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The algorithm is non-local in the sense that in a single sweep of moves a collective update of the spin variables of the system is done. The key idea is to take an additional number of 'bond' variables, as suggested by Fortuin and Kasteleyn, who mapped the Potts model onto a [[Percolation theory|percolation]] model via the [[random cluster model]].
* Starting from a given configuration of spins, we associate to each pair of nearest neighbours on sites <math>n,m</math> a random variable <math>b_{n,m}\in \lbrace 0,1\rbrace</math> which is interpreted in the following way: if <math>b_{n,m}=0</math> there is no link between the sites <math>n</math> and <math>m</math>; if <math>b_{n,m}=1</math>, <math>\sigma_n</math> and <math>\sigma_m</math> are connected. These values are assigned according to the following (conditional) probability distribution:
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