BS-11. Find the Nth root of an Integer

Summary

This video teaches how to find the integer nth root of a number using “binary search on answers.” Starting with simple examples (third root of 27 = 3; fourth root of 69 doesn’t exist → -1), the instructor contrasts a naive linear search (O(M)) with a far more efficient binary-search approach over the answer range [1, M]. He explains mid selection, moving low/high based on comparisons, and crucial overflow handling: stop multiplying mid^n as soon as it exceeds M. The video covers implementation details, time complexity improvements, edge cases, and practical tips for reliable code.

Takeaways

• 😀 The problem involves finding the nth root of a given integer M, for example, the 3rd root of 27 is 3.
• 😀 A brute force solution involves trying every integer starting from 1 and checking if multiplying it by itself n times equals M.
• 😀 The brute force approach has a time complexity of O(M), where M is the number for which we're finding the nth root.
• 😀 A more efficient approach uses binary search to narrow down the search space for possible answers, reducing the complexity to O(log M).
• 😀 The binary search approach involves defining the search range between 1 and M, and adjusting it based on whether mid^n is less than, equal to, or greater than M.
• 😀 When applying binary search, the mid value is calculated, and the nth power is computed iteratively to check if it matches M.
• 😀 The optimal time complexity for the binary search solution is O(n * log M), where n is the number of multiplications required to compute mid^n.
• 😀 Edge cases, such as overflow when calculating large powers, are handled by detecting if the result exceeds M and adjusting the search accordingly.
• 😀 If the nth root exists, the binary search returns the exact root; otherwise, it returns -1 when no valid root is found.
• 😀 The code implementation uses a helper function for power calculation, which ensures early termination if the result exceeds M, preventing overflow errors.
• 😀 Binary search on answers helps optimize the solution by eliminating unnecessary checks on numbers that will not result in the desired nth root.

Q & A

What is the problem described in the video?

The problem is to find the nth root of a given integer M. For example, if M = 27 and n = 3, the goal is to find the value of x such that x^n = M.

What is the brute force approach for solving this problem?

The brute force approach involves trying all values starting from 1 up to M. For each value, multiply it by itself n times and check if it equals M. If found, return the value; otherwise, return -1 when no match is found.

What is the time complexity of the brute force solution?

The time complexity of the brute force solution is O(M), where M is the given number. In the worst case, the loop will run M times.

How can we optimize the brute force approach?

We can optimize the solution by using binary search on the possible values of the nth root. By narrowing the search space based on whether the current mid value is smaller or greater than the target M, we can reduce the time complexity.

Why is binary search a suitable method for this problem?

Binary search is suitable because we can confidently reduce the search space by checking whether the nth power of the mid value is smaller or greater than M. If the value is too small, we can search the higher half; if it's too large, we search the lower half.

How does binary search improve the time complexity compared to brute force?

Binary search reduces the search space in half with each iteration. The time complexity of binary search is O(log M), which is significantly faster than the brute force approach, which has a time complexity of O(M).

What is the first step when applying binary search on the nth root problem?

The first step is to define the range for the possible root values, setting the low boundary as 1 and the high boundary as M, and then applying binary search within that range.

What happens if the value of mid^n exceeds M in the binary search?

If mid^n exceeds M, we know that the nth root must be smaller, so we adjust the high boundary to mid - 1 and continue the search in the lower half.

What should be returned if no exact nth root is found during the binary search?

If no exact nth root is found, the function should return -1, indicating that no integer nth root exists for the given number M.

How does the video handle edge cases such as large numbers or potential overflows?

The video discusses potential overflow when calculating mid^n for very large numbers. Instead of directly calculating the power, the approach multiplies incrementally while checking if the value exceeds M. If it does, the calculation stops, and the search space is adjusted accordingly.