200983: [AtCoder]ARC098 F - Donation

Problem Statement

There is a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1$ through $N$, and the edges are numbered $1$ through $M$. Edge $i$ connects Vertex $U_i$ and $V_i$. Also, Vertex $i$ has two predetermined integers $A_i$ and $B_i$. You will play the following game on this graph.

First, choose one vertex and stand on it, with $W$ yen (the currency of Japan) in your pocket. Here, $A_s \leq W$ must hold, where $s$ is the vertex you choose. Then, perform the following two kinds of operations any number of times in any order:

• Choose one vertex $v$ that is directly connected by an edge to the vertex you are standing on, and move to vertex $v$. Here, you need to have at least $A_v$ yen in your pocket when you perform this move.
• Donate $B_v$ yen to the vertex $v$ you are standing on. Here, the amount of money in your pocket must not become less than $0$ yen.

You win the game when you donate once to every vertex. Find the smallest initial amount of money $W$ that enables you to win the game.

Constraints

• $1 \leq N \leq 10^5$
• $N-1 \leq M \leq 10^5$
• $1 \leq A_i,B_i \leq 10^9$
• $1 \leq U_i < V_i \leq N$
• The given graph is connected and simple (there is at most one edge between any pair of vertices).

Input

Input is given from Standard Input in the following format:

$N$ $M$
$A_1$ $B_1$
$A_2$ $B_2$
$:$
$A_N$ $B_N$
$U_1$ $V_1$
$U_2$ $V_2$
$:$
$U_M$ $V_M$

Output

Print the smallest initial amount of money $W$ that enables you to win the game.

Sample Input 1

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

Sample Output 1

6

If you have $6$ yen initially, you can win the game as follows:

• Stand on Vertex $4$. This is possible since you have not less than $6$ yen.
• Donate $2$ yen to Vertex $4$. Now you have $4$ yen.
• Move to Vertex $3$. This is possible since you have not less than $4$ yen.
• Donate $1$ yen to Vertex $3$. Now you have $3$ yen.
• Move to Vertex $2$. This is possible since you have not less than $1$ yen.
• Move to Vertex $1$. This is possible since you have not less than $3$ yen.
• Donate $1$ yen to Vertex $1$. Now you have $2$ yen.
• Move to Vertex $2$. This is possible since you have not less than $1$ yen.
• Donate $2$ yen to Vertex $2$. Now you have $0$ yen.

If you have less than $6$ yen initially, you cannot win the game. Thus, the answer is $6$.

Sample Input 2

5 8
6 4
15 13
15 19
15 1
20 7
1 3
1 4
1 5
2 3
2 4
2 5
3 5
4 5

Sample Output 2

44

Sample Input 3

9 10
131 2
98 79
242 32
231 38
382 82
224 22
140 88
209 70
164 64
6 8
1 6
1 4
1 3
4 7
4 9
3 7
3 9
5 9
2 5

Sample Output 3

582