Largest Rectangle in Histogram

Finding the largest rectangle in a histogram is a common problem that can be solved using a stack data structure. The stack helps us keep track of the bars that are bounded by the current bar.

Here’s a step-by-step guide to solve the problem:

  1. Initialize: Create an empty stack that will keep track of indices of bars.

  2. Iterate Through Bars: Iterate through the array, for each bar do the following:

    a. Check Stack: While the stack is not empty and the current bar’s height is less than the height of the bar at the top index of the stack:

     i. **Pop from Stack**: Pop the top index from the stack.

 ii. **Calculate Area**: Calculate the area using the popped index and update the maximum area if the calculated area is larger.

    b. Push to Stack: Push the current index to the stack.

  3. Remaining Bars: Process the remaining bars in the stack in a similar way to step 2.

  4. Return Result: The largest area calculated is the result.

Here’s the code:

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class Solution:
    def largestRectangleArea(self, heights: List[int]) -> int:
        max_area = 0
        stack = [] # Stack to hold indices

        for i in range(len(heights)):
            # Check if the current height is less than the height at the top index of the stack
            while stack and heights[i] < heights[stack[-1]]:
                height = heights[stack.pop()] # Height of the popped bar
                width = i if not stack else i - stack[-1] - 1 # Width of the rectangle
                max_area = max(max_area, height * width) # Update the maximum area
            stack.append(i) # Push the current index

        # Process the remaining bars in the stack
        while stack:
            height = heights[stack.pop()]
            width = len(heights) if not stack else len(heights) - stack[-1] - 1
            max_area = max(max_area, height * width)

        return max_area

This code takes a list of integers representing the heights of the bars in a histogram and returns the area of the largest rectangle that can be formed within the histogram. The use of the stack ensures that the algorithm runs in O(n) time, where n is the number of bars.

Identifying Problem Isomorphism

“Largest Rectangle in Histogram” can be approximately mapped to “Maximal Rectangle”.

Reasoning:

Both problems revolve around determining the maximum area of a rectangle given certain constraints. In the “Largest Rectangle in Histogram” problem, you are given an array of integers where each integer represents the height of a bar in the histogram, and the task is to find the largest rectangle formed by the histogram bars.

In “Maximal Rectangle”, you are given a 2D binary matrix filled with 0’s and 1’s, and the task is to find the largest rectangle containing only 1’s and return its area. Each rectangle is defined by its width and height, and the area is calculated as width multiplied by height.

Here’s the approximate mapping: The height of the bar in the “Largest Rectangle in Histogram” problem corresponds to the number of consecutive 1’s in a column in the “Maximal Rectangle” problem, and the width of the rectangle corresponds to the number of consecutive 1’s in a row.

The mapping is approximate because while they share similarities in the problem structure (finding the maximum rectangle area), the actual problem conditions and objectives are different. “Largest Rectangle in Histogram” deals with an array of integers, whereas the “Maximal Rectangle” problem deals with a 2D binary matrix.

10 Prerequisite LeetCode Problems

“84. Largest Rectangle in Histogram” involves concepts of arrays and stacks. Here are 10 problems to prepare:

  1. “Maximum Subarray” (LeetCode Problem #53): This is a fundamental problem to understand how to find the maximum sum in a contiguous subarray which forms the base of dynamic programming.

  2. “Best Time to Buy and Sell Stock” (LeetCode Problem #121): This problem will introduce the concept of finding maximum differences in an array, which is helpful for understanding the histogram problem.

  3. “Two Sum” (LeetCode Problem #1): This problem is a classic that introduces the concept of using a data structure to keep track of elements you’ve seen so far.

  4. “Trapping Rain Water” (LeetCode Problem #42): This problem has a similar theme to the histogram problem and introduces the two-pointer method.

  5. “Container With Most Water” (LeetCode Problem #11): Another problem with a similar theme, which also leverages the two-pointer method.

  6. “Min Stack” (LeetCode Problem #155): This problem introduces the stack data structure, which is crucial for the histogram problem.

  7. “Valid Parentheses” (LeetCode Problem #20): This problem will help you to understand more about the stack data structure.

  8. “Largest Number” (LeetCode Problem #179): It will help you understand how to compare and order numbers in a non-trivial way.

  9. “Sliding Window Maximum” (LeetCode Problem #239): This problem helps understand how to manage a window in an array and find the maximum in it.

  10. “Monotonic Array” (LeetCode Problem #896): Understanding how to check the monotonic property of an array is crucial for the histogram problem.

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def largestRectangleArea(self, bars: List[int]) -> int:
	st, res = [], 0
	for bar in bars + [-1]: # add -1 to have an additional iteration
		step = 0
		while st and st[-1][1] >= bar:
			w, h = st.pop()
			step += w
			res = max(res, step * h)

		st.append((step + 1, bar))

	return res

Problem Classification

This problem requires identifying the largest possible rectangular area in a histogram. It’s an optimization problem where we’re finding the maximum area, hence the category Area Optimization.

This problem is about identifying the biggest possible rectangular area in a bar chart. It’s about finding the maximum area.

Language Agnostic Coding Drills

  1. Concept: Arrays/List

    • Drill: Write a function that creates an empty list and appends three items to it. Print the list to the console.

  2. Concept: For Loops

    • Drill: Write a function that loops over a list of numbers (for example, from 1 to 5), and prints each number.

  3. Concept: Conditional Statements

    • Drill: Write a function that takes two integers as input, and prints whether the first is greater, less than or equal to the second.

  4. Concept: Stack Data Structure

    • Drill: Implement a Stack class with ‘push’ and ‘pop’ methods. Test it by pushing two items onto the stack, popping an item off the stack, and printing the remaining stack.

  5. Concept: Compound Data Types (like Tuples in Python)

    • Drill: Depending on your language of choice, define a structure or class that can hold two values. Instantiate it with two numbers and print them.

  6. Concept: Appending an Element to a List

    • Drill: Write a function that takes a list and a single value as arguments. The function should add the value to the end of the list and then print the updated list.

In most modern programming languages, these concepts will have direct equivalents, though the syntax and specific methods will vary. Practicing these drills will help solidify the basic concepts of list manipulation, loops, conditionals, stack data structure, and compound data types.

Targeted Drills in Python

Let’s break down the problem and solution into smaller parts and learn the relevant concepts.

  1. Concept: Lists

    • Drill: Write a program that creates a list and appends items to it.
    • Python Code:
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      def create_and_append():
    my_list = []
    my_list.append(1)
    my_list.append(2)
    my_list.append(3)
    print(my_list)

create_and_append()  # Output: [1, 2, 3]

  2. Concept: Adding elements to a list

    • Drill: Write a program that creates a list from another list by adding an extra element.
    • Python Code:
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      def add_to_list(old_list, element):
    new_list = old_list + [element]
    print(new_list)

add_to_list([1, 2, 3], -1)  # Output: [1, 2, 3, -1]

  3. Concept: Loops

    • Drill: Write a program that loops through the elements of a list.
    • Python Code:
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      def iterate_list(my_list):
    for element in my_list:
        print(element)

iterate_list([1, 2, 3])  # Output: 1 2 3 (each on a new line)

  4. Concept: Stacks

    • Drill: Write a program that uses a list as a stack (Last In, First Out). Push elements to the stack and pop elements from the stack.
    • Python Code:
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      def stack_operations():
    stack = []
    stack.append(1)  # push
    stack.append(2)  # push
    print(stack.pop())  # pop and print: Output 2
    print(stack)  # Output: [1]

stack_operations()

  5. Concept: Tuples

    • Drill: Write a program that creates a tuple and accesses its elements.
    • Python Code:
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      def tuple_operations():
    my_tuple = (1, 2)
    print(my_tuple[0])  # Output: 1
    print(my_tuple[1])  # Output: 2

tuple_operations()

  6. Concept: Conditional Statements

    • Drill: Write a program that compares two numbers.
    • Python Code:
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      def compare_numbers(a, b):
    if a > b:
        print("a is greater than b")
    elif a < b:
        print("a is less than b")
    else:
        print("a is equal to b")

compare_numbers(1, 2)  # Output: "a is less than b"

These drills cover the individual concepts used in the provided solution. The solution first iterates over the input list, appends items to a stack, and pops items from the stack based on certain conditions. It uses tuples to store two related pieces of information together. The solution also uses conditional statements to compare values and make decisions. By practicing these drills, you can build up your understanding of these concepts and see how they are applied in the solution.