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Description:

relativity 20.1.0

multi-index data structures

Motivation
We’re going to take a mental journey of discovery to see why
relativity was written, and how you can use it to simplify
some of the most difficult problems that come up regularly
when programming. Rather then leaping straight from programming
with python’s standard data structures to programming with
relativistic data structures, we’ll get a running start
by programming in a version of python that is missing
key data structures. Then, we will draw a line from this
deficient bad version of python to regular python, and
then extend that line on into relativity.

Dict to List
Imagine programming without hashmaps. For example, let’s say we have
a list of Restaurant objects and City objects, and we want to
get how many Restaurants are in each City.
Normally this is simple:
restaurants_in_city = {}

for restaurant in restaurants:
city = restaurant.city
restaurants_in_city[city] = restaurants_in_city.get(city, 0) + 1

def get_restaurant_count(city):
return restaurants_in_city.get(city, 0)
But, imagine how you would approach the problem if the only available
data structure was a list.
cities = []
restaurants_in_city = []

for restaurant in restaurants:
missing = True
for idx, city in enumerate(cities):
if city == restaurant.city:
restaurants_in_city[idx] += 1
missing = False
if missing:
cities.append(restaurant.city)
restaurants_in_city.append(1)

def get_restaurant_count(city):
for idx, city2 in enumerate(cities):
if city == city2:
return restaurants_in_city[idx]
return 0
Comparing the two examples, there are a few key differences:

there are more low value local values (idx)
single data structures split into multiple, which must
then be kept in sync
the code is longer, therefore harder to read,
modify, and debug

Let’s leave this dystopian data structure wasteland behind
for now and go back to regular python.

Dict to M2M
The same differences that showed up when programming with
and without hashmaps will come up again when comparing
programming with single-index hashmaps to relativistic
multi-index hashmaps.
Returning to the restaurants and cities example, what if
a restaurant can have multiple locations and we need to
keep track of which cities each restaurant is in,
as well as which restaurants are in each city.
Note that we allow a restaurant to have multiple
locations within the same city, so sets must be used
to avoid double counting.
restaurants_in_city = {}
cities_of_restaurant = {}

for restaurant in restaurants:
for location in restaurant.locations:
restaurants_in_city.setdefault(location.city, set()).add(restaurant)
cities_of_restaurant.setdefault(restaurant, set()).add(location.city)

def get_restaurants_in_city(city):
return restaurants_in_city.get(city, set())

def get_cities_of_restaurant(restaurant):
return cities_of_restaurant.get(restaurant, set())
Relativity’s most basic data structure is a many-to-many
mapping M2M. M2M is a systematic abstraction over
associating every key with a set of values, and every
value with a set of keys. See how M2M simplifies
the problem:
restaurant_city_m2m = M2M()

for restaurant in restaurants:
for location in restaurant.locations:
restaurant_city_m2m.add(restaurant, location.city)

get_restaurants_in_city = restaurant_city_m2m.inv.get
get_cities_of_restaurant = restaurant_city_m2m.get
Recall that the advantages of having single-index hashmaps
were shorter code, with fewer long lived data structures
and fewer local values. M2M doesn’t replace dict
any more than dict replaces list. Rather it is
a new layer of abstraction that can greatly simplify
a broad class of problems.
Is it possible to go further? Are there higher levels
of abstraction that can represent more complex relationships
in fewer data structures, and be manipulated with fewer
lines of code and intermediate values?

M2M to M2MGraph
Where relativity really shines is releiving the programmer
of the burden of keeping data structures consistent with updates.
Let’s consider our restaurant example if we need to be able
to add and remove locations one at a time and still be able
to query.
With M2M objects, the problem is doable, but fiddly to
implement:
restaurant_location = M2M()
location_city = M2M()

def add_location(location):
restaurant_location.add(location.restaurant, location)
location_city.add(location, location.city)

def remove_location(location):
del location_city[location]
del restaurant_location.inv[location]

def restaurants_in_city(city):
restaurants = set()
for location in location_city.inv[city]:
for restaurant in restaurant_location.inv[location]:
restaurants.add(restaurant)
return restaurants

def cities_of_restaurant(restaurant):
cities = set()
for location in restaurant_location[restaurant]:
for city in location_city[location]:
cities.add(city)
return cities
This problem can be simplified by stepping up a level of
abstraction.
Where M2M is a data structure of keys and values, M2MGraph
is a higher-level data structure of M2M s.
With M2MGraph, this problem becomes simple and
intuitive:
data = M2MGraph([('restaurant', 'location'), ('location', 'city')])

def add_location(location):
data['restaurant', 'location', 'city'].add(
location.restaurant, location, location.city)

def remove_location(location):
data.remove('location', location)

def restaurants_in_city(city):
return data.pairs('city', 'restaurant').get(city)

def cities_of_restaurant(restaurant):
return data.pairs('restaurant', 'city').get(restaurant)

Introducing Chain
Graphs are good for representing arbitrary sets of data, but they
are awkward to query overy. M2MChain``s sequences of ``M2M``s, where the keys of ``M2M n are meant to be drawn from the same pool
as the values of M2M n - 1.
A simple way to construct a chain is with the chain helper function.
students2classes = M2M([
('alice', 'math'),
('alice', 'english'),
('bob', 'english'),
('carol', 'math'),
('doug', 'chemistry')])

classmates = chain(students2clases, students2classes.inv)
By chaining the student:class map to itself, we can easily
query which students have classes together.
>>> classmates.only('alice')
M2MChain([M2M([('alice', 'math'), ('alice', 'english')]), M2M([('math', 'carol'), ('math', 'alice'), ('english', 'bob'), ('english', 'alice')])])

>>> classmates.only('alice').m2ms[1]
M2M([('math', 'carol'), ('math', 'alice'), ('english', 'bob'), ('english', 'alice')])

>>> classmates.only('alice').m2ms[1].inv.keys()
['bob', 'carol', 'alice']

Relativity and DataBases
Relativity is excellent at representing many-to-many relationships
from databases which are otherwise awkward to handle.

M2M + ORM
Let’s consider an example from Django to start.
from django.db import models

class Student(models.model):
name = models.StringField()

class Course(models.model):
name = models.StringField()
students = models.ManyToMany(Student)
Students take many courses, and each course has many students.
Construting an M2M over these relationships is very natural:
from relativity import M2M
StudentCourse = Course.students.through

enrollments = M2M(
StudentCourse.objects.all().values_list('student', 'course'))

Design Philosophy

DB Feature Sets
A typical SQL database, such as PostGres, MySQL, SQLServer, Oracle, or DB2
offers many features which can be split into four categories:

relational data model and queries
network protocol and multiple concurrent connections
transactions, atomic updates, and MVCC
persistent storage, backups, and read replicas

Let’s call these “relational”, “network”, “transactional”,
and “persistence” feature sets.

“Alternative” Databases
The most widely used alternative is probably SQLite. SQLite
has relational, transactional, and persistence feature sets but does not have
a network protocol. Instead it must be embedded
as a library inside another application.
Another example is the venerable ZODB. ZODB has
network, transactional, and persistence feature sets
but replaces the relational data model
with an object data model.
As an extreme example of how less can be more, memcached has
only network features. Data is stored ephemerally in the form of opaque blobs without
any data model. There is no atomicity of updates: there is no way to ensure that
two writes either both succeed or both fail.
The so-called “NoSQL” databases (cassandra, couchdb, mongodb, etc)
generally provide network and persistence features but lack a relational data model
and transactionality.

Relativity: Relational à la carte
In this design space, Relativity offers a relational feature set and nothing else.
Relativity allows you to build in-memory data structures that represent relationships
among arbitrary Python objects and then execute queries over those objects and
relationships via a very natural and pythonic API.

SQL
Relativity

result-set
sets and M2Ms

join
chain and attach

order by
sort and sorted

where-clause
list comprehension

Architecture
The fundamental unit of Relativity is the relation, in the form of
the M2M. All other data structures are
various types of M2M containers. An M2M is a very simple
data structure that can be represented as two dicts:
{key: set(vals)}
{val: set(keys)}
The main job of the M2M is to broadcast changes to the
underlying dict and set instances such that they are kept in
sync, and to enumerate all of the key, val pairs.
Similarly, the higher order data structures –
M2MGraph, M2MChain, and M2MStar – broadcast changes to
underlying M2M s and can return and enumerate them.

M2MChain and M2MStar: rows of relations
M2MChain and M2MStar are implemented as thin wrappers over a list
of M2M. The main feature they bring provide “row-iteration”. The difference
between them is how they defined a row. M2MChain represents relationships
that connect end-to-end. M2MStar represents relationships that all
point to the same base object, similar to a star schema.

Shared M2M s
All of the higher order data structures are concerned with the structure
between and among M2M s. The contents within a particular M2M does
not need to maintain any invariants. That is, if all of the M2M s within
one of the higher order data structures were scrambled up, the higher order
data structure would still be valid.
(Contrast with, if you were to scramble
the key sets and val sets around within an M2M, it would be totally
inconsistent.)
This has important consequences, because it means that various instances
of M2MGraph, M2MChain, and M2MStar may share their underlying
M2M s, and continue to update them. This means that all of these higher
order data structures can be treated as cheap and ephemeral.
For example, M2MGraph.chain(*cols) will construct and return a new
M2MChain over the M2M s linking the passed columns. All that
actually happens here is the M2MGraph is queried for the underlying
M2M s, then the list of M2M s is passed to the M2MChain
constructor which simply holds a reference to them.
Another way to think of M2MGraph, M2MChain and M2MStar is
as cheap views over the underlying M2M s. No matter how much data is in
the underlying M2M s, assembling one of these higher order data structures
over top has a fixed, low cost.

Relativity & Python Ecosystem

Pandas
Both Relativity and Pandas enable clean extraction of data from a SQL database
to an in-memory data structure which may be further processed. Both libraries
provide data structures that can easily express queries over the in-memory
data-set that would otherwise be very difficult and tempt a developer to go
back to the database multiple times.
This sounds like Relativity and Pandas should be in competition; but, in practice
they are complementary. Whereas Pandas is excellent at representing tabular
data in rows and columns, Relativity excels at representing the foreign key
relationships that connect rows in different tables. Pandas makes it easy
to take a SQL result set and further refine it by filtering rows and addding
columns. Relativity makes it easy to extract the foreign key relationships
among many tables and further refine them by filtering by connectedness and
adding additional relationships.

When to Use
Use Pandas for doing analysis of data within rows of a table; use
Relativity for doing analysis of the relationships between rows of
different tables.
Coming back to the students-and-classes example:
class Enrollment(models.Model):
student = models.ForeignKey(Student)
class = models.ForeignKey(Class)
grade = models.FloatField() # 0.0 - 5.0

# Pandas is great at determining each students GPA
enrollments_data_frame.group_by(['student']).mean()

Better Together
At a low-level, a Pandas Series and a Relaitivity M2M can
both represent multiple values per key, so it is easy to convert
between the two.
>>> import pandas
>>> import relativity
>>> s = pandas.Series(data=[1, 2, 2], index=['a', 'a', 'b'])
>>> s
a 1
a 2
b 2
dtype: int64
>>> m2m = relativity.M2M(s.items())
>>> m2m
M2M([('a', 1L), ('a', 2L), ('b', 2L)])
>>> keys, vals = zip(*m2m.iteritems())
>>> s2 = pandas.Series(data=vals, index=keys)
>>> s2
a 1
a 2
b 2
dtype: int64

NetworkX
NetworkX is the “graph theory library” of Python:
“NetworkX is a Python package for the creation, manipulation,
and study of the structure, dynamics, and functions of complex networks.”
NetworkX is great at representing arbitrarily connections among a group
of nodes. Relativity has relationship-centric APIs and data-structures,
wehere the M2M represents a single relationship, and M2MChain,
M2MStar, and M2MGraph build higher order connections.
Underneath, both are backed by dict.