102555: [AtCoder]ABC255 F - Pre-order and In-order

Problem Statement

Consider a binary tree with $N$ vertices numbered $1, 2, \ldots, N$. Here, a binary tree is a rooted tree where each vertex has at most two children. Specifically, each vertex in a binary tree has at most one left child and at most one right child.

Determine whether there exists a binary tree rooted at Vertex $1$ satisfying the conditions below, and present such a tree if it exists.

• The depth-first traversal of the tree in pre-order is $(P_1, P_2, \ldots, P_N)$.
• The depth-first traversal of the tree in in-order is $(I_1, I_2, \ldots, I_N)$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $N$ is an integer.
• $(P_1, P_2, \ldots, P_N)$ is a permutation of $(1, 2, \ldots, N)$.
• $(I_1, I_2, \ldots, I_N)$ is a permutation of $(1, 2, \ldots, N)$.

Input

Input is given from Standard Input in the following format:

$N$
$P_1$ $P_2$ $\ldots$ $P_N$
$I_1$ $I_2$ $\ldots$ $I_N$

Output

If there is no binary tree rooted at Vertex $1$ satisfying the conditions in Problem Statement, print $-1$.
Otherwise, print one such tree in $N$ lines as follows. For each $i = 1, 2, \ldots, N$, the $i$-th line should contain $L_i$ and $R_i$, the indices of the left and right children of Vertex $i$, respectively. Here, if Vertex $i$ has no left (right) child, $L_i$ ($R_i$) should be $0$.
If there are multiple binary trees rooted at Vertex $1$ satisfying the conditions, any of them will be accepted.

$L_1$ $R_1$
$L_2$ $R_2$
$\vdots$
$L_N$ $R_N$

Sample Input 1

6
1 3 5 6 4 2
3 5 1 4 6 2

Sample Output 1

3 6
0 0
0 5
0 0
0 0
4 2

The binary tree rooted at Vertex $1$ shown in the following image satisfies the conditions.

Sample Input 2

2
2 1
1 2

Sample Output 2

-1

No binary tree rooted at Vertex $1$ satisfies the conditions, so $-1$ should be printed.

问题描述

考虑一个具有$N$个顶点的二叉树，顶点编号为$1, 2, \ldots, N$。这里，二叉树是一种有根树，其中每个顶点最多有两个子节点。具体来说，二叉树中的每个顶点最多有一个 左子节点 和一个 右子节点。

确定是否存在一个以顶点$1$为根的二叉树满足以下条件，并在存在时给出这样的树。

• 树的 前序遍历 为$(P_1, P_2, \ldots, P_N)$。
• 树的 中序遍历 为$(I_1, I_2, \ldots, I_N)$。

约束条件

• $2 \leq N \leq 2 \times 10^5$
• $N$ 是整数。
• $(P_1, P_2, \ldots, P_N)$ 是$(1, 2, \ldots, N)$的排列。
• $(I_1, I_2, \ldots, I_N)$ 是$(1, 2, \ldots, N)$的排列。

输入

输入从标准输入以以下格式给出：

$N$
$P_1$ $P_2$ $\ldots$ $P_N$
$I_1$ $I_2$ $\ldots$ $I_N$

输出

如果不存在以顶点$1$为根的二叉树满足问题描述中的条件，则输出$-1$。
否则，按以下方式在$N$行中输出这样的树。 对于每个$i = 1, 2, \ldots, N$，第$i$行应包含$L_i$和$R_i$，即顶点$i$的左子节点和右子节点的索引。 如果顶点$i$没有左（右）子节点，则$L_i$（$R_i$）应为$0$。
如果存在多个以顶点$1$为根的二叉树满足条件，任何其中之一都将被接受。

$L_1$ $R_1$
$L_2$ $R_2$
$\vdots$
$L_N$ $R_N$

样例输入 1

6
1 3 5 6 4 2
3 5 1 4 6 2

样例输出 1

3 6
0 0
0 5
0 0
0 0
4 2

以顶点$1$为根的二叉树如图所示满足条件。

样例输入 2

2
2 1
1 2

样例输出 2

-1

不存在以顶点$1$为根的二叉树满足条件，因此应输出$-1$。