Fun bonus: they can be interchangeable, e.g. increasing space to reduce time.
This is the basis for rainbow tables: precomputed tables for mapping hashes to passwords, with a space-saving “hash chaining” trick to effect a constant factor reduction in table size. Such tables are the reason why passwords must be hashed with a unique salt when stored in a database.
For example, an in-place algorithm like Bubble Sort would have a O(1) space complexity, because it requires no extra memory (and 0 memory is a constant). Merge sort on the other hand is O(n) because it always uses additional memory for its intermediate stages, and that additional memory scales with n.
Doing a quick google, the first few sites I find seem to use a similar understanding https://www.geeksforgeeks.org/time-and-space-complexity-anal...
space complexity is O(n) but auxiliary space complexity uses Theta for notation instead.
But people aren't too picky on the notation and usually say something like "O(1) extra space" instead of using theta.
Saying something is O(n) tells you it grows at most linearly, but this would also admit e.g. log n.
Saying something is Theta(n) tells you it grows exactly linearly: that is, it is not a slower growth rate like log n, nor a faster growth rate like n^2.
But yeah I guess space complexity vs auxiliary space complexity is just a bit ambigous.
Heavily simplified due to caches etc. To the point where people sometimes measure in cache misses instead as that is usually what actually matters.
Now move the compile time code to runtime.
I call it O(n) in time and memory. What do you call it?
If your say O(n) in time and memory - why is moving the code changing its complexity?
If you say still O(1) - then everything is O(1), thus proving P=NP among other things :)
Either way your interpretation isn't very useful.
And if your implementation of an algorithm allocates more space in the big-oh sense than it can actually touch (eg. O(n) space for O(log n) time or whatever), that's just a wasteful implementation. Doesn't make the algorithm itself require more space than it has time to actually use.