Largest Rectangular Area in Histogram

Largest Rectangular Area in a Histogram Using Stack | Step-by-Step Guide

In this video, we’ll explore how to find the largest rectangular area in a histogram using a stack-based approach. This problem is a classic computational geometry problem where we need to determine the maximum area that can be formed between the bars of a histogram. Using a stack helps to solve the problem efficiently, reducing the time complexity compared to a brute-force approach. By the end of this tutorial, you will understand the stack-based method for solving this problem and how to apply it in real-world scenarios.

What is the Largest Rectangular Area in a Histogram?

In a histogram, we are given the heights of multiple bars, each having a uniform width of 1 unit. The goal is to find the largest rectangular area that can be enclosed by these bars. The rectangle must be formed by contiguous bars, and the height of the rectangle is limited by the shortest bar in the selected range.

For example, if you have the following histogram with heights:

Height array: [6, 2, 5, 4, 5, 1, 6]

The largest rectangular area that can be formed here is 12, formed by bars of heights 5, 4, and 5.

Key Points Covered:

Brute-Force Approach and Limitations: A brute-force method would involve checking every possible rectangle formed by different sets of bars, calculating their areas, and keeping track of the maximum area. However, this approach has a time complexity of O(n2)O(n^2)O(n2), which is inefficient for large histograms.

Optimized Stack-Based Approach: The stack-based approach significantly improves the efficiency of the algorithm by reducing the time complexity to O(n)O(n)O(n). The stack helps in maintaining a list of bar indices in a way that allows us to efficiently calculate the maximum rectangular area.

Steps in the Stack-Based Approach:

Iterating Over Histogram Bars:

• Traverse through each bar in the histogram one by one.
• If the current bar is taller than or equal to the bar at the top of the stack, push the index of the current bar onto the stack.
• If the current bar is shorter, we start popping from the stack to calculate the area of the rectangle that can be formed with the bar at the top of the stack as the shortest bar.

Calculating Area When Bars Are Popped:

• For each bar popped from the stack, calculate the width of the rectangle using the difference between the current index and the index of the bar now at the top of the stack.
• Multiply this width by the height of the bar being popped to calculate the area.

Finalizing the Calculations:

• After processing all bars, we continue popping any remaining bars in the stack and calculate the corresponding areas until the stack is empty.

Example Walkthrough:

Consider the histogram with heights [6, 2, 5, 4, 5, 1, 6].

• We push the first index (0) since the first bar (6) is taller than the stack (which is empty initially).
• We continue pushing indices as long as the current bar is taller than the one at the top of the stack.
• When we encounter a shorter bar, we pop from the stack and calculate the area of the rectangle formed by the popped bar. This process continues until we find the largest possible rectangular area.

Edge Cases:

• Single Bar: If the histogram contains only one bar, the largest rectangle is simply the height of that bar.
• All Bars of Equal Height: If all bars are of the same height, the largest rectangle spans the entire width of the histogram.
• Decreasing Heights: If the bars are arranged in strictly decreasing order, each smaller bar defines the width of a potential rectangle.

Why Use a Stack for This Problem?

The stack helps to efficiently manage the heights and indices of the bars, ensuring that for each bar, we calculate the largest possible rectangle that can be formed. By pushing and popping indices in a controlled manner, we can compute the areas in linear time, which is much faster than the brute-force approach.

Topics Included:

Understanding the Problem: Learn how to find the largest rectangular area in a histogram and why the brute-force approach is inefficient.

Stack-Based Solution: Detailed explanation of the stack-based approach to solving the problem, including how to calculate areas efficiently.

Example Walkthrough: Step-by-step example demonstrating the stack-based approach to find the largest rectangular area.

Edge Cases and Optimizations: Handling edge cases like single bars, bars of equal height, and bars in decreasing order, along with optimizations for larger datasets.

For a detailed guide and further explanation, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/largest-rectangular-area-in-a-histogram-using-stack/.