Doesn't the number of buckets grow with the number of elements in the hash? If so, why would it take longer as more elements are added? The number of elements in a bucket should remain less than some bound, no? O(bound) = O(1). I don't know how hashes are implemented specifically in Perl, but I heard they were O(1) for lookup. (Again, not worse case.) Wikipedia
says the same for hashes in general:
Hash tables are often used to implement associative arrays, sets and caches. Like arrays, hash tables provide constant-time O(1) lookup on average, regardless of the number of items in the table. (O(1) means that it takes a constant amount of time independent of the number of items involved.)
From where did you get O(log N)?
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