Information AboutAc-3 |
| CATEGORIES ABOUT AC-3 ALGORITHM | |
| algorithms | |
| constraint satisfaction | |
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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 Problem s (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 s, Variable s, 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''. For illustration, here is an example of a very simple constraint problem: ''X'' (a variable) has the possible values {0, 1, 2, 3, 4, 5} -- the set of these values are the domain of ''X'', or D(''X''). The variable ''Y'' likewise has the domain D(''Y'') = {0, 1, 2, 3, 4, 5}. Together with the constraints ''C1'' = "''X'' must be even" and ''C2'' = "''X'' + ''Y'' must equal 4" we have a CSP which AC-3 can solve. Notice that the actual constraint graph representing this problem must contain two edges between ''X'' and ''Y'' since ''C2'' is undirected but the graph representation being used by AC-3 is directed. It does so by first removing the non-even values out of the domain of ''X'' as required by ''C1'', leaving D(''X'') = { 0, 2, 4 }. It then examines the arcs between ''X'' and ''Y'' implied by ''C2''. Only the pairs (''X''=0, ''Y''=4), (''X''=2, ''Y''=2), and (''X''=4, ''Y''=0) match the constraint ''C2''. AC-3 then terminates, with D(''X'') = {0, 2, 4} and D(''Y'') = {0, 2, 4}. AC-3 is expressed in pseudocode as follows: Input: A set of Variable s X A set of Domain s D(x) for each variable x in X. D(x) contains vx0, vx1... vxn, the possible values of x A set of unary constraints R1(x) on variable x that must be satisfied A set of binary constraints R2(x, y) on variables x and y that must be satisfied Output: Arc consistent domains for each variable. function ac3 (X, D, R1, R2) ''// Initial domains are made consistent with unary constraints.'' for each x '''in''' X | ||
| Worklist : | { (x, y) there exists a relation R2(x, y) or a relation R2(y, x) } |
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| Worklist : | worklist + { (z, x) z != y } |
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