Binary TreesBinary Trees36
  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Inorder Traversal of a Binary Tree using Recursion
  4. 4Inorder Traversal of a Binary Tree using Iteration
  5. 5Postorder Traversal of a Binary Tree Using Recursion
  6. 6Postorder Traversal of a Binary Tree using Iteration
  7. 7Level Order Traversal of a Binary Tree using Recursion
  8. 8Level Order Traversal of a Binary Tree using Iteration
  9. 9Reverse Level Order Traversal of a Binary Tree using Iteration
  10. 10Reverse Level Order Traversal of a Binary Tree using Recursion
  11. 11Find Height of a Binary Tree
  12. 12Find Diameter of a Binary Tree
  13. 13Find Mirror of a Binary Tree
  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

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 sumVisualization
[-1, -2, -3, -4] -1 [-1] All elements are negative; pick the largest single elementVisualization
[4, -1, 2, 1] 6 [4, -1, 2, 1] Maximum sum comes from a mix of positive and negativeVisualization
[-2, -3, 4, -1, -2, 1, 5, -3] 7 [4, -1, -2, 1, 5] Best subarray lies in the middleVisualization
[5] 5 [5] Single-element arrayVisualization
[] 0 [] Empty array has no subarrays, return 0 or empty resultVisualization
[0, 0, 0] 0 [0] All zeros, choose first zero as max sum subarrayVisualization
[2, -1, 2, 3, 4, -5] 10 [2, -1, 2, 3, 4] Subarray before the negative dipVisualization

Visualization Player

Solution

Understanding the Problem

We are given an array of integers that may include positive numbers, negative numbers, or zeros. Our goal is to find the contiguous subarray that has the highest possible sum. This is commonly known as the "Maximum Subarray" problem.

In simple terms, a subarray is a continuous portion of the original array. We are not allowed to skip elements. We need to return both the maximum sum and the subarray that gives this sum.

Let’s consider an example:

Input: [-2, 1, -3, 4, -1, 2, 1, -5, 4]

We want to find the subarray with the maximum sum. In this case, the correct answer is:

Subarray: [4, -1, 2, 1]  
Maximum Sum: 6

Step-by-Step Approach

To solve this problem, we use a modified version of Kadane's Algorithm. Here's how we will go step-by-step:

  1. Initialize: We start with variables to track the maximum sum so far, the current sum while traversing, and the indices of the start and end of the best subarray.
  2. Traverse: For each element:
    • Add it to the current sum.
    • If the current sum is greater than the maximum so far, update the maximum and store the current subarray boundaries.
    • If the current sum becomes negative, reset it to 0 and mark the next index as a potential start of the next subarray.
  3. Return: At the end of the loop, return the subarray between the best start and end indices, and the maximum sum.

Edge Cases and How We Handle Them

  • All Positive Elements: The sum increases as we include more elements. The whole array is the answer.
  • All Negative Elements: Since adding any two elements will decrease the sum, we pick the single element with the least negative value.
  • Mixed Elements: This is the typical use case. Kadane's algorithm handles it by resetting the sum when needed and updating the subarray bounds.
  • Empty Array: No elements mean no subarray. We return a sum of 0 and an empty subarray.
  • All Zeros: Although all values are the same (0), we still pick a single element. Usually, the first zero is returned with a sum of 0.

Explanation

Imagine you’re walking through the array with a basket. You keep adding numbers to your basket. If the total weight (sum) becomes negative, you drop everything and start a new basket. But every time your basket has the best total so far, you note down the items inside it. By the end, you’ll have the best basket you picked up along the way — the one with the highest total.

Final Result

With the help of this step-by-step strategy and edge case handling, we can always return the correct maximum subarray and its sum, even in tricky cases like all negatives or empty arrays.

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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