How to Find the Nth Root of an Integer Using Binary Search

TL;DR

To find the nth root of an integer M using binary search, define the search range between 1 and M. Check the midpoint raised to the power of n: if it matches M, return the midpoint; if it's greater, search the left range; if less, search the right range. The optimized solution has a time complexity of O(log(M) * log(N)).

Transcript

hey everyone welcome back to the channel I hope you guys are doing extremely well so we will be continuing with our binary search playlist which is the part of the Strivers A to Z DSA course in case you haven't checked it out yet there's a link in the description you can definitely check it out so today's problem will be find the nth root of a give... Read More

Key Insights

• 💭 Linear search is a simple but inefficient solution for finding the nth root of a given integer.
• 👨‍🔬 Binary search allows for faster elimination of search space.
• 👨‍🔬 The time complexity of binary search on answers is logarithmic, which is more efficient than linear search.
• 👨‍🔬 Binary search can handle cases where the target integer is not possible to find the root for.

Questions & Answers

Q: What is the problem being discussed?

The problem is to find the nth root of a given integer.

Q: How does linear search solve the problem?

Linear search involves trying each number from 1 to M and checking if the number raised to the power of n is equal to the target integer. Time complexity: O(M).

Q: What is the time complexity of the optimized solution?

The optimized solution using binary search has a time complexity of O(log(M)*log(N)), where M is the given integer and N is the root to be found.

Q: How does binary search optimize the solution?

Binary search allows for faster elimination of search space by comparing the midpoint raised to the power of n with the target integer. This helps in reducing the time complexity compared to linear search.

Q: What is the range of the search space in binary search?

The search space range is defined from 1 to M, where M is the given integer.

Q: What are the steps involved in binary search on answers?

The steps are as follows:

1. Define the range of the search space and assign low and high values.
2. Compute the midpoint and check if it is a possible answer by raising it to the power of n.
3. Depending on the result, eliminate the right or left portion of the search space and update low or high accordingly.
4. Repeat steps 2 and 3 until the search space is reduced to a single number or the target is found.

Q: What happens if the target integer is not possible to find the root for?

In such cases, the binary search will return -1 to indicate that there is no nth root for the given integer.

Q: What is the main advantage of binary search on answers over linear search?

The main advantage is that binary search eliminates portions of the search space faster, resulting in a more efficient solution with a lower time complexity.

Key Insights:

• Linear search is a simple but inefficient solution for finding the nth root of a given integer.
• Binary search allows for faster elimination of search space.
• The time complexity of binary search on answers is logarithmic, which is more efficient than linear search.
• Binary search can handle cases where the target integer is not possible to find the root for.
• Dealing with overflow during the computation of powers is important in binary search on answers.

Summary & Key Takeaways

• The problem is to find the nth root of a given integer.

• Linear search can be used to solve the problem, but it has a time complexity of O(M).

• An optimized solution can be achieved using binary search on the possible answers.

• Binary search allows for faster elimination of search space by comparing the midpoint raised to the power of n with the target integer.

• If the midpoint is greater than the target integer, the right side of the search space can be eliminated, and vice versa.

• The time complexity of the optimized solution is O(log(M)*log(N)).