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Consider a situation where you don't have function to calculate power (pow() function in C) and you need to calculate x^n where x can be any number and n is a positive integer. What can be the best possible time complexity of your power function?
- a)O(n)
- b)O(nLogn)
- c)O(LogLogn)
- d)O(Logn)
Correct answer is option 'D'. Can you explain this answer?
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Consider a situation where you dont have function to calculate power (...
We can calculate power using divide and conquer in O(Logn) time.
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Consider a situation where you dont have function to calculate power (...
Understanding Power Calculation without pow() Function
To calculate x^n efficiently without using the pow() function in C, we can employ a method called "Exponentiation by Squaring." This approach significantly reduces the time complexity of the operation.
Exponentiation by Squaring
- The key idea is to break down the exponentiation process based on whether n is even or odd.
- If n is even, we can express x^n as (x^(n/2))^2.
- If n is odd, we can express it as x * x^(n-1).
Time Complexity Analysis
- Each time we halve the exponent n, reducing the problem's size substantially.
- This leads us to a logarithmic number of operations relative to n.
Steps to Calculate x^n
1. Initialize Result: Start with the result set to 1.
2. Loop Until n is 0:
- If n is even: Update n to n/2 and square x.
- If n is odd: Multiply the result by x, then reduce n by 1.
3. Return Result: Once n is 0, return the accumulated result.
Final Time Complexity
- Each iteration effectively reduces n by half when it is even.
- This results in a maximum of log2(n) iterations, leading to a final time complexity of O(Log n).
In conclusion, using the Exponentiation by Squaring technique allows us to compute x^n with a time complexity of O(Log n), making it the most efficient method for this calculation when n is a positive integer.
To calculate x^n efficiently without using the pow() function in C, we can employ a method called "Exponentiation by Squaring." This approach significantly reduces the time complexity of the operation.
Exponentiation by Squaring
- The key idea is to break down the exponentiation process based on whether n is even or odd.
- If n is even, we can express x^n as (x^(n/2))^2.
- If n is odd, we can express it as x * x^(n-1).
Time Complexity Analysis
- Each time we halve the exponent n, reducing the problem's size substantially.
- This leads us to a logarithmic number of operations relative to n.
Steps to Calculate x^n
1. Initialize Result: Start with the result set to 1.
2. Loop Until n is 0:
- If n is even: Update n to n/2 and square x.
- If n is odd: Multiply the result by x, then reduce n by 1.
3. Return Result: Once n is 0, return the accumulated result.
Final Time Complexity
- Each iteration effectively reduces n by half when it is even.
- This results in a maximum of log2(n) iterations, leading to a final time complexity of O(Log n).
In conclusion, using the Exponentiation by Squaring technique allows us to compute x^n with a time complexity of O(Log n), making it the most efficient method for this calculation when n is a positive integer.
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Consider a situation where you dont have function to calculate power (pow() function in C) and you need to calculate x^n where x can be any number and n is a positive integer. What can be the best possible time complexity of your power function?a)O(n)b)O(nLogn)c)O(LogLogn)d)O(Logn)Correct answer is option 'D'. Can you explain this answer?
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