Complexity
The direct implementation of unit propagation takes time quadratic in the total size of the set to check, which is defined to be the sum of the size of all clauses, where the size of each clause is the number of literals it contains.
Unit propagation can however be done in linear time by storing, for each variable, the list of clauses in which each literal is contained. For example, the set above can be represented by numbering each clause as follows:
and then storing, for each variable, the list of clauses containing the variable or its negation:
This simple data structure can be built in time linear in the size of the set, and allows finding all clauses containing a variable very easily. Unit propagation of a literal can be performed efficiently by scanning only the list of clauses containing the variable of the literal. More precisely, the total running time for doing unit propagation for all unit clauses is linear in the size of the set of clauses.
Read more about this topic: Unit Propagation
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