Editing 2016 Jan Platinum Problem 3 Lights Out (section)

== Solution ==

The solution to this problem involves two main steps: First, precomputing the shortest path between each pair of switches using the Floyd-Warshall algorithm. Second, using dynamic programming to find the minimum number of steps.

In the precomputing step, because the cows can be in either of two states (on or off), we calculate the shortest path twice: once when the switch is on and once when it's off. We initialize the 3D array for this step such that the distance between each switch and itself is 0, and the distance between all other switches is infinity. Then we update the array based on the given connections, and finally we run the Floyd-Warshall algorithm to calculate the shortest paths.

In the dynamic programming step, we define dp[i][mask] as the minimum number of steps needed to turn off all the switches from switch i when the switches indicated by the mask are still on. We initialize this dp array to infinity, and then update it iteratively. We iterate over each mask in increasing order of the number of switches that are on. Then for each switch that is still on, we calculate the minimum number of steps to turn off all the switches by visiting switch i last, considering all possible previous switches we could have visited.

This approach ensures that we consider all possibilities and find the minimum number of steps that Bessie needs to take.