edge $$$1-2$$$ is recovered and the second edge $$$1-3$$$ is destroyed. Here $$$3$$$ becomes an isolation vertex. If Alice starts at $$$3$$$, she cannot jump. If Alice starts at $$$1$$$, she cannot jump either as $$$1$$$ is the root. If Alice starts at $$$2$$$, she cannot jump either as $$$2$$$ does not have grandparent. The parent of $$$2$$$ is the root $$$1$$$ but $$$1$$$ does not have a parent. Therefore, no matter where Alice starts, Bob will win.

Test case 2:

8

1 3 2 1 3 1 1 2

1 2

1 3

2 4

2 5

4 8

5 6

5 7

Output:

Alice

Bob

Bob

Alice

Bob

Bob

Bob

For example, if you delete the first edge $$$1-2$$$, Alice will win if she chooses to start at $$$8$$$.

If you delete the fourth edge $$$2-5$$$, Alice will win if she chooses to start at $$$6$$$ or $$$7$$$.

Теги tree, graph, game theory

История

 
 
 
 
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en16 Английский Aveiro_quanyue 2023-05-15 14:29:27 6
en15 Английский Aveiro_quanyue 2023-05-15 14:26:17 9 Tiny change: 'd edge of the tree $T$.\n\nO' -> 'd edge of $T$.\n\nO'
en14 Английский Aveiro_quanyue 2023-05-15 14:25:37 0 (published)
en13 Английский Aveiro_quanyue 2023-05-15 14:25:15 15
en12 Английский Aveiro_quanyue 2023-05-15 14:23:46 40
en11 Английский Aveiro_quanyue 2023-05-15 14:22:55 53 Tiny change: 'Alice and ' -> 'Link: https://ac.nowcoder.com/acm/contest/56458/E\n\nAlice and '
en10 Английский Aveiro_quanyue 2023-05-15 14:22:41 9 Tiny change: 'eger $n$, which is the numbe' -> 'eger $n$, the numbe'
en9 Английский Aveiro_quanyue 2023-05-15 14:22:25 5
en8 Английский Aveiro_quanyue 2023-05-15 14:21:37 1397
en7 Английский Aveiro_quanyue 2023-05-15 14:08:31 64
en6 Английский Aveiro_quanyue 2023-05-15 14:06:45 353
en5 Английский Aveiro_quanyue 2023-05-15 14:00:56 760
en4 Английский Aveiro_quanyue 2023-05-15 13:56:50 118
en3 Английский Aveiro_quanyue 2023-05-15 13:54:59 671
en2 Английский Aveiro_quanyue 2023-05-15 13:47:57 831
en1 Английский Aveiro_quanyue 2023-05-15 13:40:41 91 Initial revision (saved to drafts)