```Goal:
Efficient (time and space) quantiles over sliding, time-bounded windows.

Terms:
t: abstract quantity of time units over which measurements are drawn
N: maximum number of events in time window
V: the number of distinct observable values
we take this to be the integers [0,V)
(support is planned for dynamic ranges with fixed relative error)
q: number of arbitrary quantiles to be measured

N_b: number of bytes in word required to represent a value in [0,N)
V_b: number of bytes in word required to represent a value in [0,V)
take log_256(x), round up to nearest word size: 1, 2, 4, 8

w: word size of machine in bits

Requirements:
monotonic time

The algorithms here are presented in a way that depends on V for time. The
goal is O(1), but the best hypothetical we currently have is O(loglogV).

Let's briefly layout the algorithm. We will use the following window of values
for example purposes:
{ 5, 6, 5, 1, 1, 8, 9 }
sorted this is:
[ 1, 1, 5, 5, 6, 8, 9 ]

We always store at least one copy of each event along with time it occurred.
We do this in a circular buffer of length N. Events are ordered by time due to
the monotonic time requirement. This means that we always know where the next
event will go -- at the head of the list -- and where the next element may be
popped from -- the tail of the list. This is trivial to maintain in O(1) per
event.

The second structure we maintain is an associative array from values in [0,V)
to their frequency in [0,N). This is trivial to maintain in O(1) per event as
an array of values (full array, not sparse). In our example it would be:
0: 0
1: 2
2: 0
...
5: 2
6: 1
...
8: 1
9: 1
...

The bulk of the interesting work is in the maintenance of the quantiles
themselves. For each quantile, we store which quantile (as a double in [0,1])
along with the current numeric value. In addition, we store two values which
reference an imaginary flattened, sorted version of the full window of values.
We store the absolute index into this flattened version, along with the offset
into the run of values it is.

From our example, the median quantile would have the values:
index: 3
value: 5
offset: 1 (it points to the first 5)

When a new value is inserted, we may have to adjust the pseudo-index of the
current value the quantile is pointed to. If the new value is less than or
equal to the value of the quantile, the pseudo index has effectively increased
by one and we track it accordingly. Otherwise, the pseudo-index is unaffected.

We then possibly adjust the pseudo-index to match the target index of the
quantile for the size array we now have. We can move up or down in a run of
the same value in constant time with a simple decrement or increment. For
instance, if a 0 value was added to the example, the median would actually be
the first 5 and we'd only need to decrement the offset.

In the event we are in the run of the wrong number, we can detect this event
when frequency < 0 or > the count of the value, we need to search for the next
lowest or highest value in the window and set the quantiles value to that.
Naively, we must seach the entire value space of V, for O(V). We can do this
with a very low constant factor. Using Van Emde Boas trees, we can perform
this operation in O(loglogV).

With naive find prev/next:
space: O(V*N_b + N*V_b + q + 1)
time: O(Vq) per event
we speculate this to be closer to O(1) under realistic scenarios

This may require a linear scan through the universe V to find next/prev per
quantile tracked per event.

With fast find prev/next:
space: O(V*N_b + N*V_b + q + 1 + V/8)
time: O(Vq) per event, low constant => O(Vq/w)
we speculate this to be closer to O(1) under realistic scenarios

This improves the speed by maintaining an auxillary bitset marking numbers
in the range [0,V). We can skip entire any empty block the size of w with a
single comparison against 0 in this case. Inside a w-block, we can use a
single instruction like bit scan reverse if it is available. We anticipate
the constant to be on the order of 1/w potentially making this very fast.

O(Vq/w) = O(Vq), however.

With Van Emde Boas tree for finding prev/next:
space: TODO
time: O(q * loglogV) per event

This is a direct extension to maintain a Van Emde Boas tree for its fast
find previous/next functions. For small universes of values, the constants
are likely to swamp out the "fast" naive method.

```