Score : $500$ points

问题陈述

在Atcoder共和国，有编号为 $1$ 到 $N$ 的 $N$ 个城镇以及编号为 $1$ 到 $M$ 的 $M$ 条高速公路。
高速公路 $i$ 双向连接城镇 $A_i$ 和城镇 $B_i$。

高桥国王计划新建 $(N-M-1)$ 条新高速公路，以满足以下两个条件：

• 使用一些数量的高速公路可以从任意一对城镇之间通行
• 对于每 $i=1,\ldots,N$，恰好有 $D_i$ 条高速公路直接与城镇 $i$ 相连

确定是否存在这样的建设方式。如果存在，输出一种方案。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

The Republic of Atcoder has $N$ towns numbered $1$ through $N$, and $M$ highways numbered $1$ through $M$.
Highway $i$ connects Town $A_i$ and Town $B_i$ bidirectionally.

King Takahashi is going to construct $(N-M-1)$ new highways so that the following two conditions are satisfied:

• One can travel between every pair of towns using some number of highways
• For each $i=1,\ldots,N$, exactly $D_i$ highways are directly connected to Town $i$

Determine if there is such a way of construction. If it exists, print one.

Constraints

• $2 \leq N \leq 2\times 10^5$
• $0 \leq M \lt N-1$
• $1 \leq D_i \leq N-1$
• $1\leq A_i \lt B_i \leq N$
• If $i\neq j$, then $(A_i, B_i) \neq (A_j,B_j)$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$D_1$ $\ldots$ $D_N$

$A_1$ $B_1$

$\vdots$

$A_M$ $B_M$

Output

If there isn't a way of construction that satisfies the conditions, print -1.
If there is, print $(N-M-1)$ lines. The $i$-th line should contain the indices of the two towns connected by the $i$-th highway to be constructed.

Sample Input 1

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

Sample Output 1

6 2
5 6
4 5

As in the Sample Output, the conditions can be satisfied by constructing highways connecting Towns $6$ and $2$, Towns $5$ and $6$, and Towns $4$ and $5$, respectively.

Another example to satisfy the conditions is to construct highways connecting Towns $6$ and $4$, Towns $5$ and $6$, and Towns $2$ and $5$, respectively.

Sample Input 2

5 1
1 1 1 1 4
2 3

Sample Output 2

-1

Sample Input 3

4 0
3 3 3 3

Sample Output 3

-1

update @ 2024/3/10 10:19:26