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

Print Subarray with Maximum Sum using Kadane’s Algorithm - Optimal Approach



Problem Statement

Given an array of integers (which may include positive numbers, negative numbers, and zeros), your task is to find and print the subarray that has the maximum sum.

If multiple subarrays have the same maximum sum, return the one that appears first. If the array is empty, return an empty result or appropriate message.

Examples

Input Array Maximum Sum Subarray Description
[1, 2, 3, -2, 5] 9 [1, 2, 3, -2, 5] The entire array has the maximum sum
[-1, -2, -3, -4] -1 [-1] All elements are negative; pick the largest single element
[4, -1, 2, 1] 6 [4, -1, 2, 1] Maximum sum comes from a mix of positive and negative
[-2, -3, 4, -1, -2, 1, 5, -3] 7 [4, -1, -2, 1, 5] Best subarray lies in the middle
[5] 5 [5] Single-element array
[] 0 [] Empty array has no subarrays, return 0 or empty result
[0, 0, 0] 0 [0] All zeros, choose first zero as max sum subarray
[2, -1, 2, 3, 4, -5] 10 [2, -1, 2, 3, 4] Subarray before the negative dip

Solution

To find the subarray with the maximum sum, we use a modified version of Kadane's Algorithm, which not only calculates the maximum sum but also keeps track of the subarray boundaries (start and end indices).

Understanding the Problem

The array may contain positive numbers, negative numbers, or even all negatives. We are interested in the subarray (a contiguous part of the array) whose sum is the highest among all possible subarrays.

Different Scenarios

  • All Positive Elements: The whole array is the maximum subarray since every additional number only increases the sum.
  • All Negative Elements: We cannot pick multiple elements here. The maximum subarray is simply the single element with the least negative value (i.e., closest to zero).
  • Mixed Elements: This is where the real use of Kadane's algorithm shines. As we traverse, we build up the current sum, but if adding the current element makes the sum worse than just starting from the current element, we reset the current sum. While doing this, we track the subarray bounds whenever a new maximum is found.
  • Empty Array: If the array is empty, there is no subarray. We can return 0 as the maximum sum and an empty list for the subarray.
  • Zeros in Array: In cases where all values are zero, the result should be [0] with a sum of 0 — usually picking the first one.

This method works efficiently in linear time O(n) and gives us not just the value, but also the actual subarray that yields this maximum sum.

By carefully handling resets and updates of subarray bounds, we ensure that the printed subarray reflects the earliest occurrence of the maximum sum.

Visualization

Algorithm Steps

  1. Given an array arr of integers (positive, negative, or zero).
  2. Initialize max_sum = arr[0], current_sum = arr[0], start = 0, end = 0, and temp_start = 0.
  3. Iterate through the array from index 1:
  4. → If current_sum + arr[i] is less than arr[i], set current_sum = arr[i] and temp_start = i.
  5. → Else, add arr[i] to current_sum.
  6. → If current_sum > max_sum, update max_sum = current_sum, start = temp_start, end = i.
  7. After loop, the subarray from start to end gives the maximum sum.

Code

Python
JavaScript
Java
C++
C
def max_subarray_with_indices(arr):
    max_sum = current_sum = arr[0]
    start = end = temp_start = 0

    for i in range(1, len(arr)):
        if arr[i] > current_sum + arr[i]:
            current_sum = arr[i]
            temp_start = i
        else:
            current_sum += arr[i]

        if current_sum > max_sum:
            max_sum = current_sum
            start = temp_start
            end = i

    print("Maximum Sum:", max_sum)
    print("Subarray:", arr[start:end+1])

# Sample Input
arr = [-2, 1, -3, 4, -1, 2, 1, -5, 4]
max_subarray_with_indices(arr)

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(n)Even in the best case, the algorithm must scan the entire array once to track subarray indices and sums.
Average CaseO(n)Each element is visited exactly once while updating the current and maximum subarray sums.
Worst CaseO(n)Regardless of input (positive, negative, mixed), the algorithm always performs a single pass through the array.

Space Complexity

O(1)

Explanation: The algorithm uses only a fixed number of variables (like max_sum, current_sum, start, end, temp_start) and no additional data structures, hence constant space.



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