Yandex

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

GraphsGraphs46

  1. 1Breadth-First Search in Graphs
  2. 2Depth-First Search in Graphs
  3. 3Number of Provinces in an Undirected Graph
  4. 4Connected Components in a Matrix
  5. 5Rotten Oranges Problem - BFS in Matrix
  6. 6Flood Fill Algorithm - Graph Based
  7. 7Detect Cycle in an Undirected Graph using DFS
  8. 8Detect Cycle in an Undirected Graph using BFS
  9. 9Distance of Nearest Cell Having 1 - Grid BFS
  10. 10Surrounded Regions in Matrix using Graph Traversal
  11. 11Number of Enclaves in Grid
  12. 12Word Ladder - Shortest Transformation using Graph
  13. 13Word Ladder II - All Shortest Transformation Sequences
  14. 14Number of Distinct Islands using DFS
  15. 15Check if a Graph is Bipartite using DFS
  16. 16Topological Sort Using DFS
  17. 17Topological Sort using Kahn's Algorithm
  18. 18Cycle Detection in Directed Graph using BFS
  19. 19Course Schedule - Task Ordering with Prerequisites
  20. 20Course Schedule 2 - Task Ordering Using Topological Sort
  21. 21Find Eventual Safe States in a Directed Graph
  22. 22Alien Dictionary Character Order
  23. 23Shortest Path in Undirected Graph with Unit Distance
  24. 24Shortest Path in DAG using Topological Sort
  25. 25Dijkstra's Algorithm Using Set - Shortest Path in Graph
  26. 26Dijkstra’s Algorithm Using Priority Queue
  27. 27Shortest Distance in a Binary Maze using BFS
  28. 28Path With Minimum Effort in Grid using Graphs
  29. 29Cheapest Flights Within K Stops - Graph Problem
  30. 30Number of Ways to Reach Destination in Shortest Time - Graph Problem
  31. 31Minimum Multiplications to Reach End - Graph BFS
  32. 32Bellman-Ford Algorithm for Shortest Paths
  33. 33Floyd Warshall Algorithm for All-Pairs Shortest Path
  34. 34Find the City With the Fewest Reachable Neighbours
  35. 35Minimum Spanning Tree in Graphs
  36. 36Prim's Algorithm for Minimum Spanning Tree
  37. 37Disjoint Set (Union-Find) with Union by Rank and Path Compression
  38. 38Kruskal's Algorithm - Minimum Spanning Tree
  39. 39Minimum Operations to Make Network Connected
  40. 40Most Stones Removed with Same Row or Column
  41. 41Accounts Merge Problem using Disjoint Set Union
  42. 42Number of Islands II - Online Queries using DSU
  43. 43Making a Large Island Using DSU
  44. 44Bridges in Graph using Tarjan's Algorithm
  45. 45Articulation Points in Graphs
  46. 46Strongly Connected Components using Kosaraju's Algorithm

TriesTries1

Find the Nth Root of a Number
Using Binary Search



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).

  • If x = 0, then the nth root is 0.
  • If x = 1, the nth root is always 1.
  • If n = 1, then the result is just x (because the 1st root of any number is the number itself).
  • Return -1 for invalid cases, like x < 0 and even n, or if n ≤ 0.

Examples

Value (x) Root (n) Output Description
27 3 3.000000 33 = 27, so cube root of 27 is 3
16 4 2.000000 24 = 16, so 4th root is 2
10 3 2.154435 Cube root of 10 is a real number between 2 and 3
1 5 1.000000 1 raised to any power is 1
0 3 0.000000 0 raised to any positive power is 0
7 1 7.000000 1st root of any number is the number itself
0 0 -1 Invalid input: root 0 is not defined
-8 3 -2.000000 Valid: cube root of negative number is negative
-8 2 -1 Invalid: even root of negative number
-1 Empty 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.
Worst 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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