Using Stacks in Algorithms

In this post, we will explore the various ways stacks can be utilized in algorithms. Stacks are a fundamental data structure that follows the Last-In-First-Out (LIFO) principle. They allow for efficient insertion and deletion operations at one end, known as the top of the stack.

Postfix Evaluation

One common use of stacks is in evaluating postfix expressions. A postfix expression is an expression in which operators are written after their operands. For example, the infix expression "2 + 3" would be written as "2 3 +" in postfix notation.

To evaluate a postfix expression, we can use a stack to keep track of the operands and perform the operations in the correct order. Here is a step-by-step algorithm for postfix evaluation:

1. Create an empty stack.
2. Scan the postfix expression from left to right.
3. If the current element is an operand, push it onto the stack.
4. If the current element is an operator, pop the top two elements from the stack, perform the operation, and push the result back onto the stack.
5. After scanning the entire expression, the result will be the only element left on the stack.

Let's work through an example to illustrate the algorithm. Consider the postfix expression "5 2 + 4 *":

1. Create an empty stack.
2. Start scanning the expression:
   • "5": Push 5 onto the stack.
   • "2": Push 2 onto the stack.
   • "+": Pop 2 and 5 from the stack, add them, and push the result (7) onto the stack.
   • "4": Push 4 onto the stack.
   • "*": Pop 4 and 7 from the stack, multiply them, and push the result (28) onto the stack.
3. After scanning the entire expression, the only element left on the stack is the final result: 28.

This algorithm allows us to efficiently evaluate postfix expressions using a stack.

Towers of Hanoi

Another interesting problem where stacks can be utilized is the Towers of Hanoi. The problem consists of three pegs and a set of disks of different sizes, placed on one of the pegs. The goal is to move all the disks from the original peg to another peg, following these rules:

• Only one disk can be moved at a time.
• A larger disk cannot be placed on top of a smaller disk.

A recursive algorithm can be efficiently implemented using stacks to solve the Towers of Hanoi problem. Here is a simple recursive algorithm:

1. If there is only one disk, move it from the original peg to the destination peg.
2. Otherwise, recursively move n-1 disks from the original peg to an auxiliary peg, using the destination peg as an auxiliary.
3. Move the remaining disk from the original peg to the destination peg.
4. Recursively move the n-1 disks from the auxiliary peg to the destination peg, using the original peg as an auxiliary.

Let's consider an example with three disks. We have three pegs labeled A, B, and C. The initial configuration has all disks on peg A, with the largest disk at the bottom and the smallest at the top. The goal is to move all disks to peg C.

Using the algorithm described above, the sequence of moves to solve the Towers of Hanoi problem with three disks is as follows:

1. Move disk 1 from A to C.
2. Move disk 2 from A to B.
3. Move disk 1 from C to B.
4. Move disk 3 from A to C.
5. Move disk 1 from B to A.
6. Move disk 2 from B to C.
7. Move disk 1 from A to C.

By following this sequence of moves, we can efficiently solve the Towers of Hanoi problem using stacks.

Balanced Parentheses

Stacks can also be used to check if a string of parentheses is balanced or not. A string of parentheses is balanced if every opening parenthesis has a corresponding closing parenthesis in the correct order.

The algorithm to check balanced parentheses using a stack is straightforward:

1. Create an empty stack.
2. Iterate through each character in the string.
3. If the character is an opening parenthesis, push it onto the stack.
4. If the character is a closing parenthesis, check if the top of the stack contains the corresponding opening parenthesis. If it does not, or if the stack is empty, the parentheses are not balanced.
5. If all characters have been processed and the stack is empty, the parentheses are balanced. Otherwise, they are not.

Let's work through an example to illustrate the algorithm. Consider the string "((())())":

1. Create an empty stack.
2. Start iterating through the string:
   • "(": Push it onto the stack.
   • "(": Push it onto the stack.
   • ")": Pop the top of the stack, which contains "(". They are a pair.
   • "(": Push it onto the stack.
   • ")": Pop the top of the stack, which contains "(". They are a pair.
   • ")": Pop the top of the stack, which contains "(". They are a pair.
   • ")": Pop the top of the stack, which contains "(". They are a pair.
3. After iterating through the entire string, the stack is empty, indicating that the parentheses are balanced.

On the other hand, if we had the string "())(()", the stack would not be empty after iterating through the string, indicating that the parentheses are not balanced.

In conclusion, stacks have a wide range of applications in algorithmic problem-solving. We explored their usage in evaluating postfix expressions, solving the Towers of Hanoi problem, and checking for balanced parentheses. These examples demonstrate the versatility and practical applicability of stacks in various computational tasks.