Largest Rectangle in Histogram Using Stack in DSA C# - Time & Space Complexity

Time Complexity: Largest Rectangle in Histogram Using Stack

O(n)

Understanding Time Complexity

We want to understand how the time needed to find the largest rectangle in a histogram changes as the number of bars grows.

How does the algorithm's work increase when the histogram gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public int LargestRectangleArea(int[] heights) {
    Stack<int> stack = new Stack<int>();
    int maxArea = 0;
    for (int i = 0; i <= heights.Length; i++) {
        int h = (i == heights.Length) ? 0 : heights[i];
        while (stack.Count > 0 && h < heights[stack.Peek()]) {
            int height = heights[stack.Pop()];
            int width = stack.Count == 0 ? i : i - stack.Peek() - 1;
            maxArea = Math.Max(maxArea, height * width);
        }
        stack.Push(i);
    }
    return maxArea;
}
    

This code finds the largest rectangle area in a histogram by using a stack to track bars and calculate areas efficiently.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: A single loop runs through all bars once, and inside it, bars are pushed and popped from the stack.
• How many times: Each bar is pushed once and popped once, so stack operations happen at most 2n times for n bars.

How Execution Grows With Input

As the number of bars (n) increases, the total operations grow roughly in a straight line with n.

Input Size (n)	Approx. Operations
10	About 20 stack operations
100	About 200 stack operations
1000	About 2000 stack operations

Pattern observation: The operations grow linearly, doubling the bars roughly doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the largest rectangle grows directly in proportion to the number of bars.

Common Mistake

[X] Wrong: "Because there is a nested while loop inside the for loop, the time complexity must be O(n²)."

[OK] Correct: Each bar is pushed and popped only once, so the inner loop does not run for every iteration fully, keeping total operations linear.

Interview Connect

Understanding this linear time solution shows you can use stacks smartly to solve problems efficiently, a skill valued in many coding challenges.

Self-Check

"What if we used a simple nested loop to check all rectangles instead of a stack? How would the time complexity change?"

Practice

1. What is the main purpose of using a stack in the Largest Rectangle in Histogram problem?

easy

A. To keep track of bars' indices with increasing heights

B. To store the heights of all bars directly

C. To sort the histogram bars by height

D. To reverse the order of histogram bars

Solution

1. Step 1: Understand stack usage in the problem

   The stack stores indices of bars in increasing order of their heights to help calculate areas efficiently.

2. Step 2: Why store indices, not heights?

   Indices allow calculation of width between bars when popping from the stack, which is essential for area calculation.

3. Final Answer:

   To keep track of bars' indices with increasing heights -> Option A

4. Quick Check:

   Stack stores indices for area calculation [OK]

Hint: Stack holds indices of increasing height bars [OK]

Common Mistakes:

• Storing heights instead of indices
• Using stack to sort bars
• Ignoring width calculation using indices

2. Which of the following C# code snippets correctly initializes a stack to store integers for the histogram problem?

easy

A. int[] stack = new int[10];

B. Stack stack = new Stack();

C. List<int> stack = new List<int>();

D. Stack<int> stack = new Stack<int>();

Solution

1. Step 1: Identify correct stack initialization in C#

   Using generic Stack<int> is the modern and type-safe way to create a stack for integers.

2. Step 2: Why not others?

   The non-generic Stack (like "Stack stack = new Stack();") uses the older style without type safety. A List (like "List<int> stack = new List<int>();") is not a stack. An array (like "int[] stack = new int[10];") is not a stack.

3. Final Answer:

   Stack<int> stack = new Stack<int>(); -> Option D

4. Quick Check:

   Generic Stack<int> is correct stack initialization [OK]

Hint: Use Stack<int> for integer stacks in C# [OK]

Common Mistakes:

• Using non-generic Stack without type safety
• Using List instead of Stack
• Using arrays instead of stack

3. What is the output of the following C# code snippet that calculates the largest rectangle area in histogram heights = [2, 1, 5, 6, 2, 3]?

int[] heights = {2, 1, 5, 6, 2, 3};
int largestRectangleArea(int[] heights) {
    Stack<int> stack = new Stack<int>();
    int maxArea = 0, i = 0;
    while (i < heights.Length) {
        if (stack.Count == 0 || heights[i] >= heights[stack.Peek()]) {
            stack.Push(i++);
        } else {
            int top = stack.Pop();
            int width = stack.Count == 0 ? i : i - stack.Peek() - 1;
            maxArea = Math.Max(maxArea, heights[top] * width);
        }
    }
    while (stack.Count > 0) {
        int top = stack.Pop();
        int width = stack.Count == 0 ? i : i - stack.Peek() - 1;
        maxArea = Math.Max(maxArea, heights[top] * width);
    }
    return maxArea;
}

Console.WriteLine(largestRectangleArea(heights));

medium

A. 12

B. 10

C. 15

D. 9

Solution

1. Step 1: Trace the stack operations and area calculations

   For heights [2,1,5,6,2,3], the largest rectangle area is 10 (from bars 5 and 6 with width 2).

2. Step 2: Confirm final maxArea

   After processing all bars and stack cleanup, maxArea is 10.

3. Final Answer:

   10 -> Option B

4. Quick Check:

   Largest rectangle area = 10 [OK]

Hint: Largest area is max height * width using stack [OK]

Common Mistakes:

• Miscomputing width when popping stack
• Forgetting to process remaining stack after loop
• Mixing up indices and heights

4. Identify the bug in this C# code snippet for Largest Rectangle in Histogram:

int largestRectangleArea(int[] heights) {
    Stack<int> stack = new Stack<int>();
    int maxArea = 0, i = 0;
    while (i <= heights.Length) {
        int currentHeight = (i == heights.Length) ? 0 : heights[i];
        if (stack.Count == 0 || currentHeight >= heights[stack.Peek()]) {
            stack.Push(i++);
        } else {
            int top = stack.Pop();
            int width = stack.Count == 0 ? i : i - stack.Peek() - 1;
            maxArea = Math.Max(maxArea, heights[top] * width);
        }
    }
    return maxArea;
}

medium

A. Width calculation misses subtracting 1, causing wrong area

B. Stack should store heights, not indices

C. Loop condition should be i < heights.Length, not <=

D. currentHeight assignment is incorrect

Solution

1. Step 1: Check width calculation formula

   Width should be i - stack.Peek() - 1 when stack is not empty, but code misses '-1'.

2. Step 2: Effect of missing '-1'

   Missing '-1' causes width to be larger than actual, leading to incorrect maxArea.

3. Final Answer:

   Width calculation misses subtracting 1, causing wrong area -> Option A

4. Quick Check:

   Width must subtract 1 for correct area [OK]

Hint: Subtract 1 in width calculation after popping stack [OK]

Common Mistakes:

• Forgetting to subtract 1 in width
• Using stack for heights instead of indices
• Incorrect loop boundary causing index errors

5. Given a histogram with heights = [6, 2, 5, 4, 5, 1, 6], what is the largest rectangle area using the stack approach? Explain how the stack helps find this area.

hard

A. 15

B. 10

C. 12

D. 8

Solution

1. Step 1: Use stack to track indices of increasing heights

   Push indices while heights increase; pop when current height is less, calculating area with popped height and width between indices.

2. Step 2: Calculate largest area for given heights

   For [6,2,5,4,5,1,6], largest rectangle area is 12 (from bars at indices 2,3,4 with min height 4 and width 3, calculated when popping index 3).

3. Final Answer:

   12 -> Option C

4. Quick Check:

   Stack helps find max area = 12 [OK]

Hint: Stack finds max area by tracking increasing bars [OK]

Common Mistakes:

• Not popping all remaining bars after loop
• Incorrect width calculation
• Confusing heights and indices in stack