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E - Erasing Vertices 2

Score : $500$ points

Problem Statement

You are given a simple undirected graph with $N$ vertices and $M$ edges. The $i$-th edge connects Vertices $U_i$ and $V_i$. Vertex $i$ has a positive integer $A_i$ written on it.

You will repeat the following operation $N$ times.

• Choose a Vertex $x$ that is not removed yet, and remove Vertex $x$ and all edges incident to Vertex $x$. The cost of this operation is the sum of the integers written on the vertices directly connected by an edge with Vertex $x$ that are not removed yet.

We define the cost of the entire $N$ operations as the maximum of the costs of the individual operations. Find the minimum possible cost of the entire operations.

Constraints

• $1 \le N \le 2 \times 10^5$
• $0 \le M \le 2 \times 10^5$
• $1 \le A_i \le 10^9$
• $1 \le U_i,V_i \le N$
• The given graph is simple.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $A_2$ $\dots$ $A_N$

$U_1$ $V_1$

$U_2$ $V_2$

$\vdots$

$U_M$ $V_M$

Output

Print the answer.

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

3

By performing the operations as follows, the maximum of the costs of the $N$ operations can be $3$.

• Choose Vertex $3$. The cost is $A_1=3$.
• Choose Vertex $1$. The cost is $A_2+A_4=3$.
• Choose Vertex $2$. The cost is $0$.
• Choose Vertex $4$. The cost is $0$.

The maximum of the costs of the $N$ operations cannot be $2$ or less, so the solution is $3$.

7 13
464 661 847 514 74 200 188
5 1
7 1
5 7
4 1
4 5
2 4
5 2
1 3
1 6
3 5
1 2
4 6
2 7

1199