*View this file with results and syntax highlighting [here](https://mlochbaum.github.io/BQN/implementation/primitive/search.html).*

# Implementation of search functions

This page covers the [search functions](../../doc/search.md), dyadic `⊐⊒∊`, and [self-search functions](../../doc/selfcmp.md), monadic `⊐⊒∊⍷`. Generally speaking, hash tables or plain lookup tables are the fastest way to implement these functions, because they transform the problem of searching into random access, which is something computers specialize in. In some edge cases, and when the search becomes large enough that caches can't speed up random access, other methods can be relevant.

Searching is closely related to [sorting](sort.md). I've found it helpful to think of a search function as an ordered lookup according to an unspecified order. The advantage over a predefined order is that the order can be shuffled with a hash function to randomize the distribution. The fact that the wanted result is in the original argument order, not sorted, also means that sparse tables are more effective—there is never any need to traverse the table to pack ordered values as in sorting.

## Lookup tables

For the purposes of these notes, a lookup table is storage, indexed by some key, that contains at most one entry per key. This means reading the value for a given key is a simple load—differing from a hash table, which might have collisions where multiple keys indicate the same entry. Lookup table operations are very fast, but the entire table needs to be initialized and stay in cache. So they're useful when the number of possible values (that is, size of the table) is small: a 1-byte or 2-byte type, or small-range integers.

For example, a lookup table algorithm for dyadic `⊐` might traverse `𝕨`, writing each value's index to the table. Doing this step in reverse index order makes sure the lowest index "wins". Similarly, empty entries must be initialized to `≠𝕨` beforehand. Then the result is `𝕩⊏t` where `t` is the table constructed this way. A nonzero minimum value can be handled for free by subtracting it from the table pointer.

Set operations can be handled with a packed bit table, but reading a bit is slower so this should be done only if the space savings are really needed.

## Hash tables

A hash table is a more sophisticated design where there are more possible keys than table entries. For good performance it depends on not having too many *actual* keys packed into a small space, which is why this method is named after the hash function. If the data is expected to be random then no hash function is needed (the identity function can be used), but I don't think that happens much with searching. Hash tables generally degrade to the performance of a linear lookup if the hash is defeated, so it's ideal to have a way to escape and use a sorting-based method if too many hashes collide.

Hashing is really the only way to get a performant lookup on arbitrary data. For 2-byte and small-range data, lookups are better, and in several 4-byte cases, lookup with [partitioning](#partitioning) is competitive for smaller arrays and much better for large ones as the hash table outgrows the cache (>1e5 elements to be hashed or so).

While hash tables are well studied, almost all the work is focused on large persistent tables, meaning that they're not too suited for a one-shot search function. Abseil's [flat\_hash\_map](https://github.com/abseil/abseil-cpp/blob/master/absl/container/flat_hash_map.h) is fine, I guess. Roger Hui's [Index-Of, a 30-Year Quest](https://www.jsoftware.com/papers/indexof/indexof.htm) works as an introduction to hashing in APL, although it has begun to suffer from the small number of years in places, and some details have serious issues (with a power-of-two table size, multiplying by a prime causes high bits to be lost and so is hardly better than no hash). The second half of my "Sub-nanosecond Searches" talk ([video](https://dyalog.tv/Dyalog18/?v=paxIkKBzqBU), [slides](https://www.dyalog.com/user-meetings/uploads/conference/dyalog18/presentations/D08_Searches_Using_Vector_Instructions.zip)) covers a difficult 4-byte design that's very good for membership and negative lookups (in particular, it's perfect for the reverse lookup as described in the next section).

I'd take the following choices as a given for an array language hash design:
- Power-of-two size
- Open addressing
- Linear probing

The main cost for larger data is the hashing itself; [wyhash](https://github.com/wangyi-fudan/wyhash) appears to be one of the best choices at the time of writing. 4- and 8-byte lookups are where all the fancy optimizations are wanted. Hashes on these fixed sizes should be reversible and are often called mixing functions in the literature. A CRC instruction makes a good one if available.

## Reverse lookups

The classic pattern for searching is to build an index of the data to be searched, then use it for each searched-for value. This is an optimization though: the obvious way is to search for one value at a time. What I call a reverse lookup returns to this method in a sense, and is useful if the searched-in array is larger than searched-for by a factor of 2 or so.

The method is to build an index of all searched-for values, then iterate over searched-in values one at a time. For each one, check if it matches any searched-for values, update results for those accordingly, and remove that value from the index. Since each value only has one first match, the total number of removals is at most the number of searched-for values. The traversal can stop early if all these values are found, and it could also switch to a faster index if most of them are found, although I haven't tried this.

When the searched-in array is much larger, performance tends to the speed of a *set* lookup on that array, which can be several times faster than an index lookup for smaller types. The overhead for the searched-in values is usually higher than normal hash table insertion.

## Partitioning

[Robin Hood sort](https://github.com/mlochbaum/rhsort) sorts small uniform arrays quickly by considering hash tables as a way of sorting hashes. This cuts both ways: RH sort [slows down](https://github.com/mlochbaum/rhsort/blob/master/images/rand.svg) far more than other sorting methods on large arrays because of its random access patterns, and so do hash table operations. For large enough hash tables, it ought to make sense to bring in sorting-based methods in order to reduce the search size.

Of course, it's possible to implement searches using only sorting and no hashing: `∧⊸⍋⊏⍋∘⊣` with some adjustments to the binary search. A hash takes advantage of the fact that what ordering is used doesn't matter to rearrange things and get expected equal distribution of unique keys. It's usually going to be best to use a hash table as the base case, so that it's the hashes being sorted. With small element sizes and a bijective hash, only the hashes need to be compared, so the arguments can be hashed at the start and the original values discarded.

One option is [partitioning](sort.md#partitioning) as in quicksort, but the unstable version doesn't work for many functions, and comparing with a pivot is wasted effort as top bits can be used directly. Stable [radix](sort.md#radix-sort) passes are ideal here. However, they do use a lot of extra memory: twice the size of the hashed array (or equal to it if it can be reused). To undo a radix sort in a cache-friendly way, the original hash bits need to be kept around to retrace the steps. Even in cases where the original indices are known, traversing the radix-ed values and writing back to these indices makes many loops around the original array, moving too quickly for the cache to keep up.

Partitioning doesn't really have to interact with hash insertions or lookups: after the data is partitioned at a suitable size, it will just hash faster because it only accesses part of the hash table at a time. It's also possible to save space by using a smaller hash table and doing one partition with it, then the next, and so on.

There are a few tricks to avoid having to re-initialize the table on each pass. This is a big deal because it means a very low load factor can be used, which allows for faster hash or even a lookup table! First, and more generally, the table can be cleaned up *after* a pass by walking through the elements again (possibly in reverse for some hash designs). Second, if enough data is stored in the hash table to distinguish one pass from the next, then values from previous passes can be interpreted as empty so that no re-initialization is necessary.