Partially ordered sort

## Project description

An algorithm to sort a collection by a sequence of comparators.

## Comparators

A comparator takes two elements a and b and returns

0 - if they are equal a positive number if a > b a negative number if a < b

For example the natural comparator on the integers is

def small_first(a, b):

return a - b

The Python sorted function takes a comparator function as a keyword input

>>> sorted([1, 5, 2, 4, 3], cmp=small_first)
[1, 2, 3, 4, 5]


We may choose to sort by different criteria. For example we might choose to prefer even numbers

def evens_first(a, b):

return a%2 - b%2

>>> sorted([1, 5, 2, 4, 3], cmp=evens_first)
[4, 2, 5, 1, 3]


Note in the example above that all of even numbers precede all of the odd numbers; evens_first is satisfied. Also note that there are several possible solutions. In this case there are 12 possible solutions that all satisfy the comparator function. sorted returns one at random.

## posort

To obtain the solution [2, 4, 1, 3, 5] where numbers are sorted first by evenness and then by magnitude we can use the function posort.

The posort function attempts to satisfy a sequence of comparator functions of decreasing precedence.

>>> posort([1, 5, 2, 4, 3], evens_first, small_first)
[2, 4, 1, 3, 5]


The rule for even numbers dominates the rule for small numbers. Any number of comparator functions can be specified.

## Equivalent structures

A comparator function completely specifies a [partial order](http://en.wikipedia.org/wiki/Partial_order#Formal_definition).

A comparator function can describe any [directed acyclic graph (DAG)](http://en.wikipedia.org/wiki/Directed_acyclic_graph)

## Author

[Matthew Rocklin](http://matthewrocklin.com)

## History

This idea was originally developed for the [Theano project](http://github.com/theano/theano)

## Project details

This version 0.1

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