Hi design gurus,

You explored a number of different options for optimizing union and find operations, but I don't feel these optimizations were properly contextualized.

For instance, from your article it seems like you could optimize both the union and the find operations simultaneously, but optimizing them at the same time doesn't work because changing parents in the find optimization invalidates the state in the rank or the size arrays.

I would like to see better explanations of these optimizations, how to employ them, and their effects on the time complexity. Also, what is the time complexity of the un-optimized operations? This information is missing too.