# chimera

Lazy infinite compact streams with cache-friendly O(1) indexing
and applications for memoization.

Imagine having a function `f :: Word -> a`

,
which is expensive to evaluate. We would like to *memoize* it,
returning `g :: Word -> a`

, which does effectively the same,
but transparently caches results to speed up repetitive
re-evaluation.

There are plenty of memoizing libraries on Hackage, but they
usually fall into two categories:

Store cache as a flat array, enabling us
to obtain cached values in O(1) time, which is nice.
The drawback is that one must specify the size
of the array beforehand,
limiting an interval of inputs,
and actually allocate it at once.

Store cache as a lazy binary tree.
Thanks to laziness, one can freely use the full range of inputs.
The drawback is that obtaining values from a tree
takes logarithmic time and is unfriendly to CPU cache,
which kinda defeats the purpose.

This package intends to tackle both issues,
providing a data type `Chimera`

for
lazy infinite compact streams with cache-friendly O(1) indexing.

Additional features include:

- memoization of recursive functions and recurrent sequences,
- memoization of functions of several, possibly signed arguments,
- efficient memoization of boolean predicates.

## Example 1

Consider the following predicate:

```
isOdd :: Word -> Bool
isOdd n = if n == 0 then False else not (isOdd (n - 1))
```

Its computation is expensive, so we'd like to memoize it:

```
isOdd' :: Word -> Bool
isOdd' = memoize isOdd
```

This is fine to avoid re-evaluation for the same arguments.
But `isOdd`

does not use this cache internally, going all the way
of recursive calls to `n = 0`

. We can do better,
if we rewrite `isOdd`

as a `fix`

point of `isOddF`

:

```
isOddF :: (Word -> Bool) -> Word -> Bool
isOddF f n = if n == 0 then False else not (f (n - 1))
```

and invoke `memoizeFix`

to pass cache into recursive calls as well:

```
isOdd' :: Word -> Bool
isOdd' = memoizeFix isOddF
```

## Example 2

Define a predicate, which checks whether its argument is
a prime number, using trial division.

```
isPrime :: Word -> Bool
isPrime n = n > 1 && and [ n `rem` d /= 0 | d <- [2 .. floor (sqrt (fromIntegral n))], isPrime d]
```

This is certainly an expensive recursive computation and we would like
to speed up its evaluation by wrappping into a caching layer.
Convert the predicate to an unfixed form such that `isPrime = fix isPrimeF`

:

```
isPrimeF :: (Word -> Bool) -> Word -> Bool
isPrimeF f n = n > 1 && and [ n `rem` d /= 0 | d <- [2 .. floor (sqrt (fromIntegral n))], f d]
```

Now create its memoized version for rapid evaluation:

```
isPrime' :: Word -> Bool
isPrime' = memoizeFix isPrimeF
```

## Magic and its exposure

Internally `Chimera`

is represented as a *boxed* vector
of growing (possibly, *unboxed*) vectors `v a`

:

```
newtype Chimera v a = Chimera (Data.Vector.Vector (v a))
```

Assuming 64-bit architecture, the outer vector consists of 65 inner vectors
of sizes 1, 1, 2, 2<sup>2</sup>, ..., 2<sup>63</sup>. Since the outer vector
is boxed, inner vectors are allocated on-demand only: quite fortunately,
there is no need to allocate all 2<sup>64</sup> elements at once.

To access an element by its index it is enough to find out to which inner
vector it belongs, which, thanks to the doubling pattern of sizes,
can be done instantly by `ffs`

instruction. The caveat here is
that accessing an inner vector first time will cause its allocation,
taking O(n) time. So to restore *amortized* O(1) time we must assume
a dense access. `Chimera`

is no good for sparse access
over a thin set of indices.

One can argue that this structure is not infinite,
because it cannot handle more than 2<sup>64</sup> elements.
I believe that it is *infinite enough* and no one would be able to exhaust
its finiteness any time soon. Strictly speaking, to cope with indices out of
`Word`

range and `memoize`

Ackermann function,
one could use more layers of indirection, raising access time
to O(log<sup>*</sup> n).
I still think that it is morally correct to claim O(1) access,
because all asymptotic estimates of data structures
are usually made under an assumption that they contain
less than `maxBound :: Word`

elements
(otherwise you can not even treat pointers as a fixed-size data).