hackercupai/hackercup huggingface dataset:
We'll simulate the entire process in reverse, with the water level decreasing and gradually uncovering all of the platforms while rock samples need to be collected at certain times. As we go, we'll want to maintain a set of groups of platforms such that all platforms in each group are currently reachable from one another, which we can represent as a [disjoint-set data structure](https://en.wikipedia.org/wiki/Disjoint-set_data_structure). Each pair of adjacent platforms \((i_1, j_1)\) and \((i_2, j_2)\) will become directly reachable from one another when the water level reaches height \(\min(H_{i_1,j_1}, H_{i_2,j_2})\). Based on this, we can assemble a list of \(O(RC)\) events for when pairs of adjacent platforms will become reachable, combine them with another \(RC\) events for when samples need to be collected, and process all of these events in non-increasing order of height. When adjacent platforms become reachable, we'll merge their respective groups together.

When processing a sample collection event for platform \((i, j)\), the sample cannot be collected if \(S_{i,j} \ge H_{i,j}\). Otherwise, it can be, with the question being how to allocate robots to collect it and other samples. We can take a greedy approach involving reusing a previously-allocated robot to handle the sample if possible, and otherwise allocating a new robot to it. To do so, we'll augment our disjoint-set data structure to store a flag per group for whether or not a robot has already been allocated to that group, the idea being that a robot allocated to a group can also handle any more samples within the same group. When merging groups together, the resulting group is considered to have an allocated robot if at least one of the two existing groups did. All that remains is checking and potentially setting this flag for each sample's platform's group.

The most expensive part of this algorithm is sorting of the events, taking \(O(RC \log(RC))\) time.

Other solutions also exist, such as different implementations of the general concept described above (for example, by replacing the disjoint-set data structure with a gradual flood fill of which cells are reachable by robots allocated so far).

[See David Harmeyer (SecondThread)'s solution video here.](https://youtu.be/w6Xvy0c876o)