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

Find Maximum Product Subarray Dynamic Programming - Optimal Approach

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.

  • You can choose any contiguous part of the array (including a single element).
  • The subarray must have at least one element.

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

Examples

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

Visualization Player

Solution

Understanding the Problem

We are given an array of integers. Our goal is to find the maximum product subarray — that is, the contiguous subarray that gives the largest product when its elements are multiplied together.

Unlike sum problems, multiplication has unique challenges:

  • Multiplying two negative numbers gives a positive number.
  • Multiplying any number with zero resets the product to zero.
  • Negative numbers can reduce the product or, if paired well, create the maximum one.

Step-by-Step Solution Using Example

Example: nums = [2, 3, -2, 4]

We go through the array one element at a time. At each step, we keep track of:

  • maxProductEndingHere: The maximum product that ends at the current index.
  • minProductEndingHere: The minimum product that ends at the current index.

This is important because a negative number might make a small (even negative) product become a large positive one when multiplied.

Step-by-step execution:

  • Start with max = min = result = 2 (first number)
  • Next, 3:
    • tempMax = max(3, 2*3, 2*3) = 6
    • tempMin = min(3, 2*3, 2*3) = 3
    • Update result = max(6, 2) = 6
  • Next, -2:
    • tempMax = max(-2, 6*(-2), 3*(-2)) = max(-2, -12, -6) = -2
    • tempMin = min(-2, 6*(-2), 3*(-2)) = min(-2, -12, -6) = -12
    • Swap min and max because of negative
    • Update result = max(6, -2) = 6
  • Next, 4:
    • tempMax = max(4, -2*4, -12*4) = max(4, -8, -48) = 4
    • tempMin = min(4, -2*4, -12*4) = min(4, -8, -48) = -48
    • Update result = max(6, 4) = 6

Final Answer = 6

Edge Case Handling

What happens in special cases?

  • Array contains 0: Restart max and min at 0; product splits.
  • All positive: Product of full array is maximum.
  • All negative (even count): Product of entire array is maximum (negatives cancel).
  • All negative (odd count): Drop one negative (either from start or end) to get maximum product.
  • Single element: That is the result.
  • Empty array: No subarray exists. Return 0.

Key Intuition for Beginners

  • We are not just tracking the max product — we also track the min because it can become max after a negative.
  • Think of zero as a wall. You can’t continue the product across it.
  • Swapping min and max when we see a negative is a key trick.

Finally

This is a classic dynamic programming problem. At each step, you use previous results to decide your current best. You only need one pass through the array, so the time complexity is O(n).

By managing both the maximum and minimum product ending at each position, we can smartly deal with negatives and zeros to find the optimal subarray product.

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.
Worst 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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