Problem 2: Simplifying the Farm [Nathan Pinsker, 2011]
Farmer John has been taking an evening algorithms course at his local
university, and he has just learned about minimum spanning trees. However,
Farmer John now realizes that the design of his farm is not as efficient as
it could be, and he wants to simplify the layout of his farm.
The farm is currently arranged like a graph, with vertices representing
fields and edges representing pathways between these fields, each having an
associated length. Farmer John notes that for each distinct length,
at most three pathways on his farm share this length. FJ would like to
remove some of the pathways on his farm so that it becomes a tree -- that
is, so that there is one unique route between any pair of fields. Moreover,
Farmer John would like this to be a minimum spanning tree -- a tree having
the smallest possible sum of edge lengths.
Help Farmer John compute not only the sum of edge lengths in a minimum
spanning tree derived from his farm graph, but also the number of different
possible minimum spanning trees he can create.
PROBLEM NAME: simplify
INPUT FORMAT:
* Line 1: Two integers N and M (1 <= N <= 40,000; 1 <= M <= 100,000),
representing the number of vertices and edges in the farm
graph, respectively. Vertices are numbered as 1..N.
* Lines 2..M+1: Three integers a_i, b_i and n_i (1 <= a_i, b_i <= N; 1
<= n_i <= 1,000,000) representing an edge from vertex a_i to
b_i with length n_i. No edge length n_i will occur more than
three times.
SAMPLE INPUT (file simplify.in):
4 5
1 2 1
3 4 1
1 3 2
1 4 2
2 3 2
OUTPUT FORMAT:
* Line 1: Two integers representing the length of the minimal spanning
tree and the number of minimal spanning trees (mod
1,000,000,007).
SAMPLE OUTPUT (file simplify.out):
4 3
OUTPUT DETAILS:
Picking both edges with length 1 and any edge with length 2 yields a minimum
spanning tree of length 4.