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

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

Topic Contents

ProblemExamplesSolutionVisualizationCodeTime ComplexitySpace Complexity

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 ArrayMaximum SumSubarrayDescription
[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.
Average 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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