Understanding the Intuition Behind Storing Indexes in Stacks

When tackling stack problems, particularly those involving monotonic stacks, you might find yourself pondering a recurring question: why do we often store indexes instead of the values themselves? This topic can be quite perplexing, but once you grasp the underlying intuition, it becomes much clearer. In this blog post, we will explore the reasons and advantages of storing indexes in stacks, along with some examples to solidify your understanding.

The Core Argument: Indexes vs. Values

At its core, the primary reason for storing indexes rather than values is that indexes provide additional information that values alone do not. Here’s why that matters:

1. Access to Values: When you store an index, you can always retrieve the value associated with that index in the original array. Conversely, if you only have the value, you may not be able to identify its position, especially in arrays with duplicate values. Indexes serve as a reliable reference point that allows you to access relevant elements effortlessly.

2. Position Information: In many stack problems, the order and position of elements are crucial. For example, when determining how far apart two elements are or which element appeared first, storing indexes allows for straightforward calculations. If you need to find the distance between elements, it’s simple to subtract the indexes.

3. Handling Duplicates: Imagine an array with duplicate values; if you only store values, differentiating between them becomes impossible. By storing indexes, you can uniquely identify each occurrence of a value, allowing for precise operations and comparisons.

Practical Application: The Next Greater Element Problem

Let’s consider a classic algorithmic problem: finding the “next greater element” for each element in an array. This is a perfect scenario to highlight the advantages of storing indexes:

Given an array, you want to find the next element that is greater than the current one. If you were to store the values directly, it would be challenging to calculate how far you’d need to traverse to find the next greater element, especially if you encounter duplicate values. By storing indexes in your stack, you can easily track where each value is located and compute the distance to the next greater element.

Here’s a simple illustration:

def next_greater_elements(arr):
    stack = []
    result = [-1] * len(arr)

    for i in range(len(arr)):
        while stack and arr[stack[-1]] < arr[i]:
            idx = stack.pop()
            result[idx] = arr[i]
        stack.append(i)

    return result

In the code above, we store the indexes of elements in the stack. When we find a greater element, we can directly access the original array using the index, ensuring we maintain the necessary position information.

The Trade-offs: When to Store Values vs. Indexes

While storing indexes is often beneficial, it ultimately depends on the specific problem you are dealing with. Here are some scenarios to consider:

• Only Values Matter: In some cases, the actual values suffice, and you may not need to keep track of their positions. For instance, if you’re simply looking for the maximum value in a list, storing the value alone might be adequate.

• Storing Pairs: Sometimes, you may want to keep track of both the value and its index. This approach can be achieved by storing pairs (value, index) in your stack. This method allows for more complex calculations, such as determining the difference between current and previous elements, which can be particularly useful in various applications, such as calculating slopes or trends.

• Indirect Value Storage: Another option is to indirectly store values by keeping track of indexes and looking them up in the original array. This method can simplify the implementation, especially in certain programming languages.

Conclusion

The decision to store indexes instead of values in stack problems can initially seem counterintuitive. However, understanding the advantages that indexes provide—such as access to values, position information, and handling duplicates—makes it clear why this approach is frequently adopted in algorithmic challenges.

As with many programming techniques, the best choice often depends on the specific requirements of the problem at hand. By exploring various scenarios and considering the trade-offs, you can enhance your problem-solving skills and choose the most effective strategy for each challenge you encounter.

Happy coding!