ProgramGuru

DSA Visualizations /

Overview of TechniquesOverview of Techniques14
❯

  1. 1Overview of DSA Problem Solving Techniques
  2. 2Brute Force Technique in DSA
  3. 3Greedy Algorithm Technique in DSA
  4. 4Divide and Conquer Technique in DSA
  5. 5Dynamic Programming Technique in DSA
  6. 6Backtracking Technique in DSA
  7. 7Recursion Technique in DSA
  8. 8Sliding Window Technique in DSA
  9. 9Two Pointers Technique
  10. 10Binary Search Technique
  11. 11Tree / Graph Traversal Technique in DSA
  12. 12Bit Manipulation Technique in DSA
  13. 13Hashing Technique
  14. 14Heaps Technique in DSA

SortingSorting10
❯

  1. 1Bubble Sort
  2. 2Selection Sort
  3. 3Insertion Sort
  4. 4Merge Sort
  5. 5Quick Sort
  6. 6Heap Sort
  7. 7Radix Sort
  8. 8Counting Sort
  9. 9Bucket Sort
  10. 10Shell Sort

ArraysArrays32
❯

  1. 1Find Maximum and Minimum in Array using Loop
  2. 2Find Second Largest in Array
  3. 3Find Second Smallest in Array
  4. 4Reverse Array using Two Pointers
  5. 5Check if Array is Sorted
  6. 6Remove Duplicates from Sorted Array
  7. 7Left Rotate an Array by One Place
  8. 8Left Rotate an Array by K Places
  9. 9Move Zeroes in Array to End
  10. 10Linear Search in Array
  11. 11Union of Two Arrays
  12. 12Find Missing Number in Array
  13. 13Max Consecutive Ones in Array
  14. 14Find Kth Smallest Element
  15. 15Longest Subarray with Given Sum (Positives)
  16. 16Longest Subarray with Given Sum (Positives and Negatives)
  17. 17Find Majority Element in Array (more than n/2 times)
  18. 18Find Majority Element in Array (more than n/3 times)
  19. 19Maximum Subarray Sum using Kadane's Algorithm
  20. 20Print Subarray with Maximum Sum
  21. 21Stock Buy and Sell
  22. 22Rearrange Array Alternating Positive and Negative Elements
  23. 23Next Permutation of Array
  24. 24Leaders in an Array
  25. 25Longest Consecutive Sequence in Array
  26. 26Count Subarrays with Given Sum
  27. 27Sort an Array of 0s, 1s, and 2s
  28. 28Two Sum Problem
  29. 29Three Sum Problem
  30. 304 Sum Problem
  31. 31Find Length of Largest Subarray with 0 Sum
  32. 32Find Maximum Product Subarray

Arrays - Binary SearchArrays - Binary Search28
❯

  1. 1Binary Search in Array<br><span class="highlight">Using Iteration</span>
  2. 2Find Lower Bound in Sorted Array
  3. 3Find Upper Bound in Sorted Array
  4. 4Search Insert Position in Sorted Array (Lower Bound Approach)
  5. 5Floor and Ceil in Sorted Array
  6. 6First Occurrence in a Sorted Array
  7. 7Last Occurrence in a Sorted Array
  8. 8Count Occurrences in Sorted Array
  9. 9Search Element in a Rotated Sorted Array
  10. 10Search in Rotated Sorted Array with Duplicates
  11. 11Minimum in Rotated Sorted Array
  12. 12Find Rotation Count in Sorted Array
  13. 13Search Single Element in Sorted Array
  14. 14Find Peak Element in Array
  15. 15Square Root using Binary Search
  16. 16Nth Root of a Number using Binary Search
  17. 17Koko Eating Bananas
  18. 18Minimum Days to Make M Bouquets
  19. 19Find the Smallest Divisor Given a Threshold
  20. 20Capacity to Ship Packages within D Days
  21. 21Kth Missing Positive Number
  22. 22Aggressive Cows Problem
  23. 23Allocate Minimum Number of Pages
  24. 24Split Array - Minimize Largest Sum
  25. 25Painter's Partition Problem
  26. 26Minimize Maximum Distance Between Gas Stations
  27. 27Median of Two Sorted Arrays of Different Sizes
  28. 28K-th Element of Two Sorted Arrays

StringsStrings2
❯

  1. 1Remove Outermost Parentheses in String
  2. 2Edit Distance Problem

MatrixMatrix3
❯

  1. 1Set Matrix Zeroes
  2. 2Rotate Matrix by 90 Degrees Clockwise
  3. 3Print Matrix in Spiral Manner

Matrix - Binary SearchMatrix - Binary Search5
❯

  1. 1Row with Maximum Number of 1's
  2. 2Search in a Sorted 2D Matrix
  3. 3Search in Row and Column-wise Sorted Matrix
  4. 4Find Target in 2D Matrix
  5. 5Matrix Median

Binary TreesBinary Trees36
❯

  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Postorder Traversal of a Binary Tree Using Recursion
  4. 4Postorder Traversal of a Binary Tree using Iteration
  5. 5Level Order Traversal of a Binary Tree using Recursion
  6. 6Level Order Traversal of a Binary Tree using Iteration
  7. 7Reverse Level Order Traversal of a Binary Tree using Iteration
  8. 8Reverse Level Order Traversal of a Binary Tree using Recursion
  9. 9Find Height of a Binary Tree
  10. 10Find Diameter of a Binary Tree
  11. 11Find Mirror of a Binary Tree - Todo
  12. 12Inorder Traversal of a Binary Tree using Recursion
  13. 13Inorder Traversal of a Binary Tree using Iteration
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree

Binary Search TreesBinary Search Trees14
❯

  1. 1Find a Value in a Binary Search Tree
  2. 2Delete a Node in a Binary Search Tree
  3. 3Find the Minimum Value in a Binary Search Tree
  4. 4Find the Maximum Value in a Binary Search Tree
  5. 5Find the Inorder Successor in a Binary Search Tree
  6. 6Find the Inorder Predecessor in a Binary Search Tree
  7. 7Check if a Binary Tree is a Binary Search Tree
  8. 8Find the Lowest Common Ancestor of Two Nodes in a Binary Search Tree
  9. 9Convert a Binary Tree into a Binary Search Tree
  10. 10Balance a Binary Search Tree
  11. 11Merge Two Binary Search Trees
  12. 12Find the kth Largest Element in a Binary Search Tree
  13. 13Find the kth Smallest Element in a Binary Search Tree
  14. 14Flatten a Binary Search Tree into a Sorted List

Find the Nth Root of a Number
Using Binary Search

Topic Contents

ProblemExamplesSolutionVisualizationCodeTime ComplexitySpace Complexity

Problem Statement

Given a number x and a positive integer n, your task is to find the nth root of x. That is, find a real number r such that rn = x.

You should return the value of r up to a certain precision (for example, 6 decimal places).

Examples❯

Value (x)Root (n)OutputDescription
2733.00000033 = 27, so cube root of 27 is 3
1642.00000024 = 16, so 4th root is 2
1032.154435Cube root of 10 is a real number between 2 and 3
151.0000001 raised to any power is 1
030.0000000 raised to any positive power is 0
717.0000001st root of any number is the number itself
00-1Invalid input: root 0 is not defined
-83-2.000000Valid: cube root of negative number is negative
-82-1Invalid: even root of negative number
-1Empty input is invalid

Solution❯

To find the nth root of a number x, we can use a method called binary search, which efficiently narrows down the possible answer by halving the search range each time.

Understanding Different Cases

Before applying the binary search, we handle a few special cases:

  • If x = 0: The result is 0, because 0 raised to any positive power is 0.
  • If x = 1: The result is always 1, regardless of the value of n.
  • If n = 1: The answer is simply x itself.
  • If x is negative and n is even: The answer is undefined in real numbers, so we return -1.
  • If x is negative and n is odd: The result exists and will also be negative.
  • If n ≤ 0: The root is undefined or non-standard, so we return -1.

How Binary Search Helps

We know that the nth root of x lies somewhere between 0 and x (or 0 and 1 if x < 1). We set low = 0 and high = max(1, x) and repeatedly do the following:

  • Find the middle point mid.
  • Raise mid to the power n.
  • Compare it with x:
    • If midn < x, we shift our search range to the right (low = mid).
    • If midn > x, we shift the range to the left (high = mid).
    • If it's close enough to x (say, within 1e-6), we consider that our answer.

Why It Works

This approach works well because raising a number to a power is continuous, so the value of midn increases as mid increases. By halving the range at each step, we find the accurate root efficiently with a time complexity of O(log x).

This method avoids brute-force checking or floating-point rounding issues and works even for large numbers or high precision.

Visualization

Algorithm Steps

  1. Set the search range from low = 0 to high = x (or 1, whichever is higher).
  2. Run a binary search with precision (e.g., 1e-6).
  3. At each step, calculate mid and raise it to the power n.
  4. If mid^n is close enough to x, return mid.
  5. If mid^n is less than x, search in the right half.
  6. If mid^n is greater than x, search in the left half.
  7. Repeat until the desired precision is achieved.

Code

Python
Java
JavaScript
C
C++
C#
Kotlin
Swift
Go
Php
def nth_root(x, n):
    if x == 0:
        return 0.0

    low = 0.0
    high = max(1.0, x)
    eps = 1e-6  # Precision threshold

    while (high - low) > eps:
        mid = (low + high) / 2.0
        if mid ** n < x:
            low = mid  # Mid is too small
        else:
            high = mid  # Mid is too big or correct

    return round((low + high) / 2.0, 6)

x = 27
n = 3
print("Nth root is:", nth_root(x, n))

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(1)If the root is a perfect integer like sqrt(1), it returns quickly.
Average CaseO(log x)Each binary search iteration reduces the range of possible values by half.
Average CaseO(log x)Precision-based binary search continues until the result is within the specified error margin.

Space Complexity

O(1)

Explanation: The algorithm uses only a few variables and no extra memory for computation.

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