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If the expression has k variables and n operator occurrences, what is the running time complexity to check whether the expression is tautology or not?
- a)O(2kn)
- b)O(2nk)
- c)O(2kn)
- d)O(nk)
Correct answer is option 'C'. Can you explain this answer?
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If the expression has k variables and n operator occurrences, what is ...
If the expression has k variables and n operator occurrences, the table has 2k rows, and there are n columns that need to be filled out.
Therefore, the implementation of this algorithm takes O(2kn) time
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If the expression has k variables and n operator occurrences, what is ...
Understanding Tautology Check Complexity
To determine whether an expression with k variables and n operator occurrences is a tautology, we analyze the computational complexity involved.
Definition of Tautology
- A tautology is an expression that evaluates to true for all possible truth assignments of its variables.
Truth Assignments
- For k variables, there are 2^k possible combinations of truth values (true or false). This is because each variable can independently be true or false.
Evaluating the Expression
- Each of the 2^k combinations needs to be evaluated with respect to the n operators in the expression.
- Evaluating the expression for one truth assignment takes O(n) time, as you must process each operator in the expression.
Total Time Complexity
- Since there are 2^k truth assignments and each assignment requires O(n) time to evaluate, the overall complexity becomes:
- O(n) * O(2^k) = O(n * 2^k)
- However, since we often represent the total number of operator occurrences (n) as a factor of the overall evaluation, we can simplify this to:
- O(2^k * n) or O(2^kn)
Conclusion
- Given that the computation involves evaluating all possible combinations of truth assignments and processing each operator, the correct answer for the complexity to check if the expression is a tautology is:
- O(2^kn)
Thus, the correct answer is option 'C', which represents the time complexity accurately.
To determine whether an expression with k variables and n operator occurrences is a tautology, we analyze the computational complexity involved.
Definition of Tautology
- A tautology is an expression that evaluates to true for all possible truth assignments of its variables.
Truth Assignments
- For k variables, there are 2^k possible combinations of truth values (true or false). This is because each variable can independently be true or false.
Evaluating the Expression
- Each of the 2^k combinations needs to be evaluated with respect to the n operators in the expression.
- Evaluating the expression for one truth assignment takes O(n) time, as you must process each operator in the expression.
Total Time Complexity
- Since there are 2^k truth assignments and each assignment requires O(n) time to evaluate, the overall complexity becomes:
- O(n) * O(2^k) = O(n * 2^k)
- However, since we often represent the total number of operator occurrences (n) as a factor of the overall evaluation, we can simplify this to:
- O(2^k * n) or O(2^kn)
Conclusion
- Given that the computation involves evaluating all possible combinations of truth assignments and processing each operator, the correct answer for the complexity to check if the expression is a tautology is:
- O(2^kn)
Thus, the correct answer is option 'C', which represents the time complexity accurately.
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