hackercupai/hackercup huggingface dataset:
Another way to view a binary prefix code is as the [set of leaves on a unique binary tree](https://www.cs.princeton.edu/courses/archive/spr01/cs126/assignments/prefix.html). Each leaf corresponds to a codeword, given by the path of dots and dashes from the root. We can start with a binary tree consisting only of such a path to \(C_1\). To find an answer, we can continue adding nodes (without touching \(C_1\)'s leaf) until we have enough.

Implementation-wise, we can imagine a perfect binary tree that's pruned at the leaf corresponding to \(C_1\). Breadth-first search can be used to perform a level-order traversal of this tree without explicitly constructing it. Each node can be represented as a candidate codeword string, with children obtained by appending either a dot or dash.

As we BFS, we maintain all the current leaves (nodes enqueued but have yet to be processed), stopping once we hit \(N - 1\) leaves. We must be careful not to output anything on the path to \(C_1\). This can be handled by avoiding outputting prefixes of \(C_1\) or \(C_1\)-prefixed codewords, or by simply by prepending all output codewords with the opposite of \(C_1\)'s first character (essentially pruning the perfect binary tree at the start of \(C_1\) instead of the end).

Since a perfect binary tree of height \(h\) has \(2^h\) leaves, a maximum codeword length (tree height) of \(10\) still yields \(2^{10} = 1024\) leaves to work with. Since pruning once cannot get rid of more than \(1/2\) of a tree's leaves, we'll get at least \(1024/2 = 512 > 100\) leaves before hitting length \(10\).

Another approach, yielding a slightly longer total output length, is to generate bitstrings of a fixed length (\(\lceil \log_2 N \rceil + 1\) is sufficient), excluding any which prefixes or is prefixed by \(C_1\).

[See David Harmeyer's solution video here.](https://youtu.be/6Xgt70dfvNk)