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
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  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 Maximum Product Subarray
Dynamic Programming - Optimal Approach

Topic Contents

ProblemExamplesSolutionVisualizationCodeTime ComplexitySpace Complexity

Problem Statement

Given an array of integers (which may contain positive numbers, negative numbers, and zeros), your task is to find the maximum product that can be obtained from a contiguous subarray.

If the array is empty, the answer should be 0.

Examples❯

Input ArrayMaximum ProductDescription
[2, 3, -2, 4]6Subarray [2, 3] gives the max product 6
[-2, 0, -1]0Zero breaks product chains, so max product is 0
[-2, 3, -4]24Subarray [3, -4] gives 3 * -4 = -12, but [-2, 3, -4] gives 24
[0, 0, 0]0All elements are zero, so product is 0
[5]5Single element positive, product is the element itself
[-5]-5Single negative number, product is the element itself
[1, -2, -3, 4]24Subarray [-2, -3, 4] gives max product
[6, -3, -10, 0, 2]180Subarray [6, -3, -10] gives 180
[]0Empty array has no elements; product is 0

Solution❯

To find the maximum product subarray, we need to understand how multiplication behaves with different types of numbers. Unlike sum problems, multiplication is tricky because of negatives and zeros.

Key Observations

  • Multiplying a negative with another negative becomes positive.
  • Zero resets the product (anything multiplied by 0 becomes 0).
  • We are interested in the maximum product, which can be hidden between negative values or split by zeros.

How We Approach the Problem

We scan the array from left to right, and at each position, we maintain two values:

  • max_product: the maximum product ending at that index
  • min_product: the minimum product ending at that index (important for handling negatives)

If we encounter a negative number, we swap max and min because a large negative times a negative could become the new maximum.

If we encounter a zero, we reset both max and min to zero (or start fresh from the next element).

At every step, we update our final result with the highest max_product seen so far.

📦 Case-by-Case Explanation

  • All positives: The maximum product is the product of the whole array.
  • All negatives with even count: Product of the whole array will be positive and maximum.
  • All negatives with odd count: Exclude one negative (whichever gives max).
  • Zeros present: Treat zeros as split points; maximum product may lie in segments between them.
  • Single element array: That element is the maximum product.
  • Empty array: No product is possible, so return 0.

🎯 Final Thoughts

This is a classic example of using dynamic programming to track values at each step. The key trick is keeping both min and max products because negatives can flip roles. This approach ensures we always find the best subarray in O(n) time.

Visualization

Algorithm Steps

  1. Given an array arr of integers.
  2. Initialize three variables: max_product = arr[0], min_product = arr[0], and result = arr[0].
  3. Traverse the array from index 1:
  4. → If the current element is negative, swap max_product and min_product.
  5. → Update max_product as the maximum of current element and current element * max_product.
  6. → Update min_product as the minimum of current element and current element * min_product.
  7. → Update result as the maximum of result and max_product.
  8. Return result as the final maximum product subarray.

Code

Python
JavaScript
Java
C++
C
def max_product_subarray(arr):
    max_product = min_product = result = arr[0]

    for num in arr[1:]:
        if num < 0:
            max_product, min_product = min_product, max_product

        max_product = max(num, num * max_product)
        min_product = min(num, num * min_product)

        result = max(result, max_product)

    return result

# Sample Input
arr = [2, 3, -2, 4]
print("Maximum Product Subarray:", max_product_subarray(arr))

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(n)In the best case, the array contains only positive numbers, and a single pass through the array is sufficient to compute the result.
Average CaseO(n)The algorithm always traverses the entire array once to keep track of current maximum and minimum products due to possible negative values.
Average CaseO(n)Even if the array has many sign changes and zeros, the algorithm must still iterate over every element to track product variations.

Space Complexity

O(1)

Explanation: The solution uses only a constant number of variables (max_product, min_product, result), so the space used does not grow with input size.

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