Question

I want to have a dictionary that assigns a value to a set of integers.

For example key is [1 2 3] and value will have certain value.

The thing is that [3 2 1] needs to be treated the same in my case so hash needs to be equal, if I go with hash approach.

The set will have 2 to 10 items.

Sum of items is usually fixed so we cannot make hashcode according to sum, which is a first natural idea here.

Not a homework task, actually facing this problem in my code.

This set is basically IEnumerable<int> in C# so any data structure is fine to store them.

Any help appreciated. Performance is pretty important here too.

An immediate thought: we could sum up items^2 and already get some kind of better hash, but still I would like to hear some thoughts.

EDIT: hmm really sorry guys, everyone suggests ordering, didn’t come to my mind that I needed to say that actually ordering and hashing is the current solution I use and I am considering faster alternatives.

Answer 1

Basically all of the approaches here are instantiations of the same template. Map x₁, …, x_n to f(x₁) op … op f(x_n), where op is a commutative associative operation on some set X, and f is a map from items to X. This template has been used a couple of times in ways that are provably good.

• Choose a random large prime p and a random residue b in [1, p – 1]. Let f(x) = b^x mod p and let op be addition. We essentially interpret a set as a polynomial and use the Schwartz–Zippel lemma to bound the probability of a collision (= the probability that a nonzero polynomial has b as a root mod p).

• Let op be XOR and let f be a randomly chosen table. This is Zobrist hashing and minimizes in expectation the number of collisions by straightforward linear-algebraic arguments.

Modular exponentiation is slow, so don’t use it. As for Zobrist hashing, with 3 million items, the table f probably won’t fit into L2, though it does set an upper bound of one main-memory access.

I would instead take Zobrist hashing as a departure point and look for a cheap function f that behaves like a random function. This is essentially the job description of a non-cryptographic pseudorandom generator – I would try computing f by seeding a fast PRG with x and generating one value.

EDIT: given that the sets all have the same sums, don’t choose f to be a degree 1 polynomial (e.g., the step function of a linear congruential generator).