I am trying the following problems from several hours , but not able to get a correct logic for it.
You are in a planning team, which is in charge of laying
out an auxiliary electricity grid. The purpose of this
auxiliary grid is to power the lamp posts on junctions
of a small town in case of power outage. Your team is
given k generators. You can place these generators
anywhere in the grid and each generator can turn on
all the lamps that have a connection to it via the grid
(there exists a path). For each road in the town, you are
given the cost of laying down a cable in the grid to connect the two
junctions at the endpoints of that road. Given the layout of the town,
your task is to lay down the minimum cost grid and install
the generators on it such that you can turn on all the lampposts on
the junctions of the town.
Input:
The first line contains the number of cases t. Then, t cases follow.
The first line of each case contains three integer n, m, and k.
The junctions are numbered 1 to n. Then, m lines follow.
Each line contains three integers i, j, c. The integers i and j
are between 1 and n and denote the two junctions at the two endpoints of a road.
The third integer, c, is the cost
of laying down a cable in the grid on this road.
Output:
You should output t lines, one for each case. For each case output the
minimum cost grid. If this task is impossible (i.e. there is no way to
turn all the lamps on with k generators),
output “Impossible!” (quotes for clarity)
Constraints:
1 <= t <= 25
1 <= n <= 2000
0 <= m <= n(n-1)/2
0 <= c <= 1000000
Sample Input:
2
3 1 1
1 2 10
6 7 2
1 2 20
1 3 5
1 4 10
2 3 8
2 4 15
3 4 2
5 6 9
Sample Output:
Impossible!
24
Explanation:
In the first case, the junctions 1 and 2 are disconnected from
junction 3 and you cannot turn on all the lampposts with only
one generator. You need at least two generators.In the second case,
you can lay down cable on the roads (1 3), (2 3), (3 4), and (5 6) and then
install one generator on junction 1 and one generator on junction 5.
How to solve this problem efficiently?Rightnow I have no idea other than BruteForce.
Will modified Kruskal work here?
The solution you want is a collection of disconnected spanning trees. If you took all the lines not in the solution and gave them the same very large cost then you could find a minimum spanning tree and discard the lines with very large costs to find the solution.
This is obviously impossible, but Kruskal’s algorithm works by starting with the edge with least cost and carrying on in order of steadily increasing cost. So if you just stop it when you have k disconnected components you will get the same answer as if you had done this.