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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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Verified Answer
Consider a situation where you don't have function to calculate po...
We can calculate power using divide and conquer in O(Logn) time.
Most Upvoted Answer
Consider a situation where you don't have function to calculate po...
Time complexity of the power function
To calculate x^n, we can use the idea of binary exponentiation. Binary exponentiation is an efficient algorithm that reduces the number of multiplications needed to calculate the power of a number.
Binary Exponentiation Algorithm
The binary exponentiation algorithm works as follows:
1. Initialize a variable 'result' to 1.
2. Iterate through the binary representation of n from the least significant bit to the most significant bit.
3. For each bit:
- If the bit is 1, multiply 'result' by x.
- Square x.
4. Return the final value of 'result'.
Time Complexity Analysis
The time complexity of the binary exponentiation algorithm can be analyzed as follows:
- We iterate through the binary representation of n, which takes O(log n) time.
- For each bit, we perform either a multiplication or a squaring operation.
- The number of operations required depends on the number of bits in the binary representation of n.
- The number of bits in the binary representation of n is O(log n).
- Each multiplication or squaring operation can be performed in O(1) time.
Therefore, the overall time complexity of the power function using binary exponentiation is O(log n).
Conclusion
The best possible time complexity of the power function to calculate x^n, where x can be any number and n is a positive integer, is O(log n). This is achieved by using the binary exponentiation algorithm, which reduces the number of multiplications needed by operating on the binary representation of n.
To calculate x^n, we can use the idea of binary exponentiation. Binary exponentiation is an efficient algorithm that reduces the number of multiplications needed to calculate the power of a number.
Binary Exponentiation Algorithm
The binary exponentiation algorithm works as follows:
1. Initialize a variable 'result' to 1.
2. Iterate through the binary representation of n from the least significant bit to the most significant bit.
3. For each bit:
- If the bit is 1, multiply 'result' by x.
- Square x.
4. Return the final value of 'result'.
Time Complexity Analysis
The time complexity of the binary exponentiation algorithm can be analyzed as follows:
- We iterate through the binary representation of n, which takes O(log n) time.
- For each bit, we perform either a multiplication or a squaring operation.
- The number of operations required depends on the number of bits in the binary representation of n.
- The number of bits in the binary representation of n is O(log n).
- Each multiplication or squaring operation can be performed in O(1) time.
Therefore, the overall time complexity of the power function using binary exponentiation is O(log n).
Conclusion
The best possible time complexity of the power function to calculate x^n, where x can be any number and n is a positive integer, is O(log n). This is achieved by using the binary exponentiation algorithm, which reduces the number of multiplications needed by operating on the binary representation of n.
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