Correlates two sets of data by matching

# correlate

## Overview

correlate is a data analysis library. It's designed to find matches between two datasets that conceptually represent the same information, just in different formats. Its goal is to tell you "value A in the first dataset is a good match for value B in the second dataset".

To use correlate, you feed in the two datasets of (opaque) values and their associated metadata information. correlate uses the metadata to find matches between the two datasets, ranks these matches using a unique scoring heuristic, and returns the matches.

### Quick Start

This code:

    import correlate

c = correlate.Correlator()
a, b = c.datasets

greg, carol, tony, steve = "greg carol tony steve".split()
Greg, Carol, Tony, Steve = "Greg Carol Tony Steve".split()

a.set("this", greg)
a.set("is", greg)
a.set("Greg", greg, weight=5)
a.set_keys("Carol over here".split(), carol)
a.set_keys("My name is Tony".split(), tony)
a.set_keys("Hi I'm Steve".split(), steve, weight=2)

b.set_keys("gosh my name is Greg".split(), Greg)
b.set_keys("Carol is my name".split() , Carol)
b.set_keys("Pretty sure I'm still Tony".split(), Tony)
b.set_keys("I'm Steve".split(), Steve)

result = c.correlate()
for match in result.matches:
print(f"{match.score:1.3f} {match.value_a:>5} -> {match.value_b}")


produces this output:

5.750  greg -> Greg
3.800 steve -> Steve
1.286 carol -> Carol
1.222  tony -> Tony


### A Real-Life Example

There's a podcast I like. I download it as MP3 files using an RSS feed, 1990s-style. But the metadata in the RSS feed is junk--the episode titles are inconsistent, and the episode numbers are almost wholly absent.

This podcast also has a list of episodes on its website. This data is much cleaner, including clean, proper episode numbers. And it's easily scraped. But it's still not perfect. The two lists of episodes aren't exactly the same, and even the episodes that are present in both are sometimes reordered.

Obviously, I want to take the MP3s from the RSS feed, and match them up with the nice clean metadata scraped from the website. This gets me the best of both worlds.

But there are more than seven hundred episodes of this particular podcast! Matching those by hand would be a lot of work. And we get a new episode every week. And sometimes they actually add back in old episodes, or update the metadata on old episodes--changes which would mess up any hand-built ordering. And I might want to listen to more than one podcast from this same website someday! So I really didn't want to do all of this by hand.

Happily, after applying just a bit of intelligence to the two datasets, correlate did a perfect job.

### Why correlate Works So Well

The insight that inspired correlate is this: unique keys in the two datasets are probably very good matches. Let's say the key "egyptian" maps to value A1 in one dataset and value B1 in the other dataset--and it only maps to those two values. In that case, A1 and B1 are probably a match.

This leads to a virtuous cycle. Let's say the word "nickel" maps to two values in each of the two datasets: A1 and A2, and B1 and B2. We have two options to match those four values:

A1->B1 and A2->B2
or
A1->B2 and A2->B1


But the key "egyptian" already told us A1 and B1 is a good match. Since we've already matched those, that eliminates the second option. Our only remaining choice is matching A2 -> B2. Choosing a good match based on "egyptian" helped us to eliminated extraneous possibilities and make a good choice based on "nickel", too.

In short, correlate capitalizes on the relative uniqueness of keys.

## Getting Started With correlate

### Requirements

correlate requires Python 3.6 or newer. It has no other dependencies.

If you want to run the correlate test suite, you'll need to install the rapidfuzz package. (rapidfuzz is a fuzzy string matching library. It's a lot like fuzzywuzzy, except rapidfuzz is MIT licensed, wheras fuzzywuzzy is GPL licensed.)

### The High-Level Conceptual Model

To correlate two datasets with correlate, you first create a correlate.Correlator object. This object contains two members dataset_a and dataset_b; these represent the two datasets you want to correlate.

You fill each dataset with keys and values.

A value is (nearly) any Python object. Each value should represent one value from your dataset. correlate doesn't examine your values--they're completely opaque to correlate.

A key is a Python object that represents some metadata about a value. Keys "map" to values; correlate examines keys, using matching keys in the two datasets to match up the values between them.

Keys are usually strings--for example, the individual words from the title of some movie, TV show, song, or book. But keys don't have to be strings. Instances of lots of Python data types can be used as keys; they just need to be hashable.

Once you've filled in the correlate.Correlator object with your data, you call its correlate() method. This computes the matches. It returns a correlate.CorrelateResult containing those matches, and lists of any objects from the two datasets that didn't get matched.

The matches are returned as a list of correlator.CorrelatorMatch objects. Each object contains three members:

• value_a, a reference to an object from dataset_a,
• value_b, a reference to an object from dataset_b,
• and a floating-point score.

Each CorrelatorMatch object tells you that correlator thinks this value_a maps to this value_b. The score is a sort of mathematical confidence level--it's a direct result of the keys and other metadata you provide to correlate. The list of matches is sorted by score--higher scores first, as higher scores represent higher confidence in the match.

That's the basics. But correlate supports some very sophisticated behavior:

• When mapping a key to a value, you may specify an optional weight, which represents the relative importance of this key. The default weight is 1. Higher scores indicate a higher significance; a weight of 2 tells correlate that this key mapped to this value is twice as significant. (Weight is an attribute of a particular mapping of a key to a value--an "edge" in the graph of mapping keys to values.)
• A key can map to a value multiple times. Each mapping can have its own weight.
• If both datasets are ordered, this ordering can optionally influence the match scores. correlate calls this ranking. Ranking is an attribute of values, not keys.
• Keys can be "fuzzy", meaning two keys can be a partial match rather than a binary yes/no. Fuzzy keys in correlate must inherit from a custom abstract base class called correlate.FuzzyKey.

## Sample Code And infer_mv

correlate ships with some sample code for your reading pleasure. The hope is it'll help you get a feel for what it's like to use correlate. Take a look at the scripts in the tests and utilities directories.

In particular, utilities contains a script called infer_mv. infer_mv takes a source directory and a list of files and directories to rename, and produces a mapping from the former to the latter. In other words, when you run it, you're saying "here's a source directory and a list of files and directories to rename. For each file in the list of things to rename, find the filename in the source directory that most closely resembles that file, and rename the file so it's exactly like that other filename from the source directory." (If you ask infer_mv to rename a directory, it renames all the files and directories inside that directory, recursively.)

This is useful if, for example, you have a directory where you've already renamed the files the way you you like them, but then you get a fresh copy from somewhere. Simply run infer_mv with your existing directory as the "source directory" and the fresh copy as the "files". infer_mv will figure out how to rename the fresh files so they have the filenames how you like them.

Note that infer_mv doesn't actually do the work of renaming! Instead, infer_mv prints out a shell script that, if executed, performs the renames. Why? It's always a good idea to check over the output of correlate before you commit to it.

I use infer_mv like so:

% infer_mv ../old_path *
# look at output, if it's all fine run
% infer_mv ../old_path * | sh


Or you can direct the output of infer_mv into a file, then edit the file, then execute that. Or something else! Whatever works for you!

## Terminology And Requirements

### Values

Values are Python objects that represent individual elements of your two datasets. correlate doesn't examine values, and it makes very few demands on them. Here are the rules for values:

• Values must support ==.
• Value comparison must be reflexive, symmetric, transitive, and consistent. For all these examples, a b and c represent values:
• reflexive: A value must always compare as equal to itself. a == a must evaluate to True.
• symmetric: If a == b is True, then b == a must also be True.
• transitive: If a == b is True, and b == c is True, then a == c must also be True.
• consistent: If a == b is True, it must always be True, and if a == b is False it must always be False.

### Keys

Keys are Python objects that correlate uses to find matches between the two datasets. If a key maps to a value in dataset_a and also a value in dataset_b, those two values might be a good match.

Keys must obey all the same rules as values. In addition, keys must be hashable. This in turn requires that keys must be immutable.

#### Exact Keys

An "exact" key is what correlate calls any key that isn't a "fuzzy" key. Strings, integers, floats, complex, datetime objects--they're all fine to use as correlate keys, and instances of many more types too.

When considering matches, exact keys are binary--either they're an exact match or they don't match at all. To work with non-exact matches you'll have to use "fuzzy" keys.

#### Fuzzy Keys

A "fuzzy" key is a key that supports a special protocol for performing "fuzzy" comparisons--comparisons where the result can represent imperfect or partial matches.

Technically speaking, a correlate "fuzzy" key is an instance of a subclass of correlate.FuzzyKey. If a key is an instance of a subclass of that base class, it's a "fuzzy" key, and if it isn't, it's an "exact" key.

Fuzzy keys must follow the rules for keys above. Also, the type of your fuzzy keys must also obey the same rules as keys; they must be hashable, they must support ==, and their comparison must be reflexive, symmetric, transitive, and consistent.

In addition, fuzzy keys must support a method called compare with this signature: self.compare(other). other will be another fuzzy key of the same type. Your compare function should return a number (either int or float) between (and including) 0 and 1, indicating how close a match self is to other. If compare returns 1, it's saying this is a perfect match, that the two values are identical; if it returns 0, it's a perfect mismatch, telling correlate that the two keys have nothing in common.

correlate requires that compare also obey the four mathematical constraints required of comparisons between keys. In the following rules, a and b are fuzzy keys of the same type. compare must conform to these familiar four rules:

• reflexive: a.compare(a) must return 1 (or 1.0).
• symmetric: If a.compare(b) returns x, then b.compare(a) must also return x.
• transitive: If a.compare(b) returns x, and b.compare(c) returns x, then a.compare(c) must also return x.
• consistent: If a.compare(b) returns x, it must always return x.

It's important to note: fuzzy keys of two different types are automatically considered different to each other. correlate won't even bother calling compare() on them--it automatically assigns the comparison a fuzzy score of 0. This is true even for subclasses; if you declare class MyFuzzyNumber(correlate.FuzzyKey) and also class MyFuzzyInteger(MyFuzzyNumber), correlate will never compare an instance of MyFuzzyNumber and MyFuzzyKey to each other--it automatically assumes they have nothing in common.

(Internally correlate stores fuzzy keys of different types segregated from each other. This is a vital optimization!)

On a related note, correlate may optionally never actually call a.compare(a), either. That is, if the exact same key maps to a value in both dataset_a and dataset_b, correlate is permitted to skip calling compare() and instead automatically assign the comparison a fuzzy score of 1. (Currently if this situation arose it would call a.compare(a), but that wasn't true at various times during development.)

Finally, it's important to note that fuzzy keys are dramatically slower than exact keys. If you can express your problem purely using exact keys, you should do so! It'll run faster as a result. You can get a sense of the speed difference by running tests/ytjd.test.py with verbose mode on (-v). A test using the same corpus but switching everything to fuzzy keys runs about 12x slower on my computer.

## API

Correlator(default_weight=1)

The correlator class. default_weight is the weight used when you map a key to a value without specifying an explicit weight.

Correlator.dataset_a

Correlator.dataset_b

Instances of Correlator.Dataset objects representing the two sets of data you want to correlate. Initially empty.

Correlator.datasets

A list containing the two datasets: [dataset_a, dataset_b]

Correlator.correlate(*, minimum_score=0, score_ratio_bonus=1, ranking=BestRanking, ranking_bonus=0, ranking_factor=0, reuse_a=False, reuse_b=False)

Correlates the two datasets. Returns a correlate.CorrelatorResult object.

minimum_score is the minimum permissible score for a match. It must be greater than or equal to 0.

score_ratio_bonus specifies the weight of a bonus awarded to a match, based on the ratio of the actual score computed between these two values divided by the maximum possible score. For more information, consult the Score Ratio Bonus section.

ranking specifies which approch to computing ranking correlate should use. The default value of BestRanking means correlate will try all approaches and choose the one with the highest cumulative score across all matches. Other values include AbsoluteRanking and RelativeRanking.

ranking_bonus specifies the weight of the bonus awarded to a match based on the proximity of the two values in their respective datasets, as specified by their rankings. The closer the two values are to the same position in their respective datasets, the higher a percentage of the ranking_bonus will be awarded.

ranking_factor specifies the ratio of the base score of a match that is multiplied by the proximity of the two values in their respective datasets. If you ues ranking_factor=0.4, then a match only automatically keeps 60% of its original score; some percentage of the remaining 40% will be re-awarded based on the proximity of the two values.

(You can't use both a nonzero ranking_bonus and a nonzero ranking_factor in the same correlation. Pick at most one!)

For more information on all these ranking-related parameters, consult the Ranking section of this document.

reuse_a permits values in dataset_a to be matched to more than one value in dataset_b. reuse_b is the same but for values in dataset_b matching dataset_a. If you set both reuse flags to True, the correlate.CorrelatorResult.matches list returned will contain every possible match.

Correlator.print_datasets()

Prints both datasets in a human-readable form. Uses self.print to print, which defaults to print.

Correlate.Dataset()

The class for objects representing a dataset. Behaves somewhat like a write-only dict.

Correlator.Dataset.set(key, value, weight=default_weight)

You can use Dataset[key] = value as a shortcut for Dataset.set(key, value).

Correlator.Dataset.set_keys(keys, value, weight=default_weight)

Map multiple keys to a single value, all using the same weight. keys must be an iterable containing keys.

Correlator.Dataset.value(value, *, ranking=None)

Annotates a value with extra metadata. Currently only one metadatum is supported: ranking.

ranking represents the position of this value in the dataset, if the dataset is ordered. ranking should be an integer representing the ranking; if this value is the 19th in the dataset, you should supply ranking=19.

Correlator.Dataset.clear(default_weight=1, *, id=None)

Clears the Dataset. Reinitializes it to its empty state. You can optionally also change the dataset's default weight and its id.

CorrelatorResult()

The class for objects returned by Correlator.correlate(). Contains four members:

• matches, a list of CorrelatorMatch() objects, sorted with highest score first
• unmatched_a, the set of values from dataset_a that were not matched
• unmatched_b, the set of values from dataset_b that were not matched
• statistics, a dict containing human-readable statistics about the correlation

CorrelatorResult.normalize(high=None, low=None)

Normalizes the scores in matches. When normalize() is called with its default values, it adjusts every score so that they fall in the range (0, 1]. If high is not specified, it defaults to the highest score in matches. If low is not specified, it defaults to the minimum_score used for the correlation.

CorrelatorMatch()

The class for objects representing an individual match made by Correlator.correlate(). Contains three members:

• value_a, a value from dataset_a.
• value_b, a value from dataset_b.
• score, a number representing the confidence in this match. The higher the score, the higher the confidence. Scores don't have a predefined intrinsic meaning; they're a result of all the inputs to correlate.

Correlator.str_to_keys(s)

A convenience function. Converts string s into a list of string keys using a reasonable approach. Lowercases the string, converts some common punctuation into spaces, then splits the string at whitespace boundaries. Returns a list of strings.

### Getting Good Results Out Of Correlate

Unfortunately, you can't always expect perfect results with correlate every time. You'll usually have to play with it at least a little. At its heart, correlate is a heuristic, not an exact technology. It often requires a bit of tuning before it produces the results you want.

#### Ranking

Naturally, the first step with correlate is to plug in your data. I strongly encourage you to add ranking information if possible.

If the two datasets are ordered, and equivalent items should appear in roughly the same place in each of the two datasets, ranking information can make a sizeable improvement in the quality of your matches. To use ranking information, you set the ranking for each value in each dataset that you can, and specify either ranking_bonus or ranking_factor when running correlate(). Which one you use kind of depends on how much confidence you have in the ordering of your datasets. If you think your ranking information is pretty accurate, you should definitely use ranking_factor; this exerts a much stronger influence on the matches. If you have a low confidence in the ordering of your datasets, choose ranking_bonus, which only provides a little nudge.

Ranking can also speed up correlate quite a bit. If there are a lot of matches that end up with the same score, this can create a lot of work for the "match boiler" (see below), and that can get expensive quick. Even a gentle nudge from ranking information can help differentiate scores enough to result in a dramatic speedup.

#### Minimum Score

Once you've plugged in all your data, you should run the correlation, print out the result in sorted order with the best matches on top, then scroll to the bottom and see what the worst 5% or 10% of matches look like.

If literally all your matches are already perfect--congratulations! You're already getting good results out of correlate and you can stop reading here. But if you're not that lucky, you've got more work to do.

The first step in cleaning up correlate's output is usually to stop it from making bad matches by setting a minimum_score.

When you have bad matches, it's usually because the two datasets don't map perfectly to each other. If there's a value in dataset_a that has no good match in dataset_b, well, correlate doesn't really have a way of knowing that. So it may match that value to something anyway.

Look at it this way: the goal of correlate is to find matches between the two datasets. If it's made all the good matches it can, and there's only one item left in each of the the two datasets, and they have anything in common at all, correlate will match those two values together out of sheer desparation.

However! Bad matches like these tend to have a very low score. So all those low-scoring bad matches clump together at the very bottom. There'll probably be an inflection point where the scores drop off significantly and the matches go from good to bad.

This is what minimum_score is for. minimum_score tells correlate the minimum permissible score for a match. When you have a clump of bad matches at the bottom, you simply set minimum_score to be somewhere between the highest bad match and the lowest good match--and behold! No more bad matches! The values that were used in the bad matches will move to unused_a and unused_b, which is almost certainly the correct place for them.

(Technically, minimum_score isn't actually the minimum score. It's ever-so-slightly less than the lowest permitted score. As in, for a match to be considered viable, its score must be greater than minimum_score. In Python 3.9+, you can express this concept as:

actual_minimum_score = math.nextafter(minimum.score, math.inf)


The default value for minimum_score is 0, which means correlate will keep any match with a positive score.)

Unfortunately it's hard to predict what to set minimum_score to in advance. Its value really depends on your data set--how many keys you have, how good the matches are, what weights you're using, everything. It's much more straightforward to run the correlation, look over the output, find where the correlations turn bad, and set a minimum score. With large data sets there's generally a sudden and obvious dropoff in score, associated with correlate making poor matches. That makes it pretty easy: set the minimum score so it keeps the last good match and forgets the rest. But there's no predicting what that score will be in advance--every data set is different, and it's really an emergent property of your keys and weights--so you'll have to calibrate it correctly for each correlation you run.

(Sometimes there are good matches mixed in with the bad ones at the bottom. When that happens, the first step is generally to fix that, so that the bad ones are all clumped together at the bottom. I can't give any general-purpose advice on what to do here. All I can suggest is to start experimenting. Change your keys, adjust your weights, run the correlation again and see what happens. Usually when I do this, I realize something I can do to improve the data I feed in to correlate, and I can fix the problem externally.)

#### Weights

If you're still not getting the results you want, the next adjustment you should consider is increasing the weight of keys that provide a clear signal. If the datasets you're comparing have some sort of unique identifier associated with each value--like an episode number, or release date--you should experiment with giving those keys a heavier weight. Heavily-weighted keys like this can help correlate zero in on the best matches right away.

It's up to you what that weight should be; I sometimes use weights as heavy as 5 for super-important keys, which means this one single key will have the same weight as 5 normal keys. Note that a weight of 5 on the mapping in dataset_a and dataset_b means that, if those keys match, they'll have a base score of 25! If that key only appears once in each dataset, that will almost certainly result in a match.

But using weights can be a dual-edged sword. If your data has mistakes in it, heavily weighting the bad data can magnify those mistakes. One bad heavily-weighted key on the wrong value can inflate the score of a bad match over the correct match. And that can result in a vicious cycle--if value A1 should match value B1, but it get mapped to value B43 instead, that means B1 is probably going to get mismatched too. Which deprives another value of its correct match. And so on and so on and so on.

One final note on weights. The weight of a key doesn't affect how desirable it is in a match, it only affects the resulting score of that match. Consider this scenario involving weighted fuzzy keys:

FA1 and FA2 are fuzzy keys in dataset_a
FB is a fuzzy key in dataset_b
VA1 and VA2 are values in dataset_a

dataset_a.set(FA1, VA1, weight=1)
dataset_a.set(FA2, VA2, weight=5)

FA1.compare(FB) == 0.4
FA2.compare(FB) == 0.2


If correlate had to choose between these two matches, which one will it prefer? It'll prefer FA1->FB, because correlate doesn't consider weights when considering matches. It always prefers the match with the higher unweighted score. It's true, matching FA2 to FB results in a higher final score once you factor in the weights. But that doesn't make it a better match.

The best way to conceptualize this: weights don't make matches higher quality, they just make matches more interesting when true.

#### Too-Common Keys

Similarly, if there are super-common keys that aren't going to help with the correlation, consider throwing them away and not even feeding them in as data. Keys that map to most or all of the values in a dataset add little clarity, and will mainly serve just to make correlate slower. I usually throw away the word "The", and the name of the podcast or show. (When correlating filenames, I may throw away the file extension too.)

Then again, often leaving them in won't hurt anything, and it can occasionally be helpful! The way correlate works, it considers multiple maps of a key to a value as different things--if you map the key "The" to a value twice, correlate understands that those are two separate mappings. And if there's only one value in each dataset that has two "The" mappings, that can be a very strong signal indeed. So it's really up to you. Throwing away largely-redundant keys is a speed optimization, but it shouldn't affect the quality of your matches.

Note that correlate is now very efficient when it comes to matching with exact keys. For most people, the additional runtime cost for redundant or common keys is probably negligible, and not worth the additional development time or engoing support cost to make it even worth considering. It's true they provide only a tiny amount of signal--but they also have relatively little runtime cost, either in memory or CPU time. At this point it's probably not worth the bother to almost anybody.

(But here's a theoretical best of both worlds to consider: for very common keys, consider throwing away the first instance. I admit I haven't tried this experiment myself.)

As always, it's helpful to make sure your code is doing what you intend it to. Several times I've goofed up the mechanism I use to feed data sets into correlate; for example, instead of feeding in words as keys, I've occasionally fed in the individual characters in those words as keys. (Like, instead of the single key "booze", I accidentally fed in the five keys 'b', 'o', 'o', 'z', and 'e'.) However, the correlate algorithm works so well, it still did a shockingly good job! (Though it was a lot slower.)

I've learned to double-check that I'm inputting the mappings and weights I meant to, with a debugger or with Correlator.print_datasets(). Making sure you gave correlate the right data can make it not only much more accurate, it might make it faster too!

#### Normalize Strings

When using strings as keys from real-world sources, I recommend you normalize the strings: lowercase the strings, remove most or all punctuation, and break the strings up into individual keys at word boundaries. In the real world, punctuation and capitalization can both be inconsistent, so throwing it away can help dispel those sorts of inconsistencies. correlate provides a utility function called correlate.str_to_keys() that does this for you. But you can use any approach to string normalizing you like.

You might also consider interning your strings. In my limited experimentation this provided a small but measurable speedup.

If you're using fuzzy keys, make sure you sharpen your fuzzy keys. Fuzzy string-matching libraries have a naughty habit of scoring not-very-similar strings as not that much less than almost-exactly-the-same strings. If you give that data unaltered to correlate, that "everything looks roughly the same" outlook will be reflected in your results as mediocre matches.

In general, you want to force your fuzzy matches to extremes. Two good techniques:

• Specify a minimum score for fuzzy matches, and replace any fuzzy score below that minimum with 0.
• Possibly remap the remaining range to the entire range. For example, if your minimum score is 0.6, should you simply return values from 0.6 to 1? Or should you stretch the scores over the entire range with (fuzzy_score - 0.6) / (1 - 0.6)? You may need to experiment with both to find out what works well for you.
• Multiply your fuzzy score by itself. Squaring or even cubing a fuzzy score will preserve high scores and attenuate low scores. However, note that the scoring algorithm for fuzzy key matches already cubes the fuzzy score. Additional multiplying of the score by itself is probably unnecessary in most cases.

### What Do These Scores Mean?

The scores you seee in the results are directly related to the data you gave to correlate. The scores really only have as much or as little meaning as you assign to them.

If you don't enjoy the unpredictable nature of correlate scores, consider calling normalize() on your Correlate result object. This normalizes the scores as follows: the highest score measured will be adjusted to 1.0, minimum_score will be adjusted to 0.0, and every other score will be adjusted linearly between those two.

Mathematically:

score = the original score for this match
highest_score = highest score of any match
minimum_score = the minimum_score passed in to correlate()
delta = highest_score - minimum_score
normalized_score = (score - minimum_score) / delta


## Implementation Notes On The Algorithm And The Code

If the implementation is hard to explain, it's a bad idea. --The Zen Of Python by Tim Peters

What follows is an exhaustive (and exhausting!) chapter on the implementation of correlate. This is here partially for posterity, partially because I like reading this sort of thing in other people's projects, but mostly to make it easier to reaquaint myself with the code when I have to fix a bug three years from now.

### The High-Level Overview

At the heart of correlate is a brute-force algorithm. It's what computer scientists would call an O(n²) algorithm.

correlate computes every possible "match"--every mapping of a value in dataset_a to a value in dataset_b where the two values have keys in common. For exact keys, it uses set intersections to ignore pairs of values that have nothing in common, discarding early matches it knows will have a score of 0. Sadly, it can't do that for fuzzy keys, which is why fuzzy keys tend to noticably slow down correlate.

For each key that matches between the two values, correlate computes a score. It then adds all those scores together, computing the final cumulative score for the "match", which it may modifiy based on the various bonuses and factors. It then iterates over these scores in sorted order, highest score first. For every match where neither of the two values have been used in a match yet, it counts that as a "match" and adds it to the output. (This assumes reuse_a and reuse_b are both False. Also, this is a little bit of an oversimplification; see the Match Boiler section below.)

One important detail: correlate strives to be 100% deterministic. Randomness can creep in around the edges in Python programs; for example, if you ever iterate over a dictionary, the order you will see the keys will vary from run to run. correlate eliminates every source of randomness it can. As far as I can tell: given the exact same inputs, it performs the same operations in the same order and produces the same result, every time.

There are a number of concepts involved with how the correlate algorithm works, each of which I'll explain in exhausting detail in the following sub-sections.

### correlate's Six Passes And Big-O Complexity

A single correlate correlation makes six passes over its data. Here's a high-level overview of those passes, followed by deep-dives into the new terms and technical details of those passes.

Pass 1

Iterate over both datasets and compute the "streamlined" data.

Complexity: O(n)

Pass 2

Iterate over all keys and compute a sorted list of all matches that could possibly have a nonzero score. (The list represents a match with a pair of indices into the lists of values for each dataset.) This pass also performs all fuzzy key comparisons and caches their results.

Complexity: O(n²), for the fuzzy key comparisons step. For a correlate run with a lot of fuzzy keys, this is often the slowest part of the run. If your correlate is mostly exact keys, this part will be pretty quick, because all the O(n²) work is done in C code (set intersections, and sorting).

Pass 3

For every match with a nonzero score, compute subtotals for matching all fuzzy keys. We need to add some of these together to compute the final scores for fuzzy key matches.

Complexity: O(n²)

Pass 4

For every match with a nonzero score:

• compute the scores for matching all exact keys,
• finalize the scores for fuzzy key match scores,
• compute the bonuses (score_ratio_bonus, ranking),
• and store the result per-ranking.

The score for each match is now finalized.

Complexity: O(n²)

Pass 5

For every ranking approach being used, compute the final list of successful matches, using the "match boiler" and "greedy algorithm".

Complexity: O(n log n) (approximate)

Pass 6

Choose the highest-scoring ranking approach, compute unseen_a and unseen_b, and back-substitute the "indexes" with their actual values before returning.

Complexity: O(n)

Thus the big-O notation for correlate overall is O(n²). The slowest part of correlate is processing lots of fuzzy keys; if you can stick mostly to exact keys, your correlate runs will be a lot quicker.

You can see how long correlate spent in each of these passes by examining the statistic member of the CorrelatorResult object. This is a dict mapping string descriptions of passes to a floating-point number of seconds. Pass 2's sub-passes dealing with exact keys and fuzzy keys are broken out separately, as is the "match boiler" phase of Pass 5.

### Rounds

If you call correlate as follows:

c = correlate.Correlator()
o = object()
c.dataset_a.set('a', o)
c.dataset_a.set('a', o)


then key 'a' really is mapped to value o twice, and those two mappings can have different weights. Technically, the correct way to think of this is as having two edges from the same key to the same value in the dataset graph. Another way to think of it is to consider repeated keys as being two different keys--identical, but somehow distinct.

(If it helps, you can also think of it as being like two files with the same filename in two different directories. They have the same filename, but they're not the same file.)

correlate calls groups of these multiple mappings "rounds". A "round" contains all the keys from the Nth time they were repeated. Round 0 contains every key, round 1 contains the second instances of all the keys that were repeated twice, round 2 contains all the third instances of all the keys that were repeated three times, etc. Rounds are per-value, and there are as many rounds as the maximum number of redundant mappings of any single key to any particular value in a dataset.

Naturally, exact keys and fuzzy keys use a different method to determine whether or not something is "the same key". Technically both types of keys use == to determine equivalence. However, fuzzy keys don't implement a custom __eq__, so Python uses its default mechanism to determine equivalence, which is really just the is operator. Therefore: exact keys are the same if == says they're the same, and (in practice) fuzzy keys are the same if and only if they're the same object.

(Of course, you could implement your own __eq__ when you write your own fuzzy subclasses. But I don't know why you would bother.)

Consider this example:

c = correlate.Correlator()
o = object()
c.dataset_a.set('a', o, weight=1)
c.dataset_a.set('a', o, weight=3)
c.dataset_a.set('a', o, weight=5)
c.dataset_a.set('b', o)
c.dataset_a.set('b', o)
c.dataset_a.set('c', o)

o2 = object()
c.dataset_b.set('d', o2)
c.dataset_b.set('d', o2)
c.dataset_b.set('e', o2)
c.dataset_b.set('f', o2)


Here, the value o in dataset_a would have three rounds:

• Round 0 would contain the keys {'a', 'b', 'c'}.
• Round 1 would contain the keys {'a', 'b'}.
• Round 2 would contain only one key,{'a'}.

And o2 in dataset_b would have only two rounds:

• Round 0 would contain the keys {'d', 'e', 'f'}.
• Round 1 would contain only one key,{'d'}.

Again, conceptually, the "a" in round 0 is a different key from the "a" in round 1, and so on.

For exact keys, rounds are directly matched iteratively to each other; the exact keys in round 0 for a value in dataset_a are matched to the round 0 exact keys for a value in dataset_b, round 1 in dataset_a is matched to round 1 in dataset_b, and so on. If one side runs out of rounds early, you stop; if you compute the intersection of a round and they have nothing in common, you stop.

One invariant property: each subsequent round has a subset of the keys before it. The set of keys in round N+1 must be a subset of the keys in round N. (Though not necessarily a strict subset.)

What about weights? Higher weights are sorted to lower rounds. The weight for a key k in round N-1 must be greater than or equal to the weight of k in round N. In the above example, the 'a' in round 0 has weight 5, in round 1 it has weight 3, and in round 2 it has weight 1. (It doesn't matter what order you insert them in, correlate automatically sorts the weights as you add the redundant mappings.)

Thus, round 0 always contains every exact key mapped to a particular value, with the highest weights for each of those mappings.

Rounds can definitely help find the best matches. If the key "The" maps to most of your values once, that's not particularly interesting, and it won't affect the scores very much one way or another. But if there's only one value in each dataset that "The" maps to twice, that's a very strong signal indeed! correlate does an excellent job of noticing unique-ness like that and factoring it into the scoring.

### Streamlined Data

The correlate datasets store data in a format designed to eliminate redundancy and be easy to modify. But this representation is inconvenient for performing the actual correlate. Therefore, the first step ("Pass 1") is to reprocess the data into a "streamlined" format. This is an internal-only implementation detail, and in fact the data is thrown away at the end of each correlation. As an end-user you'll never have to deal with it. It's only documented here just in case you ever need to understand the implementation of correlate.

This streamlined data representation is an important optimization. It greatly speeds up computing a match between two values. And it only costs a little overhead, compared to all that matching work. Consider: if you have 600 values in dataset_a and 600 values in dataset_b, correlate will recompute 1,200 streamlined datasets. But it'll then use it in as many as 360,000 comparisons! Spending a little time precomputing the data in a convenient format saves a lot of time in the long run.

The format of the streamlined data changes as the implementation changes. And since it's an internal-only detail, it's largely undocumented here. If you need more information, you'll just have to read the code. Search the code for the word streamlined.

### The Scoring Formula, And Conservation Of Score

For each match it considers, correlate computes the intersection of the keys that map to each of those two values in the two datasets, then computes a score based on each of those key matches. This scoring formula is the heart of correlate, and it was a key insight--without it correlate wouldn't work nearly as well as it does.

In the abstract, it looks like this:

for value_a in dataset_a:
for value_b in dataset_b:
subtotal_score = 0
for key_a, weight_a that maps to value_a:
for key_b, weight_b that maps to value_b:
score = value of key_a compared to key_b
cumulative_a = the sum of all scores resulting from key_a mapping to any value in dataset_b
cumulative_b = the sum of all scores resulting from key_b mapping to any value in dataset_a
score_ratio_a = score / cumulative_a
score_ratio_b = score / cumulative_b
unweighted_score = score * score_ratio_a * score_ratio_b
score_a = weight_a * unweighted_score_a
score_b = weight_b * unweighted_score_b
final_score = score_a * score_b
subtotal_score += final_score


Two notes before we continue:

• key_a and key_b must always be per-round, for a number of reasons, the least of which is because we use their weights in computing the final_score.

• subtotal_score is possibly further adjusted by score_ratio_bonus and ranking, if used. We'll discuss that later.

This formula is how correlate computes a mathematical representation of "uniqueness". The fewer values a key maps to in a dataset, the higher it scores. A key that's only mapped once in each dataset scores 4x higher than a key mapped twice in each dataset.

This scoring formula has a virtuous-feeling mathematical property I call "conservation of score". Each key that you add to a round in a dataset adds 1 to the total cumulative score of all possible matches; when you map a key to multiple values, you divide this score up evenly between those values. For example, if the key x is mapped to three values in dataset_a and four values in dataset_b, each of those possible matches only gets 1/12 of that key's score, and the final cumulative score for all matches only goes up by 1/12. So a key always adds 1 to the sum of all scores across all possible matches, but only increases the actual final score by the amount of signal it actually communicates.

Also, now you see why repeated keys can be so interesting. They add 1 for each round they're in, but that score is only divided by the number of values they're mapped to in that round! Since there tend to be fewer and fewer uses of a key in subsequent rounds, the few keys that make it to later rounds can potentially score much higher than they did in earlier rounds, making them a more noteworthy signal.

### Matching And Scoring Exact Keys

The "streamlined" data format for exact keys looks like this:

exact_rounds[index][round] = (set(d), d)


That is, it's indexed by "index" (which represents the value), then by round number. That gives you a tuple containing a dict mapping keys to weights, and a set() of just the keys. correlate uses set.intersection() (which is super fast!) to find the set of exact keys the two values have in common for that round. The len() of this resulting set is the base cumulative score for that round, although that number is only directly useful in computing score_ratio_bonus.

Although correlate uses the same scoring formula for both exact keys and fuzzy keys in an abstract sense, scoring matches between exact keys is much simpler in practice. Let's tailor the "abstract" scoring algorithm above for exact keys. This lets us optimize the algorithm in a couple places, making it much faster!

First, with exact keys, naturally they're either an exact match or they aren't. If they're an exact match, they're the same Python value. Therefore key_a and key_b must be identical. Therefore, conceptually, we can swap them. Let's rewrite the "scoring formula" equation slightly:

cumulative_a = the sum of all scores between key_b and all keys in dataset_b
cumulative_b = the sum of all scores between key_a and all keys in dataset_a


All we've changed is: we've swapped key_a and key_b. (Why? It'll help. Hey, keep reading.)

Now consider: score for exact keys is always either 1 or 0. It's 1 when two keys are exactly the same, and 0 otherwise. If the base score for the match is 0, then the final_score will be 0 and we can skip all of it. So we only ever compute a final_score when score is 1, when the keys are identical.

Since score is only ever used as a multiplier, we can discard it.

cumulative_a and cumulative_b are similarly easy to compute. They're just the number of times that key is mapped to any value in the relevant dataset, in that round. These counts are precomputed and stored in the "streamlined" data.

So, finally: if you do the substitutions, and drop out the constant score factors, final_score for exact keys is computed like this:

final_score = (weight_a * weight_b) / (cumulative_a * cumulative_b)


Which we can rearrange into:

final_score = (weight_a / cumulative_b) * (weight_b / cumulative_a)


At the point we precompute the streamlined data for dataset_a, we know weight_a, and we can compute cumulative_b because it only uses terms in dataset_a. So we can pre-compute those terms, making the final math:

# when computing the streamlined data
precomputed_a = weight_a / cumulative_b
precomputed_b = weight_b / cumulative_a

# ...

# when computing the score for a matching exact key
final_score = precomputed_a * precomputed_b


That's a lot simpler! And these optimizations made correlate a lot faster.

### Fuzzy Keys

Let me tell you a wonderful bed-time story. Once upon a time, correlate was small and beautiful. But that version only supported exact keys. By the time fuzzy keys were completely implemented, and feature-complete, and working great, correlate was much more complex and... "practical". It's because fuzzy keys introduce a lot of complex behavior, resulting in tricky scenarios that just don't arise with exact keys.

Consider this example:

Your two datasets represent lists of farms. Both datasets list animals, but might have generic information ("horse") or might have specifics ("Clydesdale"). You create a fuzzy key subclass called AnimalKey that can handle matching these together; AnimalKey("Horse/Clydesdale") matches AnimalKey("Horse"), though with a score less than 1 because it isn't a perfect match.

The same farm, Farm X, is present in both datasets:

• In dataset_a, the key AnimalKey("Horse") maps to Farm X twice.

• In dataset_b, the keys AnimalKey("Horse/Clydesdale") and AnimalKey("Horse/Shetland Pony") map to Farm X.

Question: should one of the "Horse" keys in dataset_a match "Horse/Clydesdale" in dataset_b, and should the other "Horse" key match "Horse/Shetland Pony"?

Of course they should! But consider the ramifications: we just matched a key from round 2 in dataset_a to a key from round 1 in dataset_b. That's simply impossible with exact keys!

The scoring used for fuzzy keys is conceptually the same as the scoring for exact keys, including the concept of "rounds". In practice, fuzzy key scoring is much more complicated; there are some multipliers I elided in the description for exact keys because they're always 1, and some other things that are easy to compute for exact keys that we must do the hard way for fuzzy keys. (There's a whole section at the end of this document about the history of fuzzy key scoring in correlate, in case you're interested.)

Also, it's reasonable for a single value in a dataset to have multiple fuzzy keys of the same type, which means that now we could have multiple keys in one dataset in contention for the same key in the other dataset. In the above example with farms and horses, correlate will need to compare both AnimalKey("Horse/Clydesdale") and AnimalKey("Horse/Shetland Pony") from dataset_a to AnimalKey("Horse") in dataset_b.

But correlate doesn't add up every possible fuzzy score generated by a key; when computing the final score, a fuzzy key is only matched against one other fuzzy key. If fuzzy keys FA1 and FA2 map to value VA in dataset_a, and fuzzy key FB maps to value VB in dataset_b, correlate will consider FA1 -> FB and also FA2 -> FB and only keep the match with the highest score. A match "consumes" the two keys (one from each dataset) and they can't be matched again. (Again: when I say "key" here, I mean "this key in this round".) The flip side of this: a key that isn't matched isn't "consumed". What do we do with it? The example above with horses and farms makes it clear: unconsumed fuzzy keys should get recycled--reused in subsequent rounds.

So, where exact keys use very precise "rounds", fuzzy keys require a more dynamic approach. Precisely speaking, an unused key in round N conceptually "survives" to round N+1. That's what the above example with farms and ponies shows us; in round 0, if "Horse/Clysedale" in dataset_a gets matched to "Horse" in dataset_b, "Horse/Shetland Pony" in dataset_a goes unmatched, and survives, and advances on to round 1. This also made scoring more complicated. (For more on this, check out the test suite. There's a regression test that exercises this exact behavior.)

After a bunch of rewrites, I found the fastest way to compute fuzzy matches was: for each fuzzy type the two values have in common, compute all possible matches between all fuzzy keys mapping to the two values, even mixing between rounds. Then sort the matches, preferring higher scores to lower scores, and preferring matches in lower rounds to matches in higher rounds.

The streamlined data for fuzzy keys looks like this:

fuzzy_types[index][type] = [
[(key1, weight, round#0),  (key1, weight, round#1), ...],
[(key2, weight, round#0),  (key2, weight, round#1), ...],
]


That is, they're indexed by index (a representation of the value), then by fuzzy type. That gets you a list of lists. Each inner list is a list of tuples of

(key, weight, round_number)


where key is always the same in all entries in the list, and round_number is always the same as that tuple's index in that list.

When computing matches between fuzzy keys, correlate takes the two lists of lists and does nested for loops over them. Since the keys don't change, it only needs to look up the fuzzy score once. If the fuzzy score is greater than 0, it stores the match in an array.

Once it's done with the fuzzy key matching, it sorts this array of matches, then use the "match boiler" to reduce it down so that every per-round key is matched at most once. (The "match boiler" is discussed later; for now just assume it's a magic function that does the right thing. Though I had to ensure it was super-stable for this approach to work.)

Sorting these fuzzy key matches was tricky. They aren't merely sorted by score; we also must ensure that fuzzy key matches from earlier rounds are always consumed before matches using that key in later rounds. So we use a special sort_by tuple as the sorting key, computed as follows:

key_a, weight_a, round_a = fuzzy_types_a[index_a][type]...
key_b, weight_b, round_b = fuzzy_types_b[index_b][type]...
fuzzy_score = key_a.compare(key_b)
lowest_round_number  = min(round_a, round_b)
highest_round_number = max(round_a, round_b)
sort_by = (fuzzy_score, -lowest_round_number, -highest_round_number)


The -lowest_round_number trick is the very clever bit. This lets us sort with highest values last, which is what the "match boiler" wants. But negating it means lower round numbers are now higher numbers, which lets us prefer keys with lower round numbers.

In terms of the abstract scoring formula, score is the fuzzy score, what's returned by calling the compare() method. And cumulative_a is the sum of all fuzzy score scores for all matches using key_a.

### Score Ratio Bonus

There's a "bonus" score calculated using score_ratio_bonus. It's scored for the overall mapping of a value in dataset_a to a value in dataset_b. This bonus is one of the last things computed for a match, just before ranking.

The bonus is calculated as follows:

value_a = a value from dataset_a
value_b = a value from dataset_b
actual_a = total actual score for all keys that map to value_a in dataset_a
actual_b = total actual score for all keys that map to value_b in dataset_b
possible_a = total possible score for all keys that map to value_a in dataset_a
possible_b = total possible score for all keys that map to value_b in dataset_b
bonus_weight = score_ratio_bonus * (actual_a + actual_b) / (possible_a + possible_b)


This bonus calculated with score_ratio_bonus clears up the ambiguity when the set of keys mapping to one value is a subset of the keys mapping to a different value in the same dataset. The higher percentage of keys that match, the larger this bonus will be.

Consider this example:

c = correlate.Correlator()
c.dataset_a.set('breakin', X)

c.dataset_b.set('breakin', Y)
c.dataset_b.set_keys(['breakin', '2', 'electric', 'boogaloo'], Z)


Which is the better match, X->Y or X->Z? In early versions of correlate, both matches got the exact same score. So it was the luck of the draw as to which match correlate would choose. score_ratio_bonus disambiguates this scenario. It awards a larger bonus to X->Y than it does to X->Z, because a higher percentage of the keys matched between X and Y than matched between X and Z. That small nudge is generally all that's needed to let correlate disambiguate these situations and pick the correct match.

Two things to note. First, when I say "keys", this is another situation where the same key mapped twice to the same value is conceptually considered to be two different keys. In the example I gave in the Rounds subsection above, where value_a is o and value_b is o2, possible_a would be 6 and possible_b would be 4.

Second, the scores used to compute actual and possible are unweighted. If a match between two fuzzy keys resulted in a fuzzy score of 0.3, that adds 0.3 to both actual_a and actual_b, but each of those fuzzy keys adds 1.0 to possible_a and possible_b respectively. Weights are always ignored when computing score_ratio_bonus, just like they're ignored when comparing matches.

### Choosing Which Matches To Keep: The "Greedy Algorithm" And The "Match Boiler"

Here's a problem, presented in the abstract: if you're presented with a list of match objects called matches, where each match object M has three attributes value_a, value_b, and score, how would you compute an optimal subset of matches such that:

• every discrete value of value_a and value_b appears only once, and
• the sum of the score attributes is maximized?

Finding the perfectly optimal solution would require computing every possible set of matches, then computing the cumulative score of that set, then choosing the set with the highest score. Unfortuantely, that algorithm is O(nⁿ), which is so amazingly expensive that we can't even consider it. (You probably want your results from correlate before our sun turns into a red giant.)

Instead, correlate uses a comparatively cheap "greedy" algorithm to compute the subset. It's not guaranteed to produce the optimal subset, but in practice it seems to produce optimal results on real-world data.

Here's a short description of the correlate "greedy" algorithm:

• Sort matches with highest score first.
• For every match M in matches:
• if value_a hasn't been matched yet,
• and value_b hasn't been matched yet,
• keep M as a good match,
• remember that value_a has been matched,
• and remember that value_b has been matched.

The sorting uses Python's built-in sort (Timsort), so it's O(n log n). It's implemented in C so it's pretty quick. The for loop is O(n).

However! Late in development of correlate I realized there was a corner case where odds are good the greedy algorithm wouldn't produce an optimal result. Happily, this had a relatively easy fix, and the fix didn't make correlate any slower in the general case.

Let's start with the problem, the nasty corner case. What if two matches in the list are both viable, and they have the same score, and they have either value_a or value_b in common? It's ambiguous as to which match the greedy algorithm will choose. But choosing the wrong one could result in less-than-optimal scoring in practice.

Here's a specific example:

• dataset_a contains fuzzy keys fka1 and fka2.
• dataset_b contains fuzzy keys fkbH and fkbL. Any match containing fkbH has a higher score than any match containing fkbL. (the H means high scoring, the L means low scoring.)
• The matchesfka1->fkbH and fka2->fkbH have the same score.
• The match fka1->fkbL has a lower score than fka2->fkbL.

The cumulative score over all matches would be higher if correlate chose fka2->fkbL to fka1->fkbL. And since scores in correlate are an indicator of the quality of a match, a higher cumulative score reflects higher quality matches. Therefore we should maximize the cumulative score wherever possible.

But the greedy algorithm can only pick the higher-scoring second match if it previously picked fka1->fkbH. And there's no guarantee that it would! If two items in the list have the same score, it's ambiguous which one the greedy algorithm would choose.

To handle this properly it needs to look ahead and experiment. So that's why I wrote what I call the "match boiler", or the "boiler" for short. The boiler uses a hybrid approach. By default, when the scores for matches are unique, it uses the "greedy" algorithm. But if it encounters a group of items with matching scores, where any of those items have value_a or value_b in common, it recursively runs an experiment where it chooses each of those matches in turn. It computes the score from each of these recursive experiments and keeps the one with the highest score.

(If two or more experiments have the same score, it keeps the first one it encountered with that score--but, since the input to the "match boiler" is a list, sorted with highest scores to the end, technically it's the last entry in the list that produced the high-scoring experiment.)

With the "match boiler" in place, correlate seems to produce optimal results even in these rare ambigous situations.

I'm honestly not sure what the big-O notation is for the "match boiler". The pathological worst case is probably on the order of O(n log n), where the log n component represents the recursions. In this case, every match has the same score, and they're all connected to each other via having value_a and value_b in common. I still don't think the "match boiler" would be as bad as O(n²). The thing is, sooner or later the recursive step would cut the "group" of "connected items" in half (see next section). It's guaranteed not to recurse on every single item. So I assert that roughly cuts the number of recursive steps down to log n, in the pathological worst case that you would never see in real-world data.

#### Cheap Recursion And The "Grouper"

But wait! It gets even more complicated!

Compared to the rest of the algorithm, the recursive step of the "match boiler" is quite expensive. It does reduce the domain of the problem at every step, so it's guaranteed to complete... someday. But, if we're not careful, it'll perform a lot of expensive and redundant calculations. So there are a bunch of optimizations to the match boiler's recursive step, mainly to do with the group of matches that have the same score.

The first step is to analyze these matches and boil them out into "connected groups". A "connected group" is a set of match objects where either each object has a value_a or a value_b in common with another object in the group. These are relevant because choosing one of the matches from these groups will remove at least one other value from consideration in that group, because that value_a or value_b is now "used" and so all remaining match using those values will be discarded.

An example might help here. Let's say you have these six matches in a row all with the same score:

match[1]: value_a = A1, value_b = B1
match[2]: value_a = A1, value_b = B2
match[3]: value_a = A2, value_b = B1
match[4]: value_a = A3, value_b = B2
match[5]: value_a = A10, value_b = B10
match[6]: value_a = A10, value_b = B11


This would split into two "connected groups": matches 1-4 would be in the first group, and matches 5-6 would be in the second. Every match in the first group has one member (value_a or value_b) in common with at least one other match in the first group; every match in the second group has one member in common with at least one other match in the second group. So every match in the first group is "connected"; if you put them in a graph, every match would be "reachable" from every other match in that group, even if they aren't directly connected. For example, match[4] doesn't have any members in common with match[1], but both of them have a member in common with match[2]. But none of the matches in the first group have any member in common with any of the matches in the second group (and naturally vice-versa).

There's a utility function called grouper() that computes these connected groups. (grouper() only handles the case when reuse_a == reuse_b == False; there are alternate implementations to handle the other possible cases, e.g. grouper_reuse_a().)

The second step is to take those "connected groups" and, for every group containing only one match object, "keep" it immediately. We already know we're keeping these and it's cheaper to do that first.

Now that the only remaining connected groups are size 2 or more, the third step is to recurse over each of the values of the smallest of these connected groups. Why the smallest? It's cheaper. Let's say there are 50 items left in the list of matches. At the top are 6 match objects with the same score. There are two groups: one of length 2, the other of length 4.

The important realization is that, when we perform the experiment and recurse using each of these values, we're still going to have to examine all the remaining matches we didn't throw away. The number of operations we'll perform by looping and recursing is, roughly, NM, where N is len(group) and M is len(matches - group). So which one has fewer operations:

• 2 x 48, or
• 4 x 46?

Obviously the first one! By recursing into the smaller group, we perform fewer overall operations.

(There's a theoretical opportunity for further optimization here: when recursing, if there's more than one connected group of length 2 or greater, pass in the list of groups we didn't consume to the recursive step. That would save the recursive call from re-calculating the connected groups. In practice I imagine this happens rarely, so handling it would result in a couple of if branches that never get taken. Also, it's a little more complicated than it seems, because you'd have to re-use the grouper() on the group you're examining before passing it down, because it might split it into two groups. In practice it wouldn't speed up anybody's correlations, and it'd make the code more complicated. So let's skip it. The code is already more-or-less correct in these rare circumstances and that's good enough.)

#### The Match Boiler Reused For Fuzzy Key Scoring

Once my first version of the "match boiler" was done, I realized I could reuse it for boiling down all fuzzy key matches too. Fuzzy key matches already used basically the same "greedy" algorithm that were used for matches, and it dawned on me that the same corner case existed here too.

My first attempt was quite complicated, as the "match boiler" doesn't itself understand rounds. I added a callback which it'd call every time it kept a match, which passed in the keys that got matched. Since those keys were now "consumed", I would inject new matches using those keys from subsequent rounds (if any). This worked but the code was complicated.

And it got even more complicated later when I added the recursive step! I had to save and restore all the state of which fuzzy keys had been consumed from which rounds. I wound up building it into the subclass of MatchBoiler, which is part of why MatchBoiler clones itself when recursing. This made the code cleaner but it was still clumsy and a bit slow.

The subsequent rewrite using the sort_by tuple was a big win all around: it simplified the code, it let me remove the callback, it let the match boiler implicitly handle all the rounds without really understanding them, and it was even slightly faster!

But this is why it's so important that the boiler is super-stable. Earlier versions of the match boiler assumed it could sort the input array any time it wanted. But the array passed in was sorted by score, then by round numbers--highest score is most important, lowest round number is second-most important. And the array is sorted with highest score, then lowest round number, last. When recursing, the match boiler has to prefer the last entry in its input that produced the same score--otherwise, it might accidentally consume a key from a later round before consuming that key from an earlier round.

I didn't want to teach the boiler to understand this sort_by tuple. Happily, I didn't have to. It wasn't much work to ensure that the boiler was super-stable, and once that was true it always produced correct results. (Not to mention... faster!)

#### Theoretical Failings Of The Match Boiler

Even with the "match boiler", you can still contrive scenarios where correlate will produce arguably sub-optimal results. The boiler only tries experiments where the matches have the same score. But it's possible that the greedy algorithm may find a local maximum that causes it to miss the global maximum.

If A and B are values in dataset_a, and X and Y are values in dataset_b, and the matches have these scores:

A->X == 10
A->Y == 9
B->X == 8
B->Y == 1


In this scenario, the boiler will pick A->X, which means it's left with B->Y. Total score: 11. But if it had picked A->Y, that means it would get to pick B->X, and the total score would be 17! Amazing!

Is that better? Your first reaction is probably "of course!". But in an abstract, hypothetical scenario like this, it's hard to say for sure. I mean, yes it's a better score. But is it a better match? Is this the output the user would have wanted? Who knows--this scenario is completely hypothetical in the first place.

I doubt this is a real problem in practice. Ensuring correlate handles the ambiguous scenario where items had identical scores is already "gilding the lily", considering how rare it happens with real-world data. And when would real data behave in this contrived way? Why would A score so highly against X and Y, but B scores high against X but low against Y? If B is a good match for X, and X is a good match for A, and A is a good match for Y, then, with real-world data, transitivity would suggest B is a good match for Y. This contrived scenario seems more and more contrived the more we look at it, and unlikely to occur in the real world.

I think pathological scenarios where the "match boiler" will fail like this aren't realistic. And the only way I can think of to fix it is with the crushingly expensive O(nⁿ) algorithm. It's just not worth it. So, relax! As we say in Python: YAGNI.

### Ranking

Ranking information can help a great deal. If a value in dataset_a is near the beginning, and the order of values is significant, then we should prefer matching it to values in dataset_b near the beginning too. When the datasets are ordered, matching the first value in dataset_a against the last value in dataset_b is probably a bad match.

Conceptually it works as follows: when scoring a match, measure the distance between the two values and let that distance influence the score. The closer the two values are to each other, the higher the resulting score.

But how do you compute that distance? What do the ranking numbers mean? correlate supports three possible interpretations of the rankings:

• Absolute ranking,
• Relative ranking, and
• Reversed Absolute ranking.

These three approaches differ in how they compare the ranking numbers, as follows:

• Absolute ranking assumes the ranking numbers are the same for both datasets. ranking=5 in dataset_a is a perfect match to ranking=5 in dataset_b. This works well when your datasets are both reasonably complete; if they're different sizes, perhaps one or both are truncated at either the beginning or end.
• Relative ranking assumes that the two datasets represent the same range of data, and uses the ratio of the ranking of a value divided by the highest ranking set in that dataset to compute its relative position. If the highest ranking we saw in a particular dataset was ranking=150, then a value that has ranking=12 set is calculated to be 8% of the way from the beginning to the end. This percentage is calculated similarly for both datasets, and the distance between two values is the distance between these two percentages. This works well if one or both of your datasets are sparse.
• Reversed Absolute is like Absolute, but starts from the end rather than from the beginning. Think about Absolute ranking this way: it's comparing the distance from beginning of the dataset to the particular value. Well, Reversed Absolute uses the distance from the end of the dataset to the particular value. Consider: if dataset_a contains 100 values, and dataset_b only contains 15 values, but they're matches for the last 15 values of dataset_a, neither Absolute nor Relative are a good fit. What you'd want there is Reversed Absolute.

Here's a more concrete example of how these approaches work. Let's say dataset_a has 101 items ranked 0 to 100, dataset_b has 801 items ranked 0 to 800, and we have a value in dataset_a with ranking=50:

• With Absolute ranking, the closest value in dataset_b would have ranking=50. The two values are both 50 elements after the first (lowest-ranked) value.
• With Relative ranking, the closest value in dataset_b would have ranking=400. The two values are in the middle of the rankings for their respective datasets.
• With Reversed Absolute ranking, the closest value in dataset_b would have ranking=750. The two value are both 50 elements in front of the last (highest-ranked) value.

Which one does correlate use? It's configurable with the ranking parameter to correlate(). By default it uses the "best" ranking. "Best" ranking means correlate compute a score using all methods and chooses the one with the highest score. You can override this by supplying a different value to ranking but this shouldn't be necessary. (Theoretically it should be faster to use only one ranking approach. Unfortunately this hasn't been optimized yet, so using only one ranking doesn't currently speed things up.)

Ranking is the last step in computing the score of a match. As for how ranking affects the score, it depends on whether you use ranking_bonus or ranking_factor.

semifinal_scores_sum = sum of all "semifinal" scores above
ranking_a = the ranking value computed for value_a
ranking_b = the ranking value computed for value_b
ranking_score = 1 - abs(ranking_a - ranking_b)


ranking_bonus is then calculated per-match as follows:

bonus = ranking_score * ranking_bonus
final_score = semifinal_scores_sum + bonus


ranking_factor is also calculated per-match, as follows:

unranked_portion = (1 - ranking_factor) * semifinal_scores_sum
ranked_portion = ranking_factor * semifinal_scores_sum * ranking_score
final_score = unranked_portion + ranked_portion


(If you don't use either, the final score for the match is effectively semifinal_scores_sum.)

Obviously, ranking must be set on both values in both datasets to properly compute ranking_score. If it's not set on both values being considered for a match, correlate still applies ranking_bonus or ranking_factor as usual, but it skips the initial four calculations and just uses a ranking_score of 0.

## Final Random Topics

### Debugging

When all else fails... what next?

correlate can optionally produce an enormous amount of debug output. The main feature is showing every match it tests, and the score arrived at for that match, including every step along the way. This log output quickly gets very large; even a comparison of 600x600 elements will produce tens of megabytes of output.

Unfortunately, producing this much debugging output incurred a measurable performance penalty--even when you had logging turned off! It was mostly in computing the "f-strings" for the log, but the calls to the logging functions definitely added overhead too.

My solution: by default, each of the debug print statements is commented out. correlate ships with a custom script preprocessor called debug.py that can toggle debugging on and off, by uncommenting and re-commenting the debug code.

How does it know which lines to uncomment? Each line of the logging-only code ends with the special marker "#debug".

To turn on this logging, run the debug.py script in the same directory as correlate's __init__.py script. Each time you run it, it'll toggle (comment / uncomment) the debug print statements. Note that the debug feature in correlate requires Python 3.8 or higher, because it frequently uses 3.8's beloved "equals sign inside f-strings" syntax.

By default the logging is sent to stdout. If you want to override where it's sent, write your own print function, and assign it to your Correlator object before calling correlate().

The format of the log is undocumented and subject to change. Good luck! The main thing you'll want to do is figure out the "index" of the values in dataset_a and dataset_b that you want to compare, then search for " (index_a) x (index_b) ". For example, if the match you want to see is between value index 35 in dataset_a and value index 51 in dataset_b, search in the log for " 35 x 51 ". (The leading and trailing spaces means your search will skip over, for example, 235 x 514.)

### Alternate Fuzzy Scoring Approaches That Didn't Work

I find the math behind fuzzy scoring a bit surprising. If you boil down the formula to its constituent factors, you'll notice one of the factors is fuzzy_score cubed. Why is it cubed?

The simplest answer: that's the first approach that seemed to work properly. To really understand why, you'll need to understand the history of fuzzy scoring in correlate--all the approaches I tried first that didn't work.

Initially, the score for a fuzzy match was simply the fuzzy score multiplied by the weights and other modifiers. This was always a dumb idea; it meant fuzzy matches had way more impact on the score than they should have. This was particularly true when you got down to the last 10% or 20% of your matches, by which point the score contributed by exact keys had fallen off a great deal. This approach stayed in for what is, in retrospect, an embarassingly long time; I'd convinced myself that fuzzy keys were innately more interesting than exact keys, and so this comparative importance was fitting.

Once I realized how dumb that was, the obvious approach was to score them identically to exact keys--divide the fuzzy score by the product of the number of keys this could have matched against in each of the two datasets. This was obviously wrong right away. In the "YTJD" test, every value had one or more fuzzy keys, depending on the test: every value always had a fuzzy date key, and depending on the test it might have a fuzzy title key and/or a fuzzy episode number key too. So each of the 812 values in the first dataset had one fuzzy key for each fuzzy type, and each of the 724 values in the second dataset did too. Even if we got a perfect fuzzy match, the maximum score for a fuzzy match was now 1.0 / (812 * 724) which is 0.0000017. So now we had the opposite problem: instead of being super important, even a perfect fuzzy match contributed practically nothing to the final score.

After thinking about it for a while, I realized that the exact key score wasn't really being divided by the number of keys in the two datasets, per se; it was being divided by the total possible score contributed by that key in each of the two datasets. So instead of dividing fuzzy scores by the number of keys, they should be divided by the cumulative fuzzy score of all matches involving those two keys. That formula looks like fuzzy_score / (sum_of_fuzzy_scores_for_key_in_A * sum_of_fuzzy_scores_for_key_in_B).

This was a lot closer to correct! But this formula had a glaring new problem. Let's say that in your entire correlation, dataset_a only had one fuzzy key that maps to a single value, and dataset_a only had one fuzzy key that also only maps to a single value. And let's say the fuzzy score you get from matching those two keys is 0.000001--a really terrible match. Let's plug those numbers into our formula, shall we! We get 0.000001 / (0.000001 * 0.000001), which is 1000000.0. A million! That's crazy! We've taken an absolutely terrible fuzzy match and inflated its score to be nonsensically high. Clearly that's not right either.

This leads us to the formula that actually works. The insight here is that the same formula needs to work identically for exact keys. If you take this formula and compute it where every fuzzy_score is 1 (or 0), it produces the same result as the formula for exact keys. So the final trick is that we can multiply by fuzzy_score wherever we need to, because multiplying by 1 doesn't change anything. That means the resulting formula will still be consistent with the exact keys scoring formula. And what worked was the formula where we multiply by fuzzy_score three times!

Here again is the formula used to compute the score for a fuzzy match, simplified to ignore weights and rounds:

value_a = a value from dataset_a
value_b = a value from dataset_b
key_a = a fuzzy key in dataset_a that maps to value_a
key_b = a fuzzy key in dataset_b that maps to value_b
fuzzy_score = the result of key_a.compare(key_b)
cumulative_a = the cumulative score of all matches between key_a and every fuzzy key of the same type in dataset_b
cumulative_b = the cumulative score of all matches between key_b and every fuzzy key of the same type in dataset_a
score_ratio_a = fuzzy_score / cumulative_a
score_ratio_b = fuzzy_score / cumulative_b
unweighted_score = fuzzy_score * score_ratio_a * score_ratio_b


The final trick really was realizing what score_ratio_a represents. Really, it represents the ratio of how much this fuzzy match for key_a contributed to the sum of all fuzzy matches for key_a across all successful matches in dataset_a.

### Why correlate Doesn't Use The Gale-Shapley Algorithm

A friend asked me if the problem correlate solves is isomorphic to the Stable Matching Problem:

https://en.wikipedia.org/wiki/Stable_matching_problem

Because, if it was, I could use the Gale-Shapley algorithm:

https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm

I thought about this for quite a while, and I don't think the problem correlate solves maps perfectly onto the stable matching problem. correlate solves a problem that is:

1. simpler,
2. different, and
3. harder.

For inputs that are valid for both Gale-Shapley and correlate, I assert that both algorithms will return the same results, but correlate will be faster.

(Not a claim I make lightly! Gale and Shapley were both brilliant mathematicians--they each independently won the von Neumann prize!--and the Gale-Shapley algorithm is marvelous and elegant. It's just that correlate can take shortcuts Gale-Shapley cannot, because correlate is solving a simpler problem.)

In all the following examples, A, B, and C are values in dataset_a and X, Y, and Z are values in dataset_b. The expression A: XY means "A prefers X to Y". The expression A:X=1.5 means "when matching A and X, their score is 1.5". When I talk about Gale-Shapley, dataset_a will stand for the "men" and dataset_b will stand for the "women", which means A is a man and X is a woman. Where we need to talk about the matches themselves, we'll call them P and Q.

#### How is it simpler?

The stable matching problem only requires a local ordering, where the preferences of any value in either dataset are disjoint from the orderings of any other value. But correlate uses an absolute "score"--a number--to compute these preferences, and this score is symmetric; if A:X=1.5, then X:A=1.5 too.

https://en.wikipedia.org/wiki/Lattice_of_stable_matchings

we find a classic example of a tricky stable matching problem:

A: YXZ
B: ZYX
C: XZY

X: BAC
Y: CBA
Z: ACB


Gale-Shapley handles this situation with aplomb. Does correlate? The answer is... this arrangement of constraints just can't happen with correlate, because it uses scores to establish its preferences, and the scores are symmetric. There are nine possible pairings with those six values. It's impossible to assign a unique score to each of those nine pairings such that the preferences of each value match those constraints.

(And, yes, I'm pretty certain. Not only did I work my way through it, I also wrote a brute-force program that tried every possible combination. 362,880 attempts later, I declared that there was no possible solution.)

#### How is it different?

One minor difference: Gale-Shapley specifies that the two sets be of equal size; correlate permits the two sets to be different sizes. But extending Gale-Shapley to handle this isn't a big deal. Simply extend the algorithm to say that, if the size of the two datasets are inequal, swap the datasets if necessary so that the "men" are the larger group. Then, if you have an unmatched "man" who iterates through all the "women" and nobody traded up to him, he remains unmatched.

The second thing: Gale-Shapley requires that every value in each dataset expresses a strictly ordered preference for every value in the other dataset. But in correlate, two matches can have the same score.

Consider this expression of a correlate problem:

A:X=100
A:Y=1

B:X=100
B:Y=2


Gale-Shapley as originally stated can't solve that problem, because X doesn't prefer A or B--it likes them both equally. It wouldn't be hard to extend Gale-Shapley to handle this; in the case where it prefers two equally, let it arbitrarily pick one. For example, if X prefers A and B equally, say "maybe" to the first one that asks, and then let that decide for you that you prefer A to B.

#### How is it harder?

Here's the real problem.

correlate uses a numerical score to weigh the merits of each match, and seeks to maximize the cumulative score across all matches. Gale-Shapley's goals are comparatively modest--any match that's stable is fine. There may be multiple stable matchings; Gale-Shapley considers them all equally good.

In practice, I think if you apply the original Gale-Shapley algorithm to an input data set where the matches have numerical scores, it would return the set of matches with the highest cumulative score. In thinking about it I haven't been able to propose a situation where it wouldn't. The problem lies in datasets where two matches have the same score--which the original Gale-Shapley algorithm doesn't allow.

Ensuring that correlate returns the highest cumulative score in this situation required adding the sophisticated recursive step to the "match boiler". We'd have to make a similar modification to Gale-Shapley, giving it a recursive step. Gale-Shapley is already O(n²); I think the modified version would be O(n² log n). (But, like correlate, this worst-case shouldn't happen with real-world data.) Anyway, a modified Gale-Shapley that works for all correlate inputs is definitely more expensive than what correlate has--or what it needs.

#### Mapping Gale-Shapley To The Correlate Greedy Algorithm

Again, the set of valid inputs for correlate and Gale-Shapley aren't exactly the same. But there's a lot of overlap. Both algorithms can handle an input where:

• we can assign every match between a man A and a woman X a numerical score,
• every match involving any particular man A has a unique score,
• and also for every woman X,
• and there are exactly as many men as there are women.

I assert that, for these mutually acceptable inputs, both algorithms produce the same result. Here's an informal handwave-y proof.

First, since Gale-Shapley doesn't handle preferring two matches equally, we'll only consider datasets where all matches have a unique score. This lets us dispense with the match boiler's "recursive step", so all we need is the comparatively simple "greedy algorithm". (Again: this "greedy algorithm" is much cheaper than Gale-Shapley, but as I'll show, it's sufficient for the simple problem domain we face here.)

So let's run Gale-Shapley on our dataset. And every time we perform an operation, we write it down in a list--we write down "man A asked woman X" and whether her reply was maybe or no.

• First, order is significant in this list of operations. If you change the order in which particular men ask particular women, the pattern of resulting maybe and no responses will change.

• Second, the last "maybe" said by each woman is always, effectively, a yes.

So let's iterate backwards through this list of matches, and the first time we see any particular woman reply maybe, we change that answer to a yes.

Next: observe that we can swap any two adjacent operations--with one important caveat. We must maintain this invariant: for every woman X, for every operation containing X that happens after X says yes, X must say no.

Thus, if there are two adjacent operations P and Q, where P is currently first, and we want to swap them so Q is first, and if the following conditions are all true:

• The same woman X is asked in both P and Q.
• In Q the woman X says either maybe or yes.
• In P the woman X says maybe.

Then we can swap P and Q if and only if we change X's response in P to no.

Now that we can reorder all the operations, let's sort the operations by score, with the highest score first. Let's call the operation with the highest score P, and say that it matches man A with woman X.

We now know the following are true:

• X is the first choice of A. This must be true because P is the match with the highest score. Therefore A will ask X first.
• A is the first choice of X, again because P is the match with the highest score. Therefore X is guaranteed to say yes.

Since the first operation P is guaranteed to be a yes, that means that every subsequent operation involving either A or X must be a no.

We now iterate down the list to find an operation Q involving man B and woman Y. We define Q as the first operation such that:

• Q != P, and therefore Q is after P in our ordered list of operations,
• B != A, and
• Y != X.

By definition Q must also be a yes, because B and Y are each other's first choices now that A and X are unavailable for matching. If there are any operations between P and Q, these operations involve either A or X. Therefore they must be no. Therefore Q represents the highest remaining preference for both B and Y.

Observe that the whole list looks like this. Every yes is the first operation for both that man and that woman in which they weren't paired up with a woman or man (respectively) that had already said yes to someone else.

This list of operations now more or less resembles the operations performed by the correlate "greedy" algorithm. It sorts the matches by score, then iterates down that sorted list. For every man A and woman X, if neither A nor X has been matched yet, it matches A and X and remembers that they've been matched.

It's possible that there's minor variation in the list of operations; any operation involving any man or any woman after they've been matched with a yes is extraneous. So you can add or remove them all you like without affecting the results.

## Version History

1.1

Added a new ranking approach! The first two were AbsoluteRanking and RelativeRanking, this new third one is ReversedAbsoluteRanking.

1.0

No code changes. But correlate has been stable and working for a while now... it's time to mark it as 1.0.

0.8.3

A slight bugfix for print_datasets(). print_datasets() prints out the keys for each value in sorted order. But that meant sorting the keys, and if you have keys of disparate types, attempting to compare them with < or > could throw a ValueError. So print_datasets() now separates the keys by type and sorts and prints each list of keys separately.

The dataset API allows you to set values that don't actually have any keys mapping to them. (You can call dataset.value() with a value that you never pass in to set() or set_keys().) correlate() used to simply assert that every value had at least one key; now it raises a ValueError with a string that prints every value. (This can be unreadable if there are a lot! But better safe than sorry.)

0.8.2

Fixed up infer_mv. It works the same, but the comments it prints out are now much improved. In particular, there was a bug where it reported the same score for every match--the score of the lowest-ranked match--instead of the correct score for each match.

There were no other changes; the correlate algorithm is unchanged from 0.8.1.

0.8.1

Fixed compatibility with Python 3.6. All I needed to do was remove some equals-sign-in-f-strings usage in spots.

0.8

The result of loving hand-tuned optimization: correlate version 0.8 is now an astonishing 19.5% faster than version 0.7--and 27.3% faster than version 0.5!

The statistics have been improved, including some useful timing information. This really demonstrates how much slower fuzzy keys are.

(To see for yourself, run python3 tests/ytjd.test.py -v and compare the slowest test to the fastest, using the same corpus. On my computer the test without fuzzy keys is 12x faster than the one that uses fuzzy keys for everything.)

0.7

Careful micro-optimizations for both exact and fuzzy key code paths have made correlate up to 7.5% faster!

The MatchBoiler was made even more ridiculously stable. It should now always:

• return results in the same order they appeared in in matches, and
• prefer the last equivalent item when two or more items produce the same cumulative score.

0.6.1

Bugfix for major but rare bug: if there are multiple groups of len() > 1 of "connected" match objects with the same score, the match boiler would only keep the smallest one--the rest were accidentally discarded. (match_boiler_2_test() was added to tests/regression_test.py to check for this.)

0.6

Big performance boost in "fuzzy boiling"! Clever sorting of fuzzy matches, and improvements in the stability (as in "stable sort") of MatchBoiler, allowed using an unmodified boiler to process fuzzy matches. This allowed removal of FuzzyMatchBoiler and the MatchBoiler.filter() callback mechanism.

Minor performance improvement in MatchBoiler: when recursing, find the smallest group of connected matches with the same score, and only recursively check each of those, rather than all possibly-connected matches with the same score.

Removed key_reuse_penalty_factor. In the early days of correlate, it didn't understand rounds; if you mapped the same key to the same value twice, it only remembered one mapping, the one with the highest weight. Later I added rounds but they didn't seem to add much signal. I thought redundant keys were uninteresting. So I added key_reuse_penalty_factor. That let you turn down the signal they provided, in case it was adding more noise than actual useful signal. It wasn't until the realization that key->value in round 0 and key->value in round 1 were conceptually two different keys that I really understood how redundant mappings of the same key to the same value should work. And once rounds maintained distinct counts of keys / scores for the scoring formula, redundant keys in different rounds became way more informative. I now think key_reuse_penalty_factor is dumb and worse than useless and I've removed it. If you think key_reuse_penalty_factor is useful, please contact me and tell me why! Or, quietly just pre-multiply it into your weights.

The cumulative effect: a speedup of up to 30% in fuzzy match boiling, and up to 5% on YTJD tests using a lot of fuzzy keys. Match boiling got slightly faster too.

0.5.1

Bugfix release. In the original version, if a match didn't have any matches between fuzzy keys (with a positive score), it ignored the weights of its exact keys and just used the raw exact score.

0.5

Initial public release.

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