A771.Directory Traversal--Gold

Bessie the cow is surprisingly computer savvy. On her computer in the barn,
she stores all of her precious files in a collection of directories; for
example:
bessie/
folder1/
file1
folder2/
file2
folder3/
file3
file4
There is a single "top level" directory, called bessie.
Bessie can navigate to be inside any directory she wants. From a given
directory, any file can be referenced by a "relative path". In a relative
path, the symbol ".." refers to the parent directory. If Bessie were in
folder2, she could refer to the four files as follows:
../file1
file2
../../folder3/file3
../../file4
Bessie would like to choose a directory from which the sum of the lengths of
the relative paths to all the files is minimized.

The first line contains an integer N ($2 \leq N \leq 100,000$), giving the
total number of files and directories. For the purposes of input, each object
(file or directory) is assigned a unique integer ID between 1 and $N$, where
ID 1 refers to the top level directory.
Next, there will be $N$ lines. Each line starts with the name of a file or
directory. The name will have only lower case characters a-z and digits 0-9,
and will be at most 16 characters long. Following the name is an integer, $m$.
If $m$ is 0, then this entity is a file. If $m > 0$, then this entity is a
directory, and it has a total of $m$ files or directories inside it. Following
$m$ there will be $m$ integers giving the IDs of the entities in this
directory.

Output the minimal possible total length of all relative paths to files. Note
that this value may be too large to fit into a 32-bit integer.

This input describes the example directory structure given above.
The best solution is to be in folder1. From this directory, the relative paths
are:
file1
folder2/file2
../folder3/file3
../file4