Recursive algorithm for the (N-1)!! perfect matchings of Kn and incomplete listings for large N.
- Download the project files
- Run INSTALL_PM.m in the MATLAB Command Window (automatically adds project files to your MATLAB path and opens an example)
INSTALL_PM- See PM_verify.m for examples
open PM_verifyThis submission provides an efficient recursive algorithm for enumeration the (N-1)!! perfect matchings (or 1-factors) of complete graph (Kn). Perfect matchings are only possible on graphs with an even number of vertices so N must be even. The algorithm leverages previously found perfect matchings for smaller complete graphs to build up the perfect matchings for a larger Kn. Please see the image above for a visualization of the perfect matchings for K6.
The current practical limit on N is 20 since N=22 would require 210.8 Gb of RAM (see table below). Perhaps a more memory efficient implementation (noting that the uint8 class is used already) would allow for K22 since the predicted MAT file size of the result is 5217 Mb.
N (N-1)!! CPU Time Max Memory MAT Size
8 105 0.00041 - -
10 945 0.00067 - -
12 10395 0.00130 - -
14 135135 0.00669 2.1 Mb 328.0 Kb
16 2027025 0.16698 38.3 Mb 968.0 Kb
18 34459425 3.32852 0.6 Gb 17.3 Mb
20 654729075 80.39450 13.5 Gb 346.9 Mb
22 13749310575 - 210.8 Gb* 5217.6 Mb* *predicted size
A secondary function is PM_index2pm which allows for a list of numbers between 1 and (N-1)!! and returns the perfect matching entries that would occur in the matrix result from PM_perfectMatchings. This allows for perfect matching entries for N > 20 but a complete list still impractical due to the aforementioned limitations.
The inverse function of PM_index2pm is provided with PM_pm2index. Given a row vector that represents a perfect matching, return the corresponding perfect matching number in the enumerated list.
- Reference on perfect matchings: [link]
- Mathematica code-based discussion (my approach is much more efficient since it is not using any filtering, Michael Hale's answer also uses recursion) [link]
- Discussion on the number of perfect matchings in complete graphs [link]
- Double factorial function [link] (see first application) and [link]