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Mongodb supports many useful array operations such as $push and $pop but I can’t seem to find any information about their algorithmic complexity nor how they are implemented to figure out their runtime complexity. Any help would be greatly appreciated.
I think when it comes to Mongo updates, there are only three relevant cases:
1) an in-place atomic update. For example just increment an integer. This is very fast.
2) an in-place replace. The whole document has to be rewritten, but it still fits into the current space (it shrank or there is enough padding).
3) a document migration. You have to write the document to a new location.
In addition to that there is the cost of updating affected indexes (all, if the whole thing had to be moved).
What you actually do inside of the document (push around an array, add a field) should not have any significant impact on the total cost of the operation, which seem to depend mostly linearly on the size of the document (network and disk transfer costs).