Today, we embark on a journey through the functioning of stacks and their practical applications in solving algorithmic problems. Like a pile of plates, stacks allow us to add and remove data in a last-in, first-out manner. This lesson revolves around leveraging stacks to tackle two common algorithmic challenges.

Our first challenge involves ensuring that parentheses, braces, and brackets within a string are correctly matched — crucial for verifying syntax in programming languages and mathematical formulas.

Imagine devising a tool within a code editor that flags syntax errors by checking for mismatched brackets — a much-needed feature for developers to catch common errors early in the coding process.

In a naive approach, one might consider scanning the string and manually checking for matching closing brackets for each opening one. While this might initially seem straightforward, it fails to handle nested structures efficiently, leading to repeated scanning and potential $O(n^2)$ performance in the worst-case scenario.

A stack, however, is ideally suited for this task. It acts as a memory model, helping track the last opened bracket and ensuring its closure before moving on to the prior ones.

Our TypeScript implementation lays out the process in a structured, step-by-step manner:

TypeScript
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1function areBracketsBalanced(s: string): boolean {
2    const stack: string[] = []; // stack tracking brackets
3    const brackets: { [key: string]: string } = { '(': ')', '[': ']', '{': '}' };
4    
5    for (let char of s) {
6        if (brackets[char]) { // either add bracket to check...
7            stack.push(brackets[char]);
8        } else { // ... or check to find a match
9            if (stack.length === 0 || stack.pop() !== char) {
10                return false;
11            }
12        }
13    }
14    
15    // empty stack means no hanging brackets
16    return stack.length === 0;
17}
18
19console.log(areBracketsBalanced('{{()}}[]'))  // true
20console.log(areBracketsBalanced('[(())}'))  // false

It returns false in three cases:

• If at any point we find a closing bracket and the stack is empty
• If at any point we find a closing bracket and the latest opening bracket doesn't match
• If, at the end of the process, we have any brackets left in the stack

Here, we explore reversing a string by utilizing a stack's LIFO property, a classic operation underlying numerous applications, including data encoding and genetic research.

Using a stack provides an easy mechanism to reverse text. Each character is pushed onto the stack, and as we pop these characters off, the string is naturally reversed — as if we're unloading a stack of blocks, revealing the last-placed block first.

The code below demonstrates this string-reversal algorithm:

TypeScript
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1function reverseString(str: string): string {
2    const stack: string[] = [];
3    
4    // Push each character of the original string onto the stack
5    for (let char of str) {
6        stack.push(char);
7    }
8    
9    // Generate the reversed string based on the LIFO principle
10    let reversedStr = '';
11    while (stack.length) {
12        reversedStr += stack.pop();
13    }
14    
15    // Return the reversed string
16    return reversedStr;
17}
18
19console.log(reverseString("abcde"))  // edcba 

Applying this function to the string "algorithm" would return "mhtirogla", demonstrating the method's efficacy and practicality in reversing a sequence.

This lesson explored actionable applications of stacks, leveraging their LIFO behavior to solve practical problems such as validating nested structures and manipulating strings. The ability to recall the most recent addition aids in building efficient algorithms that mirror real-world actions, like tidying a stack of documents in the correct order or unstacking packages for delivery.

It's time for you to practice with stacks, applying the concepts to new problem sets that reinforce what we've learned.