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| In [[constraint satisfaction]], '''backmarking''' is a variant of the [[backtracking]] algorithm.
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| Backmarking works like backtracking by iteratively evaluating variables in a given order, for example, <math>x_1,\ldots,x_n</math>. It improves over backtracking by maintaining information about the last time a variable <math>x_i</math> was instantiated to a value and information about what changed since then. In particular:
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| [[Image:Backmarking-1.png|thumb|180px|An example, in which search has reached xi=d the first time.]]
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| # for each variable <math>x_i</math> and value <math>a</math>, the algorithm records information about the last time <math>x_i</math> has been set to <math>a</math>; in particular, it stores the minimal index <math>j<i</math> such that the assignment to <math>x_1,\ldots,x_j,x_i</math> was then [[inconsistent]];
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| # for each variable <math>x_i</math>, the algorithm stores some information relative to what changed since the last time it has evaluated <math>x_i</math>; in particular, it stores the minimal index <math>k<i</math> of a variable that was changed since then.
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| The first information is collected and stored every time the algorithm evaluates a variable <math>x_i</math> to <math>a</math>, and is done by simply checking consistency of the current assignments for <math>x_1,x_i</math>, for <math>x_1,x_2,x_i</math>, for <math>x_1,x_2,x_3,x_i</math>, etc.
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| [[Image:Backmarking-2.png|thumb|180px|When search reaches xi=d for the second time, part of the path is the same as the first time.]]
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| The second information is changed every time ''another'' variable is evaluated. In particular, the index of the "maximal unchanged variable since the last evaluation of <math>x_i</math>" is possibly changed every time another variable <math>x_j</math> changes value. Every time an arbitrary variable <math>x_j</math> changes, all variables <math>x_i</math> with <math>i>j</math> are considered in turn. If <math>k</math> was their previous associated index, this value is changed to <math>min(k,j)</math>.
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| The data collected this way is used to avoid some consistency checks. In particular, whenever backtracking would set <math>x_i=a</math>, backmarking compares the two indexes relative to <math>x_i</math> and the pair <math>x_i=a</math>. Two conditions allow to determine partial consistency or inconsistency without checking with the constraints. If <math>k</math> is the minimal index of a variable that changed since the last time <math>x_i</math> was evaluated and <math>j</math> is the minimal index such that the evaluation of <math>x_1,\ldots,x_j,x_i</math> was consistent the last time <math>x_i</math> has been evaluated to <math>a</math>, then:
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| # if <math>j<k</math>, the evaluation of <math>x_1,\ldots,x_j,x_i</math> is still inconsistent as it was before, as none of these variables changed so far; as a result, no further consistency check is necessary;
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| # if <math>j \geq k</math>, the evaluation <math>x_1,\ldots,x_k,x_i</math> is still consistent as it was before; this allows for skipping some consistency checks, but the assignment <math>x_1,\ldots,x_i</math> may still be inconsistent.
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| Contrary to other variants to backtracking, backmarking does not reduce the search space but only possibly reduce the number of constraints that are satisfied by a partial solution.
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| ==References==
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| *{{cite book
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| | first=Rina
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| | last=Dechter
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| | title=Constraint Processing
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| | publisher=Morgan Kaufmann
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| | year=2003
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| | url=http://www.ics.uci.edu/~dechter/books/index.html
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| | isbn=1-55860-890-7
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| }}
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| [[Category:Constraint programming]]
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