A925 | Switch Grass--Platinum

Farmer John has recently been experimenting with cultivating different types
of grass on his farm, realizing that different types of cows like different
types of grass. However, he must be careful to ensure that different types of
grass are planted sufficiently far away from each-other, in order to prevent
them from being inextricably mixed.
FJ's farm consists of $N$ fields ($1 \leq N \leq 200,000$), where $M$ pairs of
fields are connected by bi-directional pathways ($1 \leq M \leq 200,000$).
Using these pathways, it is possible to walk from any field to any other
field. Each pathway has an integer length in the range $1 \ldots 1,000,000$.
Any pair of fields will be linked by at most one direct pathway.
In each field, FJ initially plants one of $K$ types of grass ($1 \leq K \leq
N$). Over time, however, he might decide to switch the grass in some field to
a different type. He calls this an "update" operation. He might perform
several updates over the course of time, which are all cumulative in nature.
After each update, FJ would like to know the length of the shortest path
between two fields having different grass types. That is, among all pairs of
fields having different grass types, he wants to know which two are closest.
Ideally, this number is large, so he can prevent grass of one type from mixing
with grass of another type. It is guaranteed that the farm will always have at
least two fields with different grass types.
In 30 percent of the input cases, each field will be directly connected to at
most 10 pathways.

The first line of input contains four integers, $N$, $M$, $K$, and $Q$, where
$Q$ is the number of updates ($1 \leq Q \leq 200,000$). The next $M$ lines
describe the paths; each one contains three integers $A$, $B$, and $L$,
indicating a path from field $A$ to field $B$ (both integers in the range $1
\ldots N$) of length $L$. The next line indicates the initial type of grass
growing in each field ($N$ integers in the range $1 \ldots K$). Finally, the
last $Q$ lines each describe an update, specified by two integers $A$ and $B$,
where the grass in field $A$ is to be updated to type $B$.

For each update, print the length of the shortest path between two fields with
different types of grass, after the update is applied.