Question

Describe how a single array can be used to implement three stacks.

Short answer

Use an array divided into thirds by index patterns to simulate three stacks, keeping separate pointers for each stack.

Step-by-step solution

Understand the Problem

The task is to implement three individual stacks using a single array. A stack is a data structure that supports last in, first out (LIFO) operations. Typically, a stack supports the operations of push (adding an element), pop (removing the last added element), and peek (viewing the top element without removing it).

Design the Array Structure

To implement three stacks within a single array, we need to divide the array into three logical parts for each stack. This approach uses division and modulus to calculate indices for each stack. Assuming the array has length 'n', stack 1 occupies indices 0, 3, 6,...; stack 2 occupies indices 1, 4, 7,...; stack 3 occupies indices 2, 5, 8,...; continuing this pattern.

Define Array Indexing

The `i`-th element of stack `s` (where 0, 1, 2 represent stack 1, stack 2, and stack 3 respectively) can be stored at index `i * 3 + s` in the array. For example, the first element of stack 1 is at index 0, stack 2 at index 1, and stack 3 at index 2.

Describe Stack Operations

The 'push' operation adds an element to the calculated index for the next free element in the respective stack. The 'pop' operation removes an element from the calculated top index, and the 'peek' operation accesses the element at the calculated top index without removing it.

Manage Stack Sizes

Use three separate pointers or indices to keep track of the current top of each stack within the single array. Initially, these pointers are at the starting positions of their respective modulo groups (e.g., 0, 1, 2) and advance by steps of 3 on push operations and retract by 3 on pop operations.

Key concepts

Stacks

In computer science, a stack is a simple but powerful data structure that follows the Last In, First Out (LIFO) principle. Imagine a stack like a pile of plates; you can only add or remove the plate at the top. In a stack, we primarily perform two operations:

• Push: Add an element to the top of the stack.
• Pop: Remove the element from the top of the stack.

Additionally, stacks often support peeking, which lets you view the top element without removing it. Stacks are used in various applications, like undo mechanisms in text editors or managing function calls in programming languages. Understanding how stacks operate is crucial for learning more advanced data structures and algorithms.

Array Implementation

Using an array to implement stacks involves leveraging the contiguous memory nature of arrays to efficiently add and remove elements. Arrays offer direct index access, making operations straightforward and quick.
In the context of implementing multiple stacks in a single array, the array is logically divided and managed to simulate separate stacks. Each stack gets a particular section of the array, and this division is maintained through strategic indexing. This method provides the benefit of managing memory more effectively when compared to handling multiple separate arrays.
The challenge lies in properly calculating where each stack begins and keeping their operations independent of each other, which is where clever use of indices plays a significant role.

LIFO Operations

LIFO stands for Last In, First Out, which is a fundamental principle of how stacks operate. It means the last element added to the stack will be the first one to be removed. This principle resembles a stack of plates, where you can only take the top plate off first.
In practical applications, LIFO operations are essential in scenarios where you need to backtrack or reverse actions, such as navigating browser history or implementing recursive functions.
Specifically, when handling multiple stacks within a single array, each stack's LIFO pattern must be maintained despite them sharing the same array. Properly managing the last elements of each logical stack is crucial.

Index Calculation

Index calculation becomes a key part of handling multiple stacks in a single array. By using indices effectively, you can define where each stack should start and how it should grow or shrink.
To segment an array into multiple stacks, we use a combination of division and modulus operations. For instance, given an array split into three stacks, stack one might occupy positions 0, 3, 6, etc., while stack two occupies 1, 4, 7, and so on.
The general formula to locate the `i`-th element of stack `s` is computed as `i * 3 + s`. This calculation helps maintain each stack's identity within the shared array. Furthermore, it assists in implementing stack operations like push and pop by determining precise index positions.