AC-3 algorithm

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The AC-3 algorithm (short for Arc Consistency Algorithm #3) is one of a series of algorithms used for the solution of constraint satisfaction problems (or CSP's). It was developed by Alan Mackworth in 1977. The earlier AC algorithms are often considered too inefficient, and many of the later ones are difficult to implement, and so AC-3 is the one most often taught and used in very simple constraint solvers.

The algorithm

AC-3 operates on constraint, variables, and the variables' domains (scopes). A variable can take any of several discrete values; the set of values for a particular variable is known as its domain. A constraint is a relation that limits or constrains the values a variable may have. The constraint may involve the values of other variables.

The CSP can be viewed as a directed graph, where the nodes are the variables of the problem, with edges or arcs between variables that are related by constraints. AC-3 proceeds by examining the arcs between pairs of variables (x, y). It removes those values from the domains of x and y which aren't consistent with the constraints between x and y. The algorithm keeps a collection of arcs that are yet to be checked; when the domain of a variable has any values removed, all the arcs of constraints including that variable (except the current one) are added to the collection. Since the domains of the variables are finite and either one arc or one value are removed at each step, this algorithm is guaranteed to terminate.

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