Railway Construction
Contest: Codeforces Round 745 (Div. 1)
Difficulty rating: 3400
Tags: brute force, constructive algorithms, data structures, graphs, shortest paths

Because the railway system in Gensokyo is often congested, as an enthusiastic engineer, Kawasiro Nitori plans to construct more railway to ease the congestion.

There are $$$n$$$ stations numbered from $$$1$$$ to $$$n$$$ and $$$m$$$ two-way railways in Gensokyo. Every two-way railway connects two different stations and has a positive integer length $$$d$$$. No two two-way railways connect the same two stations. Besides, it is possible to travel from any station to any other using those railways. Among these $$$n$$$ stations, station $$$1$$$ is the main station. You can get to any station from any other station using only two-way railways.

Because of the technological limitation, Nitori can only construct one-way railways, whose length can be arbitrary positive integer. Constructing a one-way railway from station $$$u$$$ will costs $$$w_u$$$ units of resources, no matter where the railway ends. To ease the congestion, Nitori plans that after construction there are at least two shortest paths from station $$$1$$$ to any other station, and these two shortest paths do not pass the same station except station $$$1$$$ and the terminal. Besides, Nitori also does not want to change the distance of the shortest path from station $$$1$$$ to any other station.

Due to various reasons, sometimes the cost of building a new railway will increase uncontrollably. There will be a total of $$$q$$$ occurrences of this kind of incident, and the $$$i$$$-th event will add additional amount of $$$x_i$$$ to the cost of building a new railway from the station $$$k_i$$$.

To save resources, before all incidents and after each incident, Nitori wants you to help her calculate the minimal cost of railway construction.

Input format:
The first line contains three integers $$$n$$$, $$$m$$$, and $$$q$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \le m \le 3 \cdot 10^5$$$, $$$0 \le q \le 2\cdot10^5$$$).

The second line contains $$$n$$$ integers $$$w_1,w_2,\ldots,w_n$$$ ($$$1 \le w_i \le 10^9$$$).

Each of the next $$$m$$$ lines contains three integers $$$u$$$, $$$v$$$, $$$d$$$ ($$$1 \le u,v \le n$$$, $$$u \ne v$$$, $$$1 \le d \le 10^9$$$), denoting a two-way railway connecting station $$$u$$$ and station $$$v$$$, with length $$$d$$$.

The $$$i$$$-th of the next $$$q$$$ lines contains two integers $$$k_i,x_i$$$ ($$$1 \le k_i \le n, 1 \le x_i \le 4 \times 10^8$$$).

Output format:
Print $$$q+1$$$ lines, and the $$$i$$$-th of these lines contains one integer, denoting the minimal cost of railway construction after the $$$i-1$$$-th incident (especially, the $$$0$$$-th incident means no incident occurred).

Constraints:
Time limit: 2.5 s
Memory limit: 512.0 MB

Note:
In the second example, Nitori can build railways as follows: $$$1 \rightarrow 2$$$, $$$1 \rightarrow 3$$$, $$$1 \rightarrow 4$$$, $$$2 \rightarrow 8$$$, and the cost is $$$14 + 14 + 14 + 4 = 46$$$.

Example input:
5 5 1
1 1 1 1 1
1 2 1
2 3 1
2 4 1
3 5 1
4 5 1
1 2

Example output:
3
9

Example input:
8 11 0
14 4 16 15 1 3 1 14
4 2 1
1 2 3
7 5 4
2 3 1
8 6 2
8 5 5
5 4 5
7 6 7
3 5 5
1 6 6
8 1 4

Example output:
46

Editorial: