One factorization

one_factorization
Computes a 1-factorization of a list of objects
Other toolboxes required	none
Related functions	perfect_matchings
Function category	Helper functions

one_factorization is a function that returns a 1-factorization of a given list of objects. That is, for a list of $n$ objects, it returns $n-1$ ways of pairing up all of the objects such that each pair of objects occurs exactly once (and thus is essentially a 1-factorization of the complete graph on $n$ vertices).

Alternatively, this function can be thought of as scheduling a round-robin tournament: given an even number $n$ of players, it shows how each player can play against every other player by pairs of players playing $n/2$ games concurrently $n-1$ times (which is optimal).

Syntax

• FAC = one_factorization(N)

Argument descriptions

• N: Either an even integer, indicating that you would like a 1-factorization of the integers $1, 2, \ldots, N$, or a vector containing an even number of distinct entries, indicating that you would like a 1-factorization of those entries.

Examples

A 1-factorization of six objects

The following code generates a 1-factorization of the numbers $1,2,3,4,5,6$:

>> one_factorization(6)
 
ans =
 
     1     6     5     2     4     3
     2     6     1     3     5     4
     3     6     2     4     1     5
     4     6     3     5     2     1
     5     6     4     1     3     2

Each row of the output is a single perfect matching (i.e., a way of pairing up the N objects), which are read "naively" left-to-right. For example, the first row of the output above indicates the following pairing: $\{\{1,6\},\{5,2\},\{4,3\}\}$. The other rows are similar, and each pair occurs exactly once in the entire matrix (e.g., the pair $\{1,3\}$ occurs only in the 2nd row).

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  ONE_FACTORIZATION    Computes a 1-factorization of a list of objects

2. %   This function has one required argument:

3. %     N: either an even natural number (the number of objects to be

4. %        matched) or a vector containing an even number of distinct objects

5. %        to be matched

6. %

7. %   FAC = one_factorization(N) is a matrix with each row corresponding to a

8. %   perfect matching of N objects (or, if N is a vector, each row of PM is

9. %   a pefect matching of the entries of N). Each perfect matching is read

10. %   "naively": for each j, PM(j,1) is matched with PM(j,2), PM(j,3) is

11. %   matched with PM(j,4), and so on. The perfect matchings that are

12. %   returned have the property that every pair of objects is included in

13. %   exactly one of the perfect matchings (i.e., these perfect matchings

14. %   together make up a 1-factorization of the complete graph).

15. %

16. %   URL: http://www.qetlab.com/one_factorization

18. %   requires: nothing

19. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

20. %   package: QETLAB

21. %   last updated: November 6, 2014

23. function fac = one_factorization(n)

25. if(length(n) == 1)

26.     n = 1:n;

27. end

28. sz = length(n);

30. % If we were given an odd number of objects, no 1-factorization exists.

31. if(mod(sz,2) == 1)

32.     error('one_factorization:DoesNotExist','There is no 1-factorization of an odd number of objects.');

33. end

35. fac = zeros(sz-1,sz);

36. for j=1:sz-1

37.     fac(j,[1,2]) = [n(j),n(sz)];

38.     for k=2:(sz/2)

39.         fac(j,[2*k-1,2*k]) = [n(mod(j-(k-1)-1,sz-1)+1),n(mod(j+(k-1)-1,sz-1)+1)];

40.     end

41. end