Compressed Sparse Graph Routines (`scipy.sparse.csgraph`)
=========================================================

.. sectionauthor:: Jake Vanderplas <vanderplas@astro.washington.edu>

.. currentmodule:: scipy.sparse.csgraph


Example: Word Ladders
---------------------

A `Word Ladder <https://en.wikipedia.org/wiki/Word_ladder>`_ is a word game
invented by Lewis Carroll, in which players find paths between words by
switching one letter at a time. For example, one can link "ape" and "man"
in the following way:

.. math::
   {\rm ape \to apt \to ait \to bit \to big \to bag \to mag \to man}

Note that each step involves changing just one letter of the word. This is
just one possible path from "ape" to "man", but is it the shortest possible
path? If we desire to find the shortest word-ladder path between two given
words, the sparse graph submodule can help.

First, we need a list of valid words. Many operating systems have such a list
built in. For example, on linux, a word list can often be found at one of the
following locations::

    /usr/share/dict
    /var/lib/dict

Another easy source for words are the Scrabble word lists available at various
sites around the internet (search with your favorite search engine). We'll
first create this list. The system word lists consist of a file with one
word per line. The following should be modified to use the particular word
list you have available::

    >>> word_list = open('/usr/share/dict/words').readlines()
    >>> word_list = map(str.strip, word_list)

We want to look at words of length 3, so let's select just those words of the
correct length. We'll also eliminate words which start with upper-case
(proper nouns) or contain non-alphanumeric characters, like apostrophes and
hyphens. Finally, we'll make sure everything is lower-case for comparison
later::

    >>> word_list = [word for word in word_list if len(word) == 3]
    >>> word_list = [word for word in word_list if word[0].islower()]
    >>> word_list = [word for word in word_list if word.isalpha()]
    >>> word_list = list(map(str.lower, word_list))
    >>> len(word_list)
    586    # may vary

Now we have a list of 586 valid three-letter words (the exact number may
change depending on the particular list used). Each of these words will
become a node in our graph, and we will create edges connecting the nodes
associated with each pair of words which differs by only one letter.

There are efficient ways to do this, and inefficient ways to do this. To
do this as efficiently as possible, we're going to use some sophisticated
numpy array manipulation:

    >>> import numpy as np
    >>> word_list = np.asarray(word_list)
    >>> word_list.dtype   # these are unicode characters in Python 3
    dtype('<U3')
    >>> word_list.sort()  # sort for quick searching later

We have an array where each entry is three unicode characters long. We'd like
to find all pairs where exactly one character is different. We'll start by
converting each word to a 3-D vector:

    >>> word_bytes = np.ndarray((word_list.size, word_list.itemsize),
    ...                         dtype='uint8',
    ...                         buffer=word_list.data)
    >>> # each unicode character is four bytes long. We only need first byte
    >>> # we know that there are three characters in each word
    >>> word_bytes = word_bytes[:, ::word_list.itemsize//3]
    >>> word_bytes.shape
    (586, 3)    # may vary

Now, we'll use the
`Hamming distance <https://en.wikipedia.org/wiki/Hamming_distance>`_
between each point to determine which pairs of words are connected.
The Hamming distance measures the fraction of entries between two vectors
which differ: any two words with a Hamming distance equal to :math:`1/N`,
where :math:`N` is the number of letters, are connected in the word ladder::

    >>> from scipy.spatial.distance import pdist, squareform
    >>> from scipy.sparse import csr_matrix
    >>> hamming_dist = pdist(word_bytes, metric='hamming')
    >>> # there are three characters in each word
    >>> graph = csr_matrix(squareform(hamming_dist < 1.5 / 3))

When comparing the distances, we don't use an equality because this can be
unstable for floating point values. The inequality produces the desired
result, as long as no two entries of the word list are identical. Now, that our
graph is set up, we'll use a shortest path search to find the path between
any two words in the graph::

    >>> i1 = word_list.searchsorted('ape')
    >>> i2 = word_list.searchsorted('man')
    >>> word_list[i1]
    'ape'
    >>> word_list[i2]
    'man'

We need to check that these match, because if the words are not in the list,
that will not be the case. Now, all we need is to find the shortest path
between these two indices in the graph. We'll use
`Dijkstra's algorithm <https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm>`_,
because it allows us to find the path for just one node::

    >>> from scipy.sparse.csgraph import dijkstra
    >>> distances, predecessors = dijkstra(graph, indices=i1,
    ...                                    return_predecessors=True)
    >>> print(distances[i2])
    5.0    # may vary

So we see that the shortest path between "ape" and "man" contains only
five steps. We can use the predecessors returned by the algorithm to
reconstruct this path::

    >>> path = []
    >>> i = i2
    >>> while i != i1:
    ...     path.append(word_list[i])
    ...     i = predecessors[i]
    >>> path.append(word_list[i1])
    >>> print(path[::-1])
    ['ape', 'apt', 'opt', 'oat', 'mat', 'man']    # may vary

This is three fewer links than our initial example: the path from "ape" to "man"
is only five steps.

Using other tools in the module, we can answer other questions. For example,
are there three-letter words which are not linked in a word ladder? This
is a question of connected components in the graph::

    >>> from scipy.sparse.csgraph import connected_components
    >>> N_components, component_list = connected_components(graph)
    >>> print(N_components)
    15    # may vary

In this particular sample of three-letter words, there are 15 connected
components: that is, 15 distinct sets of words with no paths between the
sets. How many words are there in each of these sets? We can learn this from
the list of components::

    >>> [np.sum(component_list == i) for i in range(N_components)]
    [571, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]    # may vary

There is one large connected set and 14 smaller ones. Let's look at the
words in the smaller ones::

    >>> [list(word_list[np.nonzero(component_list == i)]) for i in range(1, N_components)]
    [['aha'],    # may vary
     ['chi'],
     ['ebb'],
     ['ems', 'emu'],
     ['gnu'],
     ['ism'],
     ['khz'],
     ['nth'],
     ['ova'],
     ['qua'],
     ['ugh'],
     ['ups'],
     ['urn'],
     ['use']]

These are all the three-letter words which do not connect to others via a word
ladder.

We might also be curious about which words are maximally separated. Which
two words take the most links to connect? We can determine this by computing
the matrix of all shortest paths. Note that, by convention, the
distance between two non-connected points is reported to be infinity, so
we'll need to remove these before finding the maximum::

    >>> distances, predecessors = dijkstra(graph, return_predecessors=True)
    >>> max_distance = np.max(distances[~np.isinf(distances)])
    >>> print(max_distance)
    13.0    # may vary

So, there is at least one pair of words which takes 13 steps to get from one
to the other! Let's determine which these are::

    >>> i1, i2 = np.nonzero(distances == max_distance)
    >>> list(zip(word_list[i1], word_list[i2]))
    [('imp', 'ohm'),    # may vary
     ('imp', 'ohs'),
     ('ohm', 'imp'),
     ('ohm', 'ump'),
     ('ohs', 'imp'),
     ('ohs', 'ump'),
     ('ump', 'ohm'),
     ('ump', 'ohs')]

We see that there are two pairs of words which are maximally separated from
each other: 'imp' and 'ump' on the one hand, and 'ohm' and 'ohs' on the other.
We can find the connecting list in the same way as above::

    >>> path = []
    >>> i = i2[0]
    >>> while i != i1[0]:
    ...     path.append(word_list[i])
    ...     i = predecessors[i1[0], i]
    >>> path.append(word_list[i1[0]])
    >>> print(path[::-1])
    ['imp', 'amp', 'asp', 'ass', 'ads', 'add', 'aid', 'mid', 'mod', 'moo', 'too', 'tho', 'oho', 'ohm']    # may vary

This gives us the path we desired to see.

Word ladders are just one potential application of scipy's fast graph
algorithms for sparse matrices. Graph theory makes appearances in many
areas of mathematics, data analysis, and machine learning. The sparse graph
tools are flexible enough to handle many of these situations.
