Maximum Product Subarray in DSA C++ - Time & Space Complexity

Time Complexity: Maximum Product Subarray

O(n)

Understanding Time Complexity

We want to understand how the time needed to find the maximum product of a subarray changes as the input array grows.

How does the number of steps grow when the array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int maxProduct(std::vector<int>& nums) {
    int maxProd = nums[0], minProd = nums[0], result = nums[0];
    for (int i = 1; i < nums.size(); i++) {
        if (nums[i] < 0) std::swap(maxProd, minProd);
        maxProd = std::max(nums[i], maxProd * nums[i]);
        minProd = std::min(nums[i], minProd * nums[i]);
        result = std::max(result, maxProd);
    }
    return result;
}
    

This code finds the maximum product of any contiguous subarray in the input array.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: A single loop that goes through each element of the array once.
• How many times: Exactly once for each element, so n times if the array has n elements.

How Execution Grows With Input

As the array size grows, the number of steps grows directly in proportion.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: Doubling the input size roughly doubles the work done.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the maximum product grows linearly with the size of the input array.

Common Mistake

[X] Wrong: "Because there are nested calculations inside the loop, the time complexity must be quadratic O(n²)."

[OK] Correct: The calculations inside the loop are constant time operations done once per element, so the loop still runs only n times, making the overall complexity linear.

Interview Connect

Understanding this linear time solution shows you can handle tricky problems efficiently, a skill valued in real coding challenges and interviews.

Self-Check

"What if we used a nested loop to check every possible subarray product? How would the time complexity change?"

Practice

1. What does the Maximum Product Subarray problem ask you to find in an integer array?

easy

A. The subarray with the smallest product of its elements

B. The contiguous subarray with the largest product of its elements

C. The longest subarray with all positive numbers

D. The subarray with the maximum sum of its elements

Solution

1. Step 1: Understand the problem statement

   The problem asks for a contiguous part of the array where the product of all numbers is the highest possible.

2. Step 2: Differentiate from similar problems

   This is different from maximum sum subarray; here we multiply elements, not add.

3. Final Answer:

   The contiguous subarray with the largest product of its elements -> Option B

4. Quick Check:

   Maximum product subarray = largest product subarray [OK]

Hint: Look for contiguous subarray with max multiplication, not sum [OK]

Common Mistakes:

• Confusing product with sum
• Ignoring negative numbers impact
• Thinking about non-contiguous subarrays

2. Which of the following C++ code snippets correctly initializes variables to track maximum and minimum products for the Maximum Product Subarray problem?

easy

A. int maxProd = 1, minProd = INT_MIN;

B. int maxProd = 0, minProd = 0;

C. int maxProd = INT_MIN, minProd = INT_MAX;

D. int maxProd = 1, minProd = 1;

Solution

1. Step 1: Understand initialization role

   We start with 1 because multiplying by 1 doesn't change product; it acts as neutral.

2. Step 2: Check each option

   int maxProd = 1, minProd = 1; uses 1 for both maxProd and minProd, which is correct for product tracking.

3. Final Answer:

   int maxProd = 1, minProd = 1; -> Option D

4. Quick Check:

   Initialize maxProd and minProd to 1 [OK]

Hint: Initialize max and min product as 1 to start multiplication [OK]

Common Mistakes:

• Using 0 which resets product
• Using INT_MIN or INT_MAX incorrectly
• Not initializing min product

3. What is the output of the following C++ code snippet?

int arr[] = {2, 3, -2, 4};
int n = 4;
int maxProd = arr[0], minProd = arr[0], result = arr[0];
for (int i = 1; i < n; i++) {
    if (arr[i] < 0) std::swap(maxProd, minProd);
    maxProd = std::max(arr[i], maxProd * arr[i]);
    minProd = std::min(arr[i], minProd * arr[i]);
    result = std::max(result, maxProd);
}
std::cout << result << std::endl;

medium

A. 6

B. 4

C. -2

D. 8

Solution

1. Step 1: Trace the loop iterations

   Start with maxProd=2, minProd=2, result=2. - i=1 (3): arr[i]=3 >0, no swap maxProd = max(3, 2*3)=6 minProd = min(3, 2*3)=3 result = max(2,6)=6 - i=2 (-2): arr[i]<0, swap maxProd and minProd maxProd=3, minProd=6 maxProd = max(-2, 3*(-2))=max(-2,-6)=-2 minProd = min(-2, 6*(-2))=min(-2,-12)=-12 result = max(6,-2)=6 - i=3 (4): arr[i]=4 >0, no swap maxProd = max(4, -2*4)=max(4,-8)=4 minProd = min(4, -12*4)=min(4,-48)=-48 result = max(6,4)=6

2. Step 2: Determine final result

   The maximum product found is 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   Maximum product subarray = 6 [OK]

Hint: Swap max and min when negative number appears [OK]

Common Mistakes:

• Not swapping max and min on negative
• Confusing max and min updates
• Ignoring initial values

4. Identify the bug in this C++ code snippet for Maximum Product Subarray:

int maxProduct(int arr[], int n) {
    int maxProd = 1, minProd = 1, result = 1;
    for (int i = 0; i < n; i++) {
        if (arr[i] < 0) std::swap(maxProd, minProd);
        maxProd = std::max(arr[i], maxProd * arr[i]);
        minProd = std::min(arr[i], minProd * arr[i]);
        result = std::max(result, maxProd);
    }
    return result;
}

medium

A. maxProd and minProd should be initialized to 0

B. Swapping maxProd and minProd should happen after updating them

C. Starting loop from i=0 causes incorrect initial values

D. Result should be initialized to INT_MIN

Solution

1. Step 1: Analyze loop start and initial values

   Starting loop at i=0 uses maxProd=1 and minProd=1 before processing first element, which can cause wrong product calculations.

2. Step 2: Correct approach for initialization and loop

   Initialize maxProd, minProd, and result to arr[0] and start loop from i=1 to correctly track products.

3. Final Answer:

   Starting loop from i=0 causes incorrect initial values -> Option C

4. Quick Check:

   Loop should start at 1 with initial values from arr[0] [OK]

Hint: Initialize with first element, start loop from second element [OK]

Common Mistakes:

• Loop starting at 0 with default initial values
• Swapping max and min at wrong time
• Wrong initialization of result

5. Given the array int arr[] = {-1, -3, -10, 0, 60};, what is the maximum product of any contiguous subarray?

hard

A. 60

B. 180

C. 0

D. -1

Solution

1. Step 1: Analyze subarrays and their products

   Check products of contiguous subarrays: - [-1] = -1 - [-1, -3] = 3 - [-1, -3, -10] = 30 - [-3, -10] = 30 - [-3, -10, 0] = 0 - [0, 60] = 0 - [60] = 60 - [-1, -3, -10, 0, 60] = 0 The maximum product is 30 from subarray [-1, -3, -10].

2. Step 2: Check if any larger product exists

   Although 60 is a single element, the product 30 from three negatives is smaller. The maximum product is 30, but 180 is possible if we consider [-3, -10, 0, 60] incorrectly. Actually, the maximum product is 60 from [60]. However, the product of [-1, -3, -10] is 30, not 180. The option 180 is incorrect as no contiguous subarray produces 180.

3. Final Answer:

   60 -> Option A

4. Quick Check:

   Max product subarray = 60 [OK]

Hint: Multiply negatives carefully; zero resets product [OK]

Common Mistakes:

• Ignoring zero breaks subarray
• Multiplying across zero incorrectly
• Not considering negative number pairs