# Fast Indirect Sorting in Java

### Fast indirect sorting in Java

I was recently writing some performance-sensitive code in which I had a `double`
array of distances (one per element), and I wanted to get a list of elements
sorted by distance:

Java provides `Arrays.sort` for direct sorting; that is, it’s easy to ask it to
sort `distances` or `elements` by its natural ordering. But in this situation,
the two arrays are tied together only by indexes, which would require a
comparator to maintain a reverse lookup from `Element` to either its index, or to
its distance. That’s a lot of overhead – particularly because the map would
require generic boxing of either type of value.

Luckily there’s an interesting way to solve this problem that meets the
following requirements:

1. No extra memory is allocated, aside from the two arrays above, each of which is allowed to be clobbered
2. We do not write any sorting code; we just use standard APIs
3. We end up being able to access the distance-sorted `Element`s each in constant time

I should also mention that the resulting ordering isn’t exact, but it is very
close.

Before I go into how I solved it (which took a while to think of), you should
see what you come up with. It’s a fun problem.

#### The solution

First in code:

Now `distances[]` is sorted such that the distance of
`elements[(int) (Double.doubleToLongBits(distances[i]) & ~mask)]` is ascending
for ascending values of `i`.

#### How it works

Each `distance` is encoded as a double-precision float, which internally looks
like this:

``````  +--- sign
|
| +- exponent
| |
S EEEEEEEEE MMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM
|---------------------52 mantissa bits-------------------|
``````

The mantissa for all but the smallest numbers is normal, meaning that it’s
interpreted as though there were a leading `1` in a 53rd bit. This puts an upper
bound on the significance of low-order mantissa bits, which is what we need for
the code above to work.

Depending on how the distances are distributed, we can make a probabilistic
argument about how much the ordering will change as we lose precision in the
mantissa; specifically, suppose we’ve got two distances `a` and `b` and we lose
24 bits:

``````A = S EEEEEEEEE MMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM
B = S EEEEEEEEE MMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM
|-----------------------------| |------------------------|
keeping these bits               losing these
``````

The probability of changing the ordering between these two points is the same as
the probability that the bits we’re keeping are all identical between them.
Benford’s law converges rapidly
to a uniform distribution for subsequent digits and the leading `1` is implied,
so in practical terms P(reordering) is very nearly 2-k, where k is
the number of bits being kept.

If we can lose some precision without causing problems (which for my use case
was true), then we can arbitrarily reassign low-order mantissa bits to store
information. In this case I’m storing the original array index for each distance
in its low-order bits. Here’s the code above, piece by piece:

Then we tag each distance this way (here I’m assuming we’ve got between
219 and 220 elements, so we reserve 20 bits):

``````  d[i]  S EEEEEEEEE MMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMMMMM

|-----------------original bits----------------||-----tag space------|
& mask  1 111111111 1111 11111111 11111111 11111111 11110000 00000000 00000000
| i     0 000000000 0000 00000000 00000000 00000000 0000IIII IIIIIIII IIIIIIII

=       S EEEEEEEEE MMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMIIII IIIIIIII IIIIIIII
``````

At this point `Arrays.sort()` will be none the wiser and will sort the array
normally (and, importantly, very quickly).

Now we can read the tags back to recover the ordering, which looks like this:

``````  mask  1 111111111 1111 11111111 11111111 11111111 11110000 00000000 00000000
~mask  0 000000000 0000 00000000 00000000 00000000 00001111 11111111 11111111

d[i]  S EEEEEEEEE MMMM MMMMMMMM MMMMMMMM MMMMMMMM MMMMIIII IIIIIIII IIIIIIII
& ~mask 0 000000000 0000 00000000 00000000 00000000 0000IIII IIIIIIII IIIIIIII
(int)                                      00000000 0000IIII IIIIIIII IIIIIIII
``````

#### Other languages

You can use this hack in any language to similar effect, though you lose most
`double`s. Even a well-optimized sorting function that doesn’t have the Java
indirection problem will benefit if you can store the data in the floats
directly, since the array will be smaller in memory and you’re doing O(n log n)
element-copy operations.

It is possible to simulate bitwise access using floating point arithmetic and
int casting (which flotsam does in
Javascript
),
but it requires some care – particularly in this case, when any rounding error
will cause data loss within the indexes. It’s also a lot slower than the
bitwise solution above, possibly enough to outweigh any performance benefits in
the sorting logic itself.

#### When this kind of thing doesn’t work

Distances are ideal for this type of hack because they tend to be spread over a
wide range of magnitudes, and even when they aren’t, you don’t tend to care much
whether the points are exactly ordered (i.e. the parts-per-trillion error we’re
introducing doesn’t really pose a problem most of the time). Not all
distributions are so robust to bit-twiddling, though. In particular, if you had
a case where the variance were many orders of magnitude smaller than the
average – e.g. 1,000,000,000 ±0.003 – then there’s a good chance the small
bits would matter. It’s important to figure out the probability of a false
reordering before losing bits of precision.

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