This is an interesting read. You can solve the same problem with Range Min Query Tree. The query for balanced or unbalanced parentheses is O(log N). A two or three level RMQTree can represent billions of parentheses already (a two-level tree of 65536 branch factor = 4B parentheses). The query is O(65536 + 65536) or effectively O(1). For a four-level tree of 256 branch factor, the query is O(256 + 256 + 256 + 256) or…
I've heard that phrase couple of times, and cannot stop noting that then every real-world algorithm is "effectively O(1)", because in real world we have bounded inputs and RAM. E.g. every integer is O(2^64) = O(1).
If we need to say that something is really fast, let's just say that. E.g if a CPU needs to iterate something 65536 + 65536 times, we can just say that it would take about 0.1ms on 1GHz CPU, no need to involve asymptotic.
E.g. Scala boasts that their Vector implementation is "effectively constant", while in doing up to 5 non-consecutive RAM accesses, that screws up the CPU cache. But if we can bound something to "no more than 5 operations", then I can say that any array in 32-bit arch is "no more than 2^32 operations", which is equal to O(1).