17.5 Equality, Ordering, and Hashing

17.5.1 Converting Values to Ordered Values

In Making Sets Grow on Trees, we noted that a single comparison needs to eliminate an entire set of values. With numbers, we were able to accomplish that easily: every bigger or smaller number was excluded by a comparison. But what if the data in the set are not actually numbers? Then we have to convert an arbitrary datum into a datatype that permits such comparison. This is known as hashing.

A hash function consumes an arbitrary value and produces a comparable representation of it (its hash)—most commonly (but not strictly necessarily), a number. A hash function must naturally be deterministic: a fixed value should always yield the same hash (otherwise, we might conclude that an element in the set is not actually in it, etc.). Particular uses may need additional properties, as we discuss in Equality and Ordering.

Let us now consider how one can compute hashes. If the input datatype is a number, it can serve as its own hash. Comparison simply uses numeric comparison (e.g., <). Then, transitivity of < ensures that if an element \(A\) is less than another element \(B\), then \(A\) is also less than all the other elements bigger than \(B\).

Suppose instead the input is a string. We can of course use the principle above for strings: e.g., replacing number inequality with string inequality. Strings have a lexicographic (or “alphabetic”) ordering that permit them to be treated similar to numbers.

But what if we are handed more complex datatypes?

fun hash-of(s :: String):
  fold({(a :: Number, b :: Number): a + b},
    0,
    string-to-code-points(s))
end

check:
  hash-of("Hello") is 500
  hash-of("World!") is 553
  hash-of("🏴‍☠️") is 195692
end

Now let us consider more general datatypes. The principle of hashing will be similar. If we have a datatype with several variants, we can order the variants lexicographically, and use a numeric tag to represent the variants, and recursively encode the datum and the variant tag. For each field of a record, we need an ordering of the fields—the lexicographic ordering of the field names suffices—and must hash their contents recursively; having done so, we get in effect a string of numbers, which we have shown how to handle.

Exercise

Why do we care about these two properties? Think about what would could go wrong if each one was violated.

17.5.2 Hashing in Practice

In practice, programmers do not want hash functions to do what we have described above. While Gödel encoding is extremely expensive, even computing hash-of takes time linear in the size of a string, which can get quite expensive if strings are large or we compute hashes often or both.

Instead, many programming languages do something very pragmatic. They need a value that can be compared for equality and ordering [Equality and Ordering]. Integers, we’ve already seen, already fit this bill very nicely. But how to obtain an integer out of arbitrary values, even datatype instances, quickly?

Simple: They just use the memory address of the datum. Every value has a memory address, and the language can obtain it in constant time by looking up the directory. Granted, these values may be allocated anywhere with respect to each other, but that’s okay—we only want consistency, not “meaningfulness”.

In practice, however, things are not quite so simple. For instance, suppose we want two structurally equivalent values to have the same hash. If they are allocated in different addresses, they will hash differently. Therefore, many languages that use such a strategy also allow programmers to write their own hashing functions, often to work in conjunction with this built-in notion of hashing. These end up looking not too different from the hashing strategies we described above. Therefore, some of that complexity is inescapable, especially if a programmer wants structural rather than reference equality—which they very often do.

In the rest of this material, we will therefore continue with the simple hash function above, for multiple reasons. First, it is sufficient to illustrate how hashing works. Second, in practice, when built-in hashing does not suffice, we do write (more complex versions of) functions like the above. And finally, because it’s all laid bare, it’s easy for us to experiment with.

17.5.3 Equality and Ordering

What we’ve seen [A Fine Balance: Tree Surgery] for the construction of balanced binary search trees is that we need some way of putting elements in order. In the examples we used numbers because they’re a very friendly datatype: they have several properties that we take for granted. However, not all data have these properties.

The critical property that numbers have is that they are orderable. This follows because they are comparable, and the comparison is ternary: it produces three answers, “less than”, “equal to”, and “greater than”.

However, not all data have this property. What are data that might not have these properties? Actually, there are multiple possible properties here: Is something orderable? Is something even comparable?

So…life is complicated.

That means you could potentially misuse a BBST on the wrong kind of data. Ideally, we would want to know if we’re doing this. In Pyret’s type system we chose not to build this in, but in some languages, the type system actually lets you capture these properties.