Todo

Explain Vietoris-Rips complex.

There is an elementary example in `examples/rips/rips.py`

that computes a
Rips complex of a few points with integer coordinates on a line. It illustrates
the use of Rips complexes and in particular of defining your own notion of a
Distance based on which the Rips complex is constructed.

A more useful example is given in `examples/rips/rips-pairwise.py`

(and
its C++ counterpart in `examples/rips/rips-pairwise.cpp`

). The example
takes on the command line the filename of a file with points in Euclidean space
(one point per line), and a cut off parameters for the skeleton and the
parameter for the Rips complex construction. It then constructs
the Rips complex up to these cutoff parameters, computes its persistence, and
outputs the persistence diagram (one point per line).

```
#/usr/bin/env python
from dionysus import Rips, PairwiseDistances, StaticPersistence, Filtration, points_file, \
ExplicitDistances, data_dim_cmp
from sys import argv, exit
import time
def main(filename, skeleton, max):
points = [p for p in points_file(filename)]
distances = PairwiseDistances(points)
# distances = ExplicitDistances(distances) # speeds up generation of the Rips complex at the expense of memory usage
rips = Rips(distances)
print time.asctime(), "Rips initialized"
simplices = Filtration()
rips.generate(skeleton, max, simplices.append)
print time.asctime(), "Generated complex: %d simplices" % len(simplices)
# While this step is unnecessary (Filtration below can be passed rips.cmp),
# it greatly speeds up the running times
for s in simplices: s.data = rips.eval(s)
print time.asctime(), simplices[0], '...', simplices[-1]
simplices.sort(data_dim_cmp) # could be rips.cmp if s.data for s in simplices is not set
print time.asctime(), "Set up filtration"
p = StaticPersistence(simplices)
print time.asctime(), "Initialized StaticPersistence"
p.pair_simplices()
print time.asctime(), "Simplices paired"
print "Outputting persistence diagram"
smap = p.make_simplex_map(simplices)
for i in p:
if i.sign():
b = smap[i]
if b.dimension() >= skeleton: continue
if i.unpaired():
print b.dimension(), b.data, "inf"
continue
d = smap[i.pair()]
print b.dimension(), b.data, d.data
if __name__ == '__main__':
if len(argv) < 4:
print "Usage: %s POINTS SKELETON MAX" % argv[0]
exit()
filename = argv[1]
skeleton = int(argv[2])
max = float(argv[3])
main(filename, skeleton, max)
```

The bit that sets up the Rips complex is:

```
distances = PairwiseDistances(points)
rips = Rips(distances)
simplices = Filtration()
rips.generate(skeleton, max, simplices.append)
```

The computation of persistence and output of the persistence diagram are the same as in the Alpha shape example. The example also incorporates the Speed-up suggestions given in the Brief Tutorial.

Warning

This section is not finished.

The example given in `examples/rips/rips.cpp`

illustrates how one can use
the library to compute persistence of a Vietoris-Rips complex for a given set of
distances. At the top of the file a struct Distances is defined. The
particular distances in the example are trivial (they are points with integer
coordinates on a real line), however, the struct illustrates the basic
requirements of any such class to be passed to the Rips<Distances> class.

The Rips complex itself is generated in the line:

```
rips.generate(2, 50, make_push_back_functor(complex));
```

which tells it to generate a 2-skeleton of the Rips complex up to distance value of 50, and insert the simplices into the previously defined vector complex.

Subsequent sort is unnecessary since Bron-Kerbosch algorithm that generates the complex will actually generate the simplices in lexicographic order; it’s there for illustration purposes only (the simplices must be sorted lexicographically).

The following “paragraph” sets up the filtration with respect to simplex sizes (specified by Generator::Comparison(distances)), and computes its persistence:

```
// Generate filtration with respect to distance and compute its persistence
Fltr f(complex.begin(), complex.end(), Generator::Comparison(distances));
Persistence p(f);
p.pair_simplices();
```