See for example, the quadratic behavior in this dotted path example HELP-47436. The situation may have improved for $project, but has not improved for $match.

SERVER-81868 Here's another example.

Note - this task is somewhat open-ended, and may need to be split into multiple tasks. Whoever takes this one, please feel free to split it up (:

Result - we can run this scenario in the testing harness against the placeholder algorithm.