Performance at Scale
====================

:doc:`Sorted Containers<index>` scales extremely well. This page discusses why
both in theory and in practice. The design of :doc:`Sorted Containers<index>`
permits testing through tens of billions of items and benchmarks present
results at this scale. The results of this page would be difficult and
expensive to replicate with other implementations.

Two methods are most important for analysis. Each stresses editing while
maintaining sorted order. The first is :func:`SortedList.add` which identifies
the insertion position using binary search. The second is
:func:`SortedList.__delitem__` which identifies the deletion position using
indexing. Most other methods are either symmetric or a subset of the runtime
complexity of these methods.

To fully grasp the content below, its strongly recommended that you read the
:doc:`implementation details<implementation>`
first. :doc:`Sorted Containers<index>` maintains a B-tree-like data structure
limited to two levels of nodes.

Theory
------

To discuss the runtime complexity of :doc:`Sorted Containers<index>`, we must
consider two values:

1. `n` -- the size of the sorted list.
2. `m` -- the "load" or sublist length.

In a :doc:`sortedlist` with `n` elements, there will be a top-level list of
pointers to :math:`\frac{n}{m}` sublists each of which are approximately `m`
elements long. There will also be a "maxes" index which is the max of each
sublist. The index will contain :math:`\frac{n}{m}` references.

Now adding an element using :func:`SortedList.add` requires these steps:

1. Bisect the "maxes" index to find the appropriate sublist --
   :math:`O(\log_2{\frac{n}{m}})`

2. Bisect the sublist to find the appropriate insertion position --
   :math:`O(\log_2{m})`

3. Insert the element in the sublist -- :math:`O(m)`

4. A sublist may be split and inserted into the top-level list --
   :math:`O(m+\frac{n}{m})`

Consider how often step #4 may occur if we begin with a :doc:`sortedlist` of
`n` elements and add `n` more elements. The top-level list length will grow
from :math:`\frac{n}{m}` to :math:`2*\frac{n}{m}` meaning that step #4 was
executed :math:`\frac{n}{m}` times.

Altogether, growing a :doc:`sortedlist` of `n` elements by `n` more elements
has the time complexity:

.. math::

   T_{add}(n, m) = O(n * \log_2{\frac{n}{m}} + n * \log_2{m} + n * m +
   \frac{n}{m} * (m + \frac{n}{m}))

Now consider an extreme case:

.. math::

   T_{add}(n|m \propto 1) = O(n * \log_2{\frac{n}{1}} + n * \log_2{1} + n * 1 +
   \frac{n}{1} * (1 + \frac{n}{1}))

   T_{add}(n|m \propto 1) = O(n * \log_2{n} + n + n + n * n)

   T_{add}(n|m \propto 1) = O(n^2)

The above is why some consider :doc:`Sorted Containers<index>`
unscalable. Because the load is not chosen dynamically, in theoretical
analysis, it must be treated as a constant with respect to `n` which makes it
proportional to 1.

In practice, the default load is 1,000 which is generally closer to the square
root or cube root of `n`. Consider where `m` is proportional to the square root
of `n`:

.. math::

   T_{add}(n|m \propto n^\frac{1}{2}) = O(n * \log_2{\frac{n}{n^\frac{1}{2}}} +
   n * \log_2{n^\frac{1}{2}} + n * n^\frac{1}{2} + \frac{n}{n^\frac{1}{2}} *
   (n^\frac{1}{2} + \frac{n}{n^\frac{1}{2}}))

   T_{add}(n|m \propto n^\frac{1}{2}) = O(n * \log_2{n^\frac{1}{2}} + n *
   \log_2{n^\frac{1}{2}} + n*n^\frac{1}{2} + n^\frac{1}{2}*n^\frac{1}{2})

   T_{add}(n|m \propto n^\frac{1}{2}) = O(n * n^\frac{1}{2})

The amortized cost of adding an individual item is therefore proportional to
the square root of `n`.

Our best bounds will be to use the cube root of `n`:

.. math::

   T_{add}(n|m \propto n^\frac{1}{3}) = O(n * \log_2{\frac{n}{n^\frac{1}{3}}} +
   n * \log_2{n^\frac{1}{3}} + n * n^\frac{1}{3} + \frac{n}{n^\frac{1}{3}} *
   (n^\frac{1}{3} + \frac{n}{n^\frac{1}{3}}))

   T_{add}(n|m \propto n^\frac{1}{3}) = O(n * \log_2{n^\frac{2}{3}} + n *
   \log_2{n^\frac{1}{3}} + n*n^\frac{1}{3} + n^\frac{2}{3}*n^\frac{1}{3})

   T_{add}(n|m \propto n^\frac{1}{3}) = O(n * n^\frac{1}{3})

Now the amortized cost of adding an individual item is proportional to the cube
root of `n`.

Alternative tree-based implementations have a runtime complexity proportional
to :math:`log_2{n}` for adding elements. The logarithm grows much more slowly
than the cube root for large values of `n`. However, in practice we never reach
those large values and the constant factors involved have a significant
impact. Consider a billion elements:

.. math::

   \log_2{1,000,000,000} \approx 33

   (1,000,000,000)^\frac{1}{3} \approx 1,000

The constant factor between those is :math:`1,000 / 33 \approx 33`. So if the
operations for tree-based implementations are more than 33 times slower, then
:doc:`Sorted Containers<index>` may be faster. Below I'll make an argument for
why that occurs in practice.

Now deleting an element using :func:`SortedList.__delitem__` requires these
steps:

1. Build the index if not present -- :math:`O(\frac{n}{m})`

2. Traverse the index to resolve the internal location --
   :math:`O(\log_2{\frac{n}{m}})`

3. Delete the element in the sublist -- :math:`O(m)`

4. Update the index -- :math:`O(\log_2{\frac{n}{m}})`

5. A sublist may be combined with a neighboring sublist if it becomes too
   small. When this happens, the index is deleted -- :math:`O(m+\frac{n}{m})`

Consider how often steps #1 and #5 may occur if we begin with a
:doc:`sortedlist` of `n` elements and delete all `n` elements. The top-level
list will shrink from :math:`\frac{n}{m}` to zero meaning that steps #1 and #5
were executed :math:`\frac{n}{m}` times.

Altogether, deleting `n` elements from a :doc:`sortedlist` of `n` elements has
the time complexity:

.. math::

   T_{del}(n, m) = O(\frac{n}{m} * \frac{n}{m} + n * \log_2{\frac{n}{m}} + n *
   m + n * \log_2{\frac{n}{m}} + \frac{n}{m} * (m + \frac{n}{m}))

Most terms are the same for adding and deleting elements. But the first term is
different. Rebuilding the index takes:

.. math::

   T_{index}(n, m) = O(\frac{n}{m} * \frac{n}{m})

Furthermore index lookups and updates are proportional to :math:`n *
\log_2{\frac{n}{m}}`. All these terms are minimized with :math:`n = m`. However
that maximizes the cost of step (3), :math:`O(n * m)`.

Once again our best bounds will be to use the cube root of `n`:

.. math::

   T_{del}(n|m \propto n^\frac{1}{3}) = O(\frac{n}{n^\frac{1}{3}} *
   \frac{n}{n^\frac{1}{3}} + n * \log_2{\frac{n}{n^\frac{1}{3}}} + n *
   n^\frac{1}{3} + n * \log_2{\frac{n}{n^\frac{1}{3}}} +
   \frac{n}{n^\frac{1}{3}} * (n^\frac{1}{3} + \frac{n}{n^\frac{1}{3}}))

   T_{del}(n|m \propto n^\frac{1}{3}) = O(n^\frac{2}{3} * n^\frac{2}{3} +
   n * \log_2{n^\frac{2}{3}} + n * n^\frac{1}{3} + n * \log_2{n^\frac{2}{3}} +
   n^\frac{2}{3} * (n^\frac{1}{3} + n^\frac{2}{3}))

   T_{del}(n|m \propto n^\frac{1}{3}) = O(n * n^\frac{1}{3})

When deleting elements by index, the amortized time complexity is proportional
to the cube root of `n`.

Although using :math:`m \propto \sqrt[3]{n}` is the best theoretical time
complexity, index lookups, updates, and building are composed of expensive
operations. In practice, the square root of `n` works better when doing a lot
of numerical indexing.

Python Implementations
----------------------

I've now said that some operations are more expensive than others while still
considering each to take :math:`O(1)` time. To understand this, we have to look
at how Python is implemented.

The most popular implementation of Python is `CPython`_. CPython implements
lists as arrays of pointers and integers as allocated memory objects. This
means that shifting elements in lists is very fast. It's akin to a mem-move
operation for which modern processors are well optimized. The memory access
pattern is entirely sequential.

In 64-bit builds of CPython, integers require approximately thirty bytes
each. This severely limits the number of integers we can hold in memory. In
2016, the largest commercial servers support up to terabytes of memory which
can hold only hundreds of billions of integers. While large in practice, the
number is small in theory. Doing integer math in CPython requires a memory
allocation which, while still :math:`O(1)`, is quite a bit more costly than a
processor-supported integer.

An optimized implementation of Python is `PyPy`_. PyPy improves on CPython in
many ways but one of the most important to our discussion is the use of "tagged
pointers." Tagged pointers are capable of storing integers within the pointer
itself. This greatly reduces memory consumption so that many integers in PyPy
take only eight bytes.

Lists of integers in PyPy are therefore packed densely together. When storing
integers in a sorted list, both the "maxes" index and positional index are
densely packed lists of integers. This improves locality for various processor
cache features.

The access pattern of both indexes is also optimized for modern
processors. Traversing both the "maxes" index and the sublist uses bisect which
while initially random, narrows locality with each iteration. Likewise the
positional index is a tree, densely stored in a list. The memory access pattern
locality is very good initially and then becomes random, the exact opposite of
bisect.

The benchmarks below use PyPy, without loss of generality, to maximize memory
utilization and performance.

.. _`CPython`: https://www.python.org/
.. _`PyPy`: http://pypy.org/

Sampling
--------

As described in the theory section above, some costs in sorted lists are
amortized over many operations. Amortized algorithms present unique
difficulties in measuring performance as, by design, expensive operations are
avoided.

For example, consider measuring the expected value of the lottery without
knowing the total jackpot. Purchasing a thousand tickets may still result in no
winnings which would conclude incorrectly an expected value of zero.

More practically, consider the list data type in CPython. Lists grow and shrink
as necessary but the underlying implementation is restricted to static
allocations. For this reason, lists are often over-allocated so that most
appends may occur immediately. Occassionally, the list must be reallocated and
possibly copied, which takes linear time. If we sampled performance by
initializing lists of various sizes and appending an element, we may never
observe a resize operation and so over-estimate performance.

One solution for both Python lists and :doc:`Sorted Containers<index>` sorted
lists would be to double the size or remove all elements from the initialized
list as was done in the Theory section above. Unfortunately, that method is too
expensive to be practical. Doing so would require weeks and months of time
incurring hundreds and thousands of dollars in machine costs.

To shorten the measured time, two techniques are used. The first constructs
sorted lists very quickly by initializing private member variables
directly. The latter uses sampling in representative scenarios to perform a
hundredth of the operations needed to double the size or remove all elements.

The problem solved here is similar to that faced by binary tree
implementations, like red-black trees, which do not maintain a binary tree in
perfect balance. In fact, the maximum height of a red-black tree is :math:`2 *
log_2{n + 1}`. While still :math:`O(log_2{n})`, the constant factor can have a
big impact on performance analysis. When using sampling to measure the
performance of red-black trees, trees of various shapes and heights must be
used. The same is done here with :doc:`Sorted Containers<index>`.

Consider a sorted list initialized from an iterable of random values. Those
values are sorted using the "sorted" builtin function and the resulting list is
chopped into sublists of the given load. The "maxes" index is simply the last
element of each sublist. If we plotted sublist length as a histogram, there
would be one tall bar at the load size. In this scenario, all sublists are the
same length which works very well in practice but is misleading for sampling
performance.

To more accurately measure performance, we must consider sublist lengths as
random values are added individually. The video below displays a histogram of
sublist lengths as random values are added to a sorted list. The load is one
thousand.

.. raw:: html

The histogram of sublist lengths is in blue while a normal curve fitted to the
histogram is plotted in green. The size of the sorted list grows to millions of
elements. Notice the fit of the normal curve improves with time. The sublist
lengths grow from one to two thousand elements, at which point sublists are
split and the process repeats. At the boundaries, bimodal distributions occur
which may be approximated as a normal distribution that wraps-around at the
limits.

Consider also a sorted list with a million values each of which is removed at
random. The video below displays a histogram of sublist lengths as values are
random deleted from the sorted list. The load is again one thousand.

.. raw:: html

The sorted list was initialized with a million values. Notice all sublists
begin with the same length represented by a spike in the histogram. As elements
are deleted, the spike moves toward five hundred. When sublists become too
small they are combined with neighboring sublists. Those neighboring sublists
may be any size between 500 and 2,000. This behavior results in new peaks at
1,000, 1,500, and 2,000. Each of those peaks then begins traversing left and
the process repeats. The overall effect is like watching ripples. Over time
each ripple starts as a sharp-looking normal curve and then flattens out.

In modeling each of the above cases, a normal curve is used to represent the
sublist lengths. When adding elements, the range of the curve is bounded by
:math:`load` and :math:`load * 2`. While deleting elements the curve is bounded
by :math:`\frac{load}{2}` and :math:`load`. The curve wraps-around these
limits. Normal distributions have two parameters: :math:`\mu` and
:math:`\sigma` which are the mean and standard deviation. Several means are
tested to improve sampling, each called a "moment." When adding elements, there
are ten moments evenly distributed in the range. And when deleting elements,
there are five moments evenly distributed in the range. The parameter,
:math:`\sigma`, is given as a tenth of the load, :math:`\frac{load}{10}`.

With this information about the distribution of sublist lengths, we can very
quickly construct large sorted lists. To do so, we sample lengths from a normal
distribution and construct the sublists from sequential integers up to the
desired size. The "maxes" index is simply given as the last element in each
sublist. Because sequential integers are used, sublists are already sorted.

After constructing a sorted list at each of the moments in the sublist length
range, operations are performed to total a hundredth of the total size. The
total time is the sum of the time at each moment. This process is repeated five
times and the median is selected from the measurements. The median is often
more accurate than the minimum due to cache effects. Details of the memory
cache hierarchy are described below.

Benchmarks
----------

Two benchmarks are measured. The first is adding random values to a sorted list
and the second is deleting random indices from a sorted list. When adding
values, the load is set to the cube root of the list size. And when deleting
random indices, the load is set to the square root of the list size.

Each table below displays: the method used, the initial size of the sorted
list, the number of operations performed, the sum of the times at each moment
as the total time, the operations completed per second, and the ratio of the
previous Ops/Sec to the current Ops/Sec.

In theory, using a load equal to the cube root of the list size should yield an
algorithmic time complexity of :math:`n * \sqrt[3]{n}`. With a bit of math, we
can calculate the expected Ops/Sec ratio as 2.154. Similarly, with a load equal
to the square root of the list size, the time complexity should be :math:`n *
\sqrt{n}`. In that scenario, the Ops/Sec ratio should be 3.162.

Tree-based sorted list implementations often advertise :math:`n * \log_2{n}`
time complexity for which, at extremely large sizes, the Ops/Sec ratio would
approach one. However, at the sizes discussed below, the ratio is closer to
1.136. This means that as we grow from one million to one billion elements, we
expect a net ratio of ~2. By comparison, the cubic root time complexity would
expect a net ratio of ~10. In practice, :doc:`Sorted Containers<index>` is
often five to ten times faster at smaller list sizes. So the total effect is
for performance to be equal at large list sizes. Also, tree-based
implementations have difficulty trying to realize the theoretical ratio and so
remain slower even at scale.

Local Results
.............

Measurements were made locally on a MacBook Pro (Retina, Late 2013) with 2.6
GHz Intel Core i7 processor and 16 GB of 1,600 MHz DDR3 memory. Sorted list
sizes ranged from one million to one billion elements. The benchmark required
approximately twelve gigabytes of memory.

====== ======= ============ ============== ============= =========
Method    Size   Operations           Time       Ops/Sec     Ratio
====== ======= ============ ============== ============= =========
   add   1e+06        1e+04        0.01501    666045.025       nan
   add   1e+07        1e+05        0.26612    375764.681     1.773
   add   1e+08        1e+06        4.69080    213183.298     1.763
   add   1e+09        1e+07       83.01831    120455.358     1.770
====== ======= ============ ============== ============= =========

The above table displays the performance of adding elements to a sorted
list. Notice the particularly good ratio, approximately 1.77, out-performed the
theoretically expected 2.154. This is mainly due to the different constant
times required for various operations, of which memory plays a large role and
is discussed below.

====== ======= ============ ============== ============= =========
Method    Size   Operations           Time       Ops/Sec     Ratio
====== ======= ============ ============== ============= =========
   del   1e+06        1e+04        0.00827   1208897.485       nan
   del   1e+07        1e+05        0.13309    751393.836     1.609
   del   1e+08        1e+06        3.79143    263752.866     2.849
   del   1e+09        1e+07      124.59184     80262.081     3.286
====== ======= ============ ============== ============= =========

When deleting elements, the ratio starts by out-performing the theoretically
expected 3.162 but increases with size. The limited processor caches at these
large sizes play a significant role in the performance. Traversing the
positional index will evict elements of the top-level list and sublists.

Virtual Machine Results
.......................

Virtual machine results were made on a Google Compute Engine, Haswell
generation, 2.3 GHZ Intel Xeon processor with 208 GB of memory. Sorted list
sizes ranged from one million to ten billion elements. The benchmark required
approximately 128 gigabytes of memory.

====== ======= ======= ============== ============= =========
Method    Size     Ops           Time       Ops/Sec     Ratio
====== ======= ======= ============== ============= =========
   add   1e+06   1e+04        0.02133    468884.826       nan
   add   1e+07   1e+05        0.38629    258872.924     1.811
   add   1e+08   1e+06        6.20695    161109.825     1.607
   add   1e+09   1e+07      120.24735     83161.919     1.937
   add   1e+10   1e+08     2416.60713     41380.330     2.010
====== ======= ======= ============== ============= =========

As with local results, the ratio out-performed the theoretically expected 2.154
at small list sizes. At very large sizes processor caches played more
significant roles and the ratio approached the theoretically expected value.

====== ======= ======= ============== ============= =========
Method    Size     Ops           Time       Ops/Sec     Ratio
====== ======= ======= ============== ============= =========
   del   1e+06   1e+04        0.01791    558289.343       nan
   del   1e+07   1e+05        0.26171    382097.449     1.461
   del   1e+08   1e+06        6.11150    163626.036     2.335
   del   1e+09   1e+07      171.58899     58278.798     2.808
   del   1e+10   1e+08     5493.95076     18201.838     3.202
====== ======= ======= ============== ============= =========

The virtual machine results are again similar to local measurements. At smaller
list sizes the ratio out-performs the expected 3.162 but increases at larger
sizes.

Total cost of the rented virtual machine was $33.97 for 1,011 minutes of
use. Anyone interested in funding further scaling to one hundred billion
elements should contact the :doc:`project lead<development>`.

Memory
------

Modern processors use multiple caches to improve memory performance. Caches are
organized into individual levels: L1, L2, and L3. Each successive level is
larger and slower than the previous level. For example, the size and average
latency for random memory accesses on Intel's latest Skylake i7-6700 processor:

* L1 Cache, 64 KB, ~4 cycles
* L2 Cache, 256 KB, ~12 cycles
* L3 Cache, 8 MB, ~42 cycles
* RAM, 16 GB, ~446 cycles

The exact size and latency are not important but the ratios are significant. L2
cache is about four times larger and slower than L1 cache. L3 cache is thirty
times larger and four times slower than L2 cache. And memory is two thousand
times larger and ten times slower than L3 cache. These ratios are approximate
but illustrative of the slowdowns.

Also important is the memory access pattern. These advertised latencies are
averages for random memory access. But there are two other patterns often seen
in practice: sequential and data-dependent.

Sequential memory access is faster than random due to its predictable
nature. The speedup varies but about five times faster is a reasonable
guess. Striding through memory sequentially will incur almost zero cycle
stalls.

Data-dependent memory access is slower than random because no parallelization
can occur. Each successive memory access depends on the previous and so stalls
the processor. The pattern is typical of dereferencing pointers. Again its
difficult to quantify the slowdown but five times slower is a reasonable
guess. Altogether, jumping around memory with data-dependent accesses could be
one thousand times slower than sequential accesses.

Given the size and latencies for memory in modern processors, consider the
typical cost of adding an element to a sorted list with size one billion. First
the "maxes" index will be bisected. The index will be one million integers
densely packed in a list. Using PyPy, the entire index could fit in the L3
cache. As the list is bisected, nearby indexes will be pulled into the L2 and
L1 cache and lookups will accelerate a hundred times. Once the sublist is
found, it too will be bisected. The sublist will contain only one thousand
integers and those too will be quickly pulled from memory into L3, L2, and L1
caches. Once bisected, the new value will be inserted and memory will be
traversed sequentially to make space.

For comparison, consider traversing an AVL-binary tree with one billion
elements. A highly optimized implementation will require at least 24 gigabytes
of memory. The binary tree will likely traverse thirty levels, each of which is
a data-dependent lookup. Some lookups will have good locality but most will
not. Each lookup could be hundreds to thousands of times slower than sequential
accesses. These slow lookups are why :doc:`Sorted Containers<index>` can afford
to shift a thousand sequential elements in memory and have most additions take
less time than binary tree implementations.

Due to the memory cache hierarchy, :doc:`Sorted Containers<index>` scales
extremely well. Each element in a :doc:`sortedlist` has little overhead which
increases cache utilization. Data is randomly accessed and related data is
stored together. These patterns in computing have held for decades which
promises :doc:`Sorted Containers<index>` a bright future.
