#CRA

This program implements the matrix reduction algorithm for computing the persistence of a complex, in two variants: regular and exhaustive.

#Notes on the algorithm

The implementation of the algorithms is not really aligned all that well with the common descriptions of it, namely as a matrix reduction algorithm. Instead, we think of the algorithm as working with sets, which corresponds to the boundary of simplices, or matrix columns in the traditional description. Modulo 2 addition of matrix columns becomes symmetric difference of the sets.

Here's pseudocode that reflects how the algorithms are implemented.

reduced = [0; length(simplices)]
for s in simplices
	if s is empty
		continue
	end
	m = max(s)
	if reduced[m] is empty
		reduced[m] = simplex
		continue
	else
		s = symdiff(s, reduced[m])
	end
end

#Usage

Run with --help to get information on the arguments the program takes.

$ cargo run -q -- -help

or

$ # this once:
$ cargo build --release
$ ./target/release/cra --help

This will run with debug output, which right now prints a comma separated table of which simplices are added to which. To display this nicely in the terminal, you can pipe it to column as follows (only tested on Linux):

cargo run -- --boundary data/tet.boundary -d | column -ts';'
j   faces             adds          reduced
0   []                []            [0]
0   [0]               []            [0]
1   [0]               [0]           []
2   [0]               [0]           []
3   [0]               [0]           []
5   [1, 2]            []            [1, 2]
6   [1, 3]            []            [1, 3]
7   [1, 4]            []            [1, 4]
8   [2, 3]            [6, 5]        []
9   [2, 4]            [7, 5]        []
10  [3, 4]            [7, 6]        []
11  [5, 6, 8]         []            [5, 6, 8]
12  [5, 7, 9]         []            [5, 7, 9]
13  [6, 7, 10]        []            [6, 7, 10]
14  [8, 9, 10]        [13, 12, 11]  []
15  [11, 12, 13, 14]  []            [11, 12, 13, 14]

#Statistics

By running CRA with --stats=<filename> (set filename to - to write to stdout), you get a bunch of statistics in json. This can be piped into jq to format, or filter out things, like so:

$ cra --boundary data/tet.boundary --stats=- | jq

Alternatively, you can save it in a file and run jq afterwards:

$ cra --boundary data/tet.bondary --stats=tet.json
$ jq '.histogram_add_cost_by_dim' tet.json
{
  "1": {
    "2": 3,
    "4": 3
  },
  "0": {
    "1": 3
  },
  "2": {
    "7": 1,
    "6": 1,
    "3": 1
  }
}

Use jq 'keys' to list the keys. You probably also want to run jq with -S, which sorts the keys.

#Example

You want to show the average sizes of the two simplices added together, by dimension:

cra --boundary data/tet.boundary --stats=- | jq -S '.avg_add_size_by_dim'