```
vector<int> xs(n);
vector<int> order2ix(n);
for(int i = 0; i < n; ++i) { order2ix[i] = i; }
sort(order2ix.begin(), order2ix.end(),
[](int i, int j) {
return make_pair(xs[i], i) < make_pair(xs[j], j);
};
```

where `order2ix[o]`

gives us the element with that order statistic. So,
`order2ix[0]`

contains the smallest element, `order2ix[n-1]`

contains the
largest element, etc. I've previously tried the naming conventions:
`rank: ix -> order`

, `select: order -> ix`

. The pro of this is that it
uses the rank/select naming convention.
This leads into the Galois connection aspect of it, but is otherwise not so
useful.
`order2ix: order -> ix`

, `ix2order: ix -> order`

. This uses
the order statistic naming
convention, and thereby makes it clear what the query is: you tell me the
kth order, I give you the index in the array in `order2ix(k)`

. Alternatively,
you tell me the index `i`

, and I'll tell you its order statistic in `ix2order(k).`

However, I found implementing this kind of odd. In particular, I need to
pause for a second and think about what maps to what in the `ix2order[order2ix[o]] = o;`

.
```
vector<int> xs(n);
vector<int> order2ix(n);
for(int i = 0; i < n; ++i) { order2ix[i] = i; }
sort(order2ix.begin(), order2ix.end(),
[](int i, int j) {
return make_pair(xs[i], i) < make_pair(xs[j], j);
};
// REVERSE BIJECTION
vector<int> ix2order(n);
for(int o = 0; o < n; ++o) { ix2order[order2ix[o]] = i; }
```

For me, the major source of disconnect is that this "order" feels somewhat
disconnected from the original array `xs`

. So I feel like I'm trying to reason
about these three
- The indexes
`0, 1,..,n`

- The orders
`0th,1st,2nd,..(n-1)th`

- The array values
`xs[0],xs[1],xs[2],..xs[n]`

`order2ix`

and `ix2order`

relate to each other.
`compressed`

/`decompressed`

:`compressed`

/`decompressed`

:
The idea is that `compressed`

maps the original numbers to their compressed variants.
So it's going to have a signature `compressed: ix -> smalluniv`

, where it
compresses the element `xs[i]`

into `[0..n-1]`

. `decompressed`

is the inverse
function, which takes a number in the smaller universe, and returns its
index in the original array. So we have `decompressed: smalluniv -> ix`

.
`compressed`

/`decompressed`

do and why they should be inverses. I feel
it also matches `[0..(n-1)]`

.
Furthermore, the very `compressed`

is the compressed version of `xs`

)
and that `decompressed`

is the decompressed version of `[0..(n-1)]`

). This makes it clear how they're reated
to the original array linguistically, which I quite like.