However the nasty unbalanced hash trees only occur if you have not used a good enough hashing function; the implementation, above some table size, could arrange to use a stronger cryptographic function (e.g. SHA512, but still SHA1 is still very resistant) which will ensure a really flattened distribution, instead of a simple modular arithmetic function with the multiplication by known static prime (like the simple Knut's hashing function).
SHA-512 is not very fast when hashing small values. SHA-512 pads the value to at least 1024 bits, and then evaluate an 80 round function to complete the hash, which takes over 1000 cycles on Intel Skylake for instance. Hashing random integers, floats and short strings will be very slow because you always have to hash at least 128 bytes, which is mostly padding and a length field.
Some newer Intel and AMD processors have specialized instructions to speed up SHA-256, but even then you have to process at least 512bits over 64 rounds.
Almost-universal hash families are a better candidate for this kind of thing if your threat model includes hash collision DOS attacks. You do not need most of the properties of hash functions like SHA-512 to build a DOS attack resistant hashmap.
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