- type is Tiered or Leveled and explains compaction into that level
- fanout is the size of a sorted run in Lk relative to a sorted run from Lk-1, a real and >= 1
- runs is the number of sorted runs in that level, an integer and >= 1
Given the above how many valid configurations exist for an LSM tree? There are additional constraints that can be imposed on the 3-tuple but I will ignore most of them except for limiting fanout and runs to be <= 20. The answer is easy - there are an infinite number of configurations because fanout is a real.
The question is more interesting when fanout is limited to an integer and the number of levels is limited to between 1 and 10. I am doing this to explain the size of the search space but I don't think that fanout should be limited to an integer.
There are approximately 2^11 configurations only considering compaction type, which has 2 values, and 1 to 10 levels because there are 2^N configurations of compaction types for a tree with N levels and the sum of 2^1 + 2^2 + ... + 2^9 + 2^10 = 2^11 - 1
But when type, fanout and runs are considered then there are 2 x 20 x 20 = 800 choices per level and 800^N combinations for an LSM tree with N levels. Considering LSM trees with 1 to 10 levels then the number of valid configurations is the sum 800^1 + 800^2 + ... + 800^9 + 800^10. That is a large number of configurations if exhaustive search were to be used to find the best configuration. Note that I don't think exhaustive search should be used.