Add blogpost on flox heuristics

src/posts/flox-smart/index.md

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+---
+title: 'flox: GroupBy, now with smarts!'
+date: '2024-05-31'
+authors:
+  - name: Deepak Cherian
+    github: dcherian
+
+summary: 'flox adds heuristics for automatically choosing an appropriate strategy with dask arrays!'
+---
+
+## TL;DR
+
+`flox>=0.9` adds heuristics for automatically choosing an appropriate strategy with dask arrays! Here I describe how.
+
+## What is flox?
+
+[`flox` implements](https://flox.readthedocs.io/) grouped reductions for chunked array types like [cubed](https://cubed-dev.github.io/cubed/) and [dask](https://docs.dask.org/en/stable/array.html) using tree reductions.
+Tree reductions ([example](https://people.csail.mit.edu/xchen/gpu-programming/Lecture11-reduction.pdf)) are a parallel-friendly way of computing common reduction operations like `sum`, `mean` etc.
+Without flox, Xarray effectively shuffles — sorts the data to extract all values in a single group — and then runs the reduction group-by-group.
+Depending on data layout or "chunking" this shuffle can be quite expensive.
+![shuffle](https://flox.readthedocs.io/en/latest/_images/new-split-apply-combine-annotated.svg)
+With flox installed, Xarray instead uses its parallel-friendly tree reduction.
+In many cases, this is a massive improvement.
+Notice how much cleaner the graph is in this image:
+![map-reduce](https://flox.readthedocs.io/en/latest/_images/new-map-reduce-reindex-True-annotated.svg)
+See our [previous blog post](https://xarray.dev/blog/flox) for more.
+
+Two key realizations influenced the development of flox:
+
+1. Array workloads frequently group by a relatively small in-memory array. Quite frequently those arrays have patterns to their values e.g. `"time.month"` is exactly periodic, `"time.dayofyear"` is approximately periodic (depending on calendar), `"time.year"` is commonly a monotonic increasing array.
+2. Chunk sizes (or "partition sizes") for arrays can be quite small along the core-dimension of an operation. This is an important difference between arrays and dataframes!
+
+These two properties are particularly relevant for "climatology" calculations (e.g. `groupby("time.month").mean()`) — a common Xarray workload in the Earth Sciences.
+
+## Tree reductions can be catastrophically bad
+
+Consider `ds.groupby("time.year").mean()`, or the equivalent `ds.resample(time="Y").mean()` for a 100 year long dataset of monthly averages with chunk size of **1** (or **4**) along the time dimension.
+This is a fairly common format for climate model output.
+The small chunk size along time is offset by much larger chunk sizes along the other dimensions — commonly horizontal space (`x, y` or `latitude, longitude`).
+
+A naive tree reduction would accumulate all averaged values into a single output chunk of size 100 — one value per year for 100 years.
+Depending on the chunking of the input dataset, this may overload the final worker's memory and fail catastrophically.
+More importantly, there is a lot of wasteful communication — computing on the last year of data is completely independent of computing on the first year of the data, and there is no reason the results for the two years need to reside in the same output chunk.
+This issue does not arise for regular reductions where the final result depends on the values in all chunks, and all data along the reduced axes are reduced down to one final value.
+
+## Avoiding catastrophe
+
+Thus `flox` quickly grew two new modes of computing the groupby reduction.
+
+First, `method="blockwise"` which applies the grouped-reduction in a blockwise fashion.
+This is great for `resample(time="Y").mean()` where we group by `"time.year"`, which is a monotonic increasing array.
+With an appropriate (and usually quite cheap) rechunking, the problem is embarrassingly parallel.
+![blockwise](https://flox.readthedocs.io/en/latest/_images/new-blockwise-annotated.svg)
+
+Second, `method="cohorts"` which is a bit more subtle.
+Consider `groupby("time.month")` for the monthly mean dataset i.e. grouping by an exactly periodic array.
+When the chunk size along the core dimension "time" is a divisor of the period; so either 1, 2, 3, 4, or 6 in this case; groups tend to occur in cohorts ("groups of groups").
+For example, with a chunk size of 4, monthly mean input data for the "cohort" Jan/Feb/Mar/Apr are _always_ in the same chunk, and totally separate from any of the other months.
+Here is a schematic illustration where each month is represented by a different shade of red:
+![monthly cohorts](https://flox.readthedocs.io/en/latest/_images/cohorts-month-chunk4.png)
+This means that we can run the tree reduction for each cohort (three cohorts in total: `JFMA | MJJA | SOND`) independently and expose more parallelism.
+Doing so can significantly reduce compute times and in particular memory required for the computation.
+
+Importantly if there isn't much separation of groups into cohorts; example, the groups are randomly distributed, then it's hard to do better than the standard `method="map-reduce"`.
+
+## Choosing a strategy is hard, and harder to teach.
+
+These strategies are great, but the downside is that some sophistication is required to apply them.
+Worse, they are hard to explain conceptually! I've tried! ([example 1](https://discourse.pangeo.io/t/optimizing-climatology-calculation-with-xarray-and-dask/2453/20?u=dcherian), [example 2](https://discourse.pangeo.io/t/understanding-optimal-zarr-chunking-scheme-for-a-climatology/2335)).
+
+What we need is to choose the appropriate strategy automatically.
+
+## Problem statement
+
+Fundamentally, we know:
+
+1.  the data layout or chunking.
+2.  the array we are grouping by, and can detect possible patterns in that array.
+
+We want to find all sets of groups that occupy similar sets of chunks.
+For groups `A,B,C,D` that occupy the following chunks (chunk 0 is the first chunk along the core-dimension or the axis of reduction)
+
+```
+A: [0, 1, 2]
+B: [1, 2, 3]
+D: [5, 6, 7, 8]
+E: [8]
+X: [0, 3]
+```
+
+We want to detect the cohorts `{A,B,X}` and `{C, D}` with the following chunks.
+
+```
+[A, B, X]: [0, 1, 2, 3]
+[C, D]: [5, 6, 7, 8]
+```
+
+Importantly, we do _not_ want to be dependent on detecting exact patterns, and prefer approximate solutions and heuristics.
+
+## The solution
+
+After a fun exploration involving such fun ideas as [locality-sensitive hashing](http://ekzhu.com/datasketch/lshensemble.html), and [all-pair set similarity search](https://www.cse.unsw.edu.au/~lxue/WWW08.pdf), I settled on the following algorithm.
+
+I use set _containment_, or a "normalized intersection", to determine the similarity the sets of chunks occupied by two different groups (`Q` and `X`).
+
+```
+C = |Q ∩ X| / |Q| ≤ 1;     (∩ is set intersection)
+```
+
+Unlike [Jaccard similarity](https://en.wikipedia.org/wiki/Jaccard_index), _containment_ [isn't skewed](http://ekzhu.com/datasketch/lshensemble.html) when one of the sets is much larger than the other.
+
+The steps are as follows:
+
+1. First determine which labels are present in each chunk. The distribution of labels across chunks
+   is represented internally as a 2D boolean sparse array `S[chunks, labels]`. `S[i, j] = 1` when
+   label `j` is present in chunk `i`.
+1. Now we can quickly determine a number of special cases:
+   1. Use `"blockwise"` when every group is contained to one block each.
+   1. Use `"cohorts"` when every chunk only has a single group, but that group might extend across multiple chunks
+   1. [and more](https://github.com/xarray-contrib/flox/blob/e6159a657c55fa4aeb31bcbcecb341a4849da9fe/flox/core.py#L408-L426)
+1. Now invert `S` to compute an initial set of cohorts whose groups are in the same exact chunks (this is another groupby!).
+   Later we will want to merge together the detected cohorts when they occupy _approximately_ the same chunks, using the containment metric.
+1. For that we first quickly compute containment for all groups `i` against all other groups `j` as `C = S.T @ S / number_chunks_per_group`.
+1. To choose between `"map-reduce"` and `"cohorts"`, we need a summary measure of the degree to which the labels overlap with
+   each other. We can use _sparsity_ --- the number of non-zero elements in `C` divided by the number of elements in `C`, `C.nnz/C.size`.
+   We use _sparsity_ --- the number of non-zero elements in `C` divided by the number of elements in `C`, `C.nnz/C.size`. When sparsity is relatively high, we use `"map-reduce"`, otherwise we use `"cohorts"`.
+1. If the sparsity is high enough, we merge together similar cohorts using a for-loop.
+1. Finally we execute one tree-reduction per cohort and concatenate the results.
+
+For more detail [see the docs](https://flox.readthedocs.io/en/latest/implementation.html#heuristics) or [the code](https://github.com/xarray-contrib/flox/blob/e6159a657c55fa4aeb31bcbcecb341a4849da9fe/flox/core.py#L336).
+Suggestions and improvements are very welcome!
+
+Here is `C` for a range of chunk sizes from 1 to 12, for computing `groupby("time.month")` of a monthly mean dataset, [the title on each image is (chunk size, sparsity)].
+![flox sparsity image](https://flox.readthedocs.io/en/latest/_images/containment.png)
+
+Given the above `C`, flox will choose:
+
+1. `"blockwise"` for chunk size 1,
+2. `"cohorts"` for (2, 3, 4, 6, 12),
+3. and `"map-reduce"` for the rest.
+
+Cool, isn't it?!
+
+Importantly this inference is fast — [400ms for the US county](https://flox.readthedocs.io/en/latest/implementation.html#example-spatial-grouping) GroupBy problem in our [previous post](https://xarray.dev/blog/flox)!
+But we have not tried with bigger problems (example: GroupBy(100,000 watersheds) in the US).
+
+## What's next?
+
+flox' ability to do such inferences relies entirely on the input chunking, a big knob.
+A recent Xarray feature makes such rechunking a lot easier for time grouping:
+
+```python
+from xarray.groupers import TimeResampler
+
+rechunked = ds.chunk(time=TimeResampler("YE"))
+```
+
+will rechunk so that a year of data is in a single chunk.
+
+Even so, it would be nice to automatically rechunk to minimize number of cohorts detected, or to a perfectly blockwise application when that's cheap.
+A challenge here is that we have lost _context_ when moving from Xarray to flox.
+The string `"time.month"` tells Xarray that I am grouping a perfectly periodic array with period 12; similarly
+the _string_ `"time.dayofyear"` tells Xarray that I am grouping by a (quasi-)periodic array with period 365, and that group `366` may occur occasionally (depending on calendar).
+But Xarray passes flox an array of integer group labels `[1, 2, 3, 4, 5, ..., 1, 2, 3, 4, 5]`.
+It's hard to infer the context from that!
+_[Get in touch](https://github.com/xarray-contrib/flox/issues) if you have ideas for how to do this inference._
+
+One way to preserve context may be be to have Xarray's new Grouper objects report ["preferred chunks"](https://github.com/pydata/xarray/blob/main/design_notes/grouper_objects.md#the-preferred_chunks-method-) for a particular grouping.
+This would allow a downstream system like `flox` or `cubed` or `dask-expr` to take this in to account later (or even earlier!) in the pipeline.
+That is an experiment for another day.