00: Introduction to Algorithms

Chapter 00: Introduction to Algorithms

Overview

Welcome to Algorithms for PHP Developers! This chapter introduces algorithms and why they matter for PHP developers. You'll learn what algorithms are, explore common problem types they solve, and develop a systematic framework for thinking algorithmically.

Understanding algorithms helps you write faster, more efficient PHP applications. Whether you're optimizing database queries, building search features, or preparing for technical interviews, algorithmic thinking is an essential skill.

By the end of this chapter, you'll have a solid foundation in algorithmic thinking and be ready to dive deep into algorithm complexity analysis in the next chapter.

Prerequisites

Before starting this chapter, you should have:

PHP 8.4+ installed and confirmed working with php --version
Completion of Chapter 02: Variables, Data Types, and Constants or equivalent understanding
Completion of Chapter 03: Control Structures or familiarity with loops and conditionals
Completion of Chapter 04: Understanding and Using Functions or basic understanding of functions and type hints
Completion of Chapter 06: Deep Dive into Arrays or familiarity with array operations

Estimated Time: ~55 minutes

Verify your setup:

bash

# Check PHP version
php --version

What You'll Build

By the end of this chapter, you will have:

Understanding of what algorithms are and why they matter for PHP developers
A systematic problem-solving framework for approaching algorithmic challenges
Knowledge of common algorithm categories (searching, sorting, data structures)
Working examples of algorithms including finding maximum values, checking duplicates, and processing strings
A development environment set up for algorithm practice and testing
Experience implementing three practice exercises with solutions

Objectives

Understand what algorithms are and why they're crucial for PHP developers
Learn to think algorithmically using a systematic problem-solving framework
Explore common algorithm categories including searching, sorting, and data structures
Develop skills to analyze algorithm efficiency and make informed design choices
Set up your development environment for algorithm practice and testing

Step 1: Understanding What Algorithms Are (~5 min)

Goal

Learn what algorithms are and see your first algorithm in action.

Actions

Understand the concept: An algorithm is a step-by-step procedure for solving a problem. You use algorithms every day:
- Following a recipe to bake a cake
- Getting directions from point A to point B
- Sorting your email by date
See a simple algorithm: In programming, an algorithm transforms input into output. Here's a complete example:

php

# filename: find-max.php
<?php

declare(strict_types=1);

// A simple algorithm to find the maximum value in an array
function findMax(array $numbers): int|float
{
    if (empty($numbers)) {
        throw new InvalidArgumentException('Array cannot be empty');
    }

    $max = $numbers[0];

    foreach ($numbers as $number) {
        if ($number > $max) {
            $max = $number;
        }
    }

    return $max;
}

$numbers = [3, 7, 2, 9, 1, 5];
echo findMax($numbers); // Output: 9

Run the example: Save this code to find-max.php and run it:

bash

# Run the algorithm example
php find-max.php

Expected Result

Why It Works

The findMax algorithm works by:

Starting with the first number as the initial maximum
Comparing each subsequent number to the current maximum
Updating the maximum when a larger number is found
Returning the final maximum value after checking all numbers

This is a linear search pattern—we examine each element once, making it an O(n) algorithm where n is the array size.

Troubleshooting

Error: "Array is empty" — The function now includes error handling; ensure you pass a non-empty array
Error: "InvalidArgumentException" — This means the array was empty; check your input data
Wrong result — Verify the array contains numeric values; strings will be compared lexicographically
Type errors — Ensure all array elements are numeric (int or float) for accurate comparison

Step 2: Understanding Why Algorithms Matter for PHP Developers (~10 min)

Goal

Learn why algorithms are essential for PHP developers, even when building web applications.

Actions

Consider the performance impact: You might think "I build web applications, not search engines. Why do I need algorithms?" Here's why they matter:
Compare algorithm performance: See how algorithm choice dramatically affects performance:

php

# filename: duplicate-check-comparison.php
<?php

declare(strict_types=1);

// Bad: O(n²) - checking for duplicates
// Note: We cache count() to avoid calling it repeatedly in the loop
function hasDuplicatesSlow(array $items): bool
{
    $n = count($items);
    for ($i = 0; $i < $n; $i++) {
        for ($j = $i + 1; $j < $n; $j++) {
            if ($items[$i] === $items[$j]) {
                return true;
            }
        }
    }
    return false;
}

// Good: O(n) - using a hash set
function hasDuplicatesFast(array $items): bool
{
    $seen = [];
    foreach ($items as $item) {
        if (isset($seen[$item])) {
            return true;
        }
        $seen[$item] = true;
    }
    return false;
}

// Test with 1,000 items
$largeArray = range(1, 1000);
$largeArray[] = 500; // Add duplicate

$start = microtime(true);
hasDuplicatesSlow($largeArray);
$slowTime = (microtime(true) - $start) * 1000;

$start = microtime(true);
hasDuplicatesFast($largeArray);
$fastTime = (microtime(true) - $start) * 1000;

echo "Slow version: {$slowTime}ms\n";
echo "Fast version: {$fastTime}ms\n";

Expected Result

Slow version: ~500ms (approximately)
Fast version: ~0.1ms (approximately)

Why It Works

Performance Optimization: With 1,000 items, the slow version does ~500,000 comparisons (n²/2). The fast version? Just 1,000 (n). That's a 500x speedup!

Better Technical Interviews: Many companies test algorithm knowledge. Understanding algorithms in PHP gives you an edge.

Framework Understanding: Laravel, Symfony, and other frameworks use sophisticated algorithms:

Laravel Collections use various sorting and filtering algorithms
Routing systems use tree-based data structures for fast lookups
Database ORMs optimize queries using algorithmic strategies

Problem-Solving Skills: Learning algorithms trains you to break down complex problems, think about edge cases, and reason about efficiency—skills that make you a better developer overall.

Troubleshooting

Performance difference not noticeable — Try with larger arrays (10,000+ items) to see the difference
Memory error with large arrays — The fast version uses more memory (O(n) space) but is much faster
Type errors — Ensure array items are comparable (same type) for accurate duplicate detection

Step 3: Understanding Algorithms vs Data Structures (~5 min)

Goal

Distinguish between algorithms and data structures, and understand how they work together.

Actions

Understand the difference:
- Algorithm: A step-by-step procedure for solving a problem (the "how")
- Data Structure: A way to organize and store data (the "what")
- They work together: algorithms operate on data structures
See examples:

php

# filename: algorithm-vs-datastructure.php
<?php

declare(strict_types=1);

// DATA STRUCTURE: Stack (how data is organized)
class Stack
{
    private array $items = [];

    public function push(mixed $item): void
    {
        $this->items[] = $item;
    }

    public function pop(): mixed
    {
        return array_pop($this->items);
    }

    public function isEmpty(): bool
    {
        return empty($this->items);
    }
}

// ALGORITHM: Reverse a string using a stack (the procedure)
function reverseWithStack(string $str): string
{
    $stack = new Stack();
    
    // Push all characters onto stack
    for ($i = 0; $i < strlen($str); $i++) {
        $stack->push($str[$i]);
    }
    
    // Pop all characters (reverses order)
    $reversed = '';
    while (!$stack->isEmpty()) {
        $reversed .= $stack->pop();
    }
    
    return $reversed;
}

echo reverseWithStack('hello'); // Output: olleh

Expected Result

You'll understand that:

Data structures organize data (arrays, stacks, trees, hash maps)
Algorithms process data (sorting, searching, traversing)
They're complementary: good algorithms need appropriate data structures

Why It Works

Think of it like cooking:

Data structure = ingredients organized in containers (bowls, pans)
Algorithm = recipe steps (mix, bake, serve)

You need both: the right containers (data structures) and the right steps (algorithms) to solve problems efficiently.

Troubleshooting

Confused about the difference — Remember: data structure = storage, algorithm = process
Not sure which to learn first — Start with algorithms on simple arrays, then learn data structures as needed
Overthinking it — For now, just know they're different but work together

Step 4: Exploring Types of Problems Algorithms Solve (~15 min)

Goal

Explore common algorithm categories and see examples of each type.

Actions

Understand algorithm categories: Algorithms solve different types of problems. Let's explore the main categories:
Searching algorithms: Finding specific data in a collection:

php

# filename: linear-search.php
<?php

declare(strict_types=1);

// Linear search - checks each item
function linearSearch(array $items, mixed $target): int|false
{
    foreach ($items as $index => $item) {
        if ($item === $target) {
            return $index;
        }
    }
    return false;
}

$users = ['Alice', 'Bob', 'Charlie', 'David'];
$result = linearSearch($users, 'Charlie');
echo $result !== false ? "Found at index: $result\n" : "Not found\n";

Sorting algorithms: Arranging data in a specific order:

php

# filename: bubble-sort.php
<?php

declare(strict_types=1);

// Bubble sort - simple but slow for large datasets
function bubbleSort(array $arr): array
{
    $n = count($arr);

    for ($i = 0; $i < $n - 1; $i++) {
        for ($j = 0; $j < $n - $i - 1; $j++) {
            if ($arr[$j] > $arr[$j + 1]) {
                // Swap elements
                [$arr[$j], $arr[$j + 1]] = [$arr[$j + 1], $arr[$j]];
            }
        }
    }

    return $arr;
}

$numbers = [64, 34, 25, 12, 22, 11, 90];
$sorted = bubbleSort($numbers);
print_r($sorted); // [11, 12, 22, 25, 34, 64, 90]

Data structure operations: Managing collections of data efficiently:

php

# filename: stack.php
<?php

declare(strict_types=1);

// Stack - Last In, First Out (LIFO)
class Stack
{
    private array $items = [];

    public function push(mixed $item): void
    {
        $this->items[] = $item;
    }

    public function pop(): mixed
    {
        if ($this->isEmpty()) {
            throw new UnderflowException('Stack is empty');
        }
        return array_pop($this->items);
    }

    public function isEmpty(): bool
    {
        return empty($this->items);
    }
}

$stack = new Stack();
$stack->push('First');
$stack->push('Second');
$stack->push('Third');
echo $stack->pop() . "\n"; // Output: Third (last in, first out)

Graph and tree traversal: Navigating hierarchical or networked data:

php

# filename: tree-traversal.php
<?php

declare(strict_types=1);

// Tree structure representing a file system
$fileSystem = [
    'name' => 'root',
    'children' => [
        ['name' => 'var', 'children' => [
            ['name' => 'log', 'children' => []],
            ['name' => 'www', 'children' => []]
        ]],
        ['name' => 'etc', 'children' => []],
        ['name' => 'home', 'children' => []]
    ]
];

// Depth-first traversal
function printTree(array $node, int $depth = 0): void
{
    echo str_repeat('  ', $depth) . $node['name'] . "\n";

    foreach ($node['children'] as $child) {
        printTree($child, $depth + 1);
    }
}

printTree($fileSystem);

String processing: Manipulating and analyzing text:

php

# filename: palindrome.php
<?php

declare(strict_types=1);

// Check if a string is a palindrome
function isPalindrome(string $str): bool
{
    $str = strtolower(preg_replace('/[^a-z0-9]/', '', $str));
    return $str === strrev($str);
}

echo isPalindrome('A man, a plan, a canal: Panama') ? "true\n" : "false\n"; // true
echo isPalindrome('hello') ? "true\n" : "false\n"; // false

Expected Result

For the tree traversal example:

root
  var
    log
    www
  etc
  home

Why It Works

Each algorithm category solves different problems:

Searching: Finds items quickly (linear search is O(n), binary search is O(log n) for sorted arrays)
Sorting: Organizes data for efficient access (bubble sort is O(n²), merge sort is O(n log n))
Data structures: Provides efficient ways to store and access data (stacks, queues, hash maps)
Tree traversal: Navigates hierarchical structures (file systems, organization charts, DOM trees)
String processing: Manipulates text data (palindromes, pattern matching, parsing)

Troubleshooting

Search returns wrong index — Ensure you're using strict comparison (===) to match exact values
Sort doesn't work — Verify array elements are comparable (same type)
Stack underflow error — Always check isEmpty() before calling pop()
Tree traversal infinite loop — Ensure tree structure doesn't have circular references

Step 5: Learning Algorithmic Thinking Framework (~10 min)

Goal

Develop a systematic approach to solving algorithmic problems.

Actions

Understand the problem: Ask yourself:
- What are the inputs?
- What should the output be?
- Are there constraints or edge cases?
Plan your approach: Consider:
- Can you break the problem into smaller steps?
- Have you solved a similar problem before?
- Can you solve a simpler version first?
Consider efficiency: Think about:
- How does performance change with input size?
- What resources (time, memory) do you have?
- Is there a trade-off between speed and simplicity?
Implement and test: Follow these steps:
- Start with a working solution (even if inefficient)
- Test with various inputs (normal, edge cases, large datasets)
- Refine and optimize based on results
Analyze and improve: After implementation:
- Measure actual performance
- Compare to alternative approaches
- Document your decisions

Expected Result

You'll have a five-step framework you can apply to any algorithmic problem.

Why It Works

This systematic approach prevents common mistakes:

Understanding first avoids solving the wrong problem
Planning helps you see the big picture before coding
Considering efficiency ensures your solution scales
Testing catches bugs early
Analyzing helps you learn and improve

Troubleshooting

Stuck on understanding — Try restating the problem in your own words
Can't break it down — Start with the simplest possible case (1-2 elements)
Performance unclear — Implement first, then measure and optimize
Too many edge cases — List them explicitly before coding

Step 6: Applying Algorithmic Thinking to a Real Problem (~10 min)

Goal

Apply the algorithmic thinking framework to solve a real-world problem: finding duplicate email addresses.

Actions

Understand the problem:
- Input: Array of user objects with email addresses
- Output: Array of duplicate email addresses
- Constraints: Could have 100,000+ users
Plan your approach: Consider three options:
- Compare each email to every other email (simple but slow)
- Sort emails and check adjacent entries (moderate complexity)
- Use a hash map to track seen emails (fast, uses more memory)
Consider efficiency:
- Approach 1: O(n²) - too slow for large datasets
- Approach 2: O(n log n) - decent
- Approach 3: O(n) - best time, acceptable memory usage
Implement the best solution:

php

# filename: find-duplicate-emails.php
<?php

declare(strict_types=1);

function findDuplicateEmails(array $users): array
{
    $seen = [];
    $duplicates = [];

    foreach ($users as $user) {
        $email = strtolower($user['email']);

        if (isset($seen[$email])) {
            if (!in_array($email, $duplicates)) {
                $duplicates[] = $email;
            }
        } else {
            $seen[$email] = true;
        }
    }

    return $duplicates;
}

$users = [
    ['name' => 'Alice', 'email' => 'alice@example.com'],
    ['name' => 'Bob', 'email' => 'bob@example.com'],
    ['name' => 'Charlie', 'email' => 'alice@example.com'], // Duplicate!
    ['name' => 'David', 'email' => 'david@example.com'],
];

$duplicates = findDuplicateEmails($users);
print_r($duplicates); // ['alice@example.com']

Run the example:

bash

# Test the duplicate finder
php find-duplicate-emails.php

Expected Result

Array
(
    [0] => alice@example.com
)

Why It Works

Time complexity: O(n) - single pass through users Space complexity: O(n) - stores emails in hash map

The algorithm:

Normalizes emails to lowercase for case-insensitive comparison
Uses a hash map ($seen) for O(1) lookup time
Tracks duplicates separately to avoid duplicates in the result array
Works efficiently even with millions of users

Troubleshooting

No duplicates found — Check that emails are normalized (lowercase) before comparison
Memory issues — For very large datasets, consider streaming or batch processing
Case sensitivity — Always normalize emails to lowercase before comparison
Empty result — Verify the input array structure matches expected format

Step 7: Understanding When NOT to Optimize (~5 min)

Goal

Learn when optimization is premature and when it's necessary.

Actions

Understand premature optimization: Not every piece of code needs optimization:

php

# filename: premature-optimization.php
<?php

declare(strict_types=1);

// ❌ PREMATURE: Over-optimizing a function that runs once
function getUserName(int $userId): string
{
    // Don't need hash map for single lookup!
    $users = [
        1 => 'Alice',
        2 => 'Bob',
        3 => 'Charlie'
    ];
    return $users[$userId] ?? 'Unknown';
}

// ✅ APPROPRIATE: Optimizing code that runs millions of times
function findDuplicatesInLargeDataset(array $items): array
{
    // This runs on 100,000+ items - optimization matters!
    $seen = [];
    $duplicates = [];
    
    foreach ($items as $item) {
        if (isset($seen[$item])) {
            $duplicates[] = $item;
        } else {
            $seen[$item] = true;
        }
    }
    
    return $duplicates;
}

Follow the optimization rule: Optimize when:
- Code runs frequently (loops, API endpoints, batch processing)
- Performance is actually a problem (measure first!)
- Data size is large (thousands+ items)
- Users report slowness
Don't optimize when:
- Code runs rarely (one-time scripts, admin tools)
- Performance is already acceptable
- You're guessing without measuring
- It makes code harder to read/maintain

Expected Result

You'll understand that optimization should be:

Measured: Profile first, optimize second
Targeted: Focus on bottlenecks, not everything
Balanced: Consider readability and maintainability

Why It Works

Premature optimization wastes time and makes code complex. Measured optimization solves real problems efficiently.

The best approach:

Write clear, working code first
Measure performance
Optimize only what's slow
Verify the optimization helped

Troubleshooting

Not sure if optimization is needed — Measure first with microtime() or profiling tools
Code is slow but don't know why — Use profiling tools (Xdebug, Blackfire) to find bottlenecks
Optimization made code worse — Sometimes simpler is better; measure before and after

Step 8: Setting Up Your Development Environment (~5 min)

Goal

Configure your development environment for algorithm practice and testing.

Actions

Verify PHP installation: You need PHP 8.4+ for modern features:

bash

# Check PHP version
php --version

If you need to install PHP, visit php.net or use your system's package manager.

Set up your code editor: We recommend Visual Studio Code with the PHP Intelephense extension for:
- Syntax highlighting
- Code completion
- Error detection
- Debugging support
Install testing framework (optional): Install PHPUnit for testing your algorithms:

bash

# Create composer.json if needed
composer init --no-interaction

# Install PHPUnit
composer require --dev phpunit/phpunit

Set up performance profiling: Use PHP's built-in microtime() for benchmarking:

php

# filename: benchmark-template.php
<?php

declare(strict_types=1);

$start = microtime(true);

// Your algorithm here
// bubbleSort($largeArray);

$end = microtime(true);
$duration = ($end - $start) * 1000; // Convert to milliseconds

echo "Execution time: {$duration}ms\n";

Expected Result

You'll have PHP 8.4+ installed, a code editor configured, and tools ready for algorithm development.

Why It Works

PHP 8.4+ provides modern type system and performance improvements
Code editor with PHP support helps catch errors early
PHPUnit enables automated testing of algorithm correctness
Benchmarking helps you measure and compare algorithm performance

Troubleshooting

PHP version too old — Update to PHP 8.4+ using your system's package manager
Composer not found — Install Composer from getcomposer.org
Editor extensions not working — Restart your editor after installing PHP extensions
Benchmark shows 0ms — Use larger datasets or run multiple iterations for accurate timing

Exercises

Before moving on, try these exercises to reinforce your understanding:

Exercise 1: Reverse a String

Goal: Practice string manipulation and array/loop operations.

Write a function that reverses a string without using strrev():

php

# filename: exercise-01-reverse-string.php
<?php

declare(strict_types=1);

function reverseString(string $str): string
{
    // Your code here
}

echo reverseString('hello'); // Should output: olleh

Validation: Test your implementation:

php

echo reverseString('hello'); // Expected: olleh
echo reverseString('PHP'); // Expected: PHP
echo reverseString(''); // Expected: (empty string)

Solution

php

# filename: exercise-01-reverse-string.php
<?php

declare(strict_types=1);

function reverseString(string $str): string
{
    $reversed = '';
    for ($i = strlen($str) - 1; $i >= 0; $i--) {
        $reversed .= $str[$i];
    }
    return $reversed;
}

// Test cases
echo reverseString('hello') . "\n"; // olleh
echo reverseString('PHP') . "\n"; // PHP
echo reverseString('') . "\n"; // (empty)

Exercise 2: Find the Second Largest Number

Goal: Practice array traversal and comparison logic.

Write a function that finds the second-largest number in an array:

php

# filename: exercise-02-second-largest.php
<?php

declare(strict_types=1);

function findSecondLargest(array $numbers): int|float|null
{
    // Your code here
}

echo findSecondLargest([3, 7, 2, 9, 1, 5]); // Should output: 7

Validation: Test your implementation:

php

echo findSecondLargest([3, 7, 2, 9, 1, 5]); // Expected: 7
echo findSecondLargest([1, 1, 1]); // Expected: null (no second largest)
echo findSecondLargest([5]); // Expected: null (need at least 2 numbers)

Solution

php

# filename: exercise-02-second-largest.php
<?php

declare(strict_types=1);

function findSecondLargest(array $numbers): int|float|null
{
    if (count($numbers) < 2) {
        return null;
    }

    $first = $second = PHP_INT_MIN;

    foreach ($numbers as $number) {
        if ($number > $first) {
            $second = $first;
            $first = $number;
        } elseif ($number > $second && $number !== $first) {
            $second = $number;
        }
    }

    return $second === PHP_INT_MIN ? null : $second;
}

// Test cases
echo findSecondLargest([3, 7, 2, 9, 1, 5]) . "\n"; // 7
var_dump(findSecondLargest([1, 1, 1])); // null
var_dump(findSecondLargest([5])); // null

Exercise 3: Count Vowels

Goal: Practice string iteration and character matching.

Count the number of vowels in a string:

php

# filename: exercise-03-count-vowels.php
<?php

declare(strict_types=1);

function countVowels(string $str): int
{
    // Your code here
}

echo countVowels('Hello World'); // Should output: 3

Validation: Test your implementation:

php

echo countVowels('Hello World'); // Expected: 3
echo countVowels('PHP'); // Expected: 0
echo countVowels('aeiou'); // Expected: 5

Solution

php

# filename: exercise-03-count-vowels.php
<?php

declare(strict_types=1);

function countVowels(string $str): int
{
    $vowels = ['a', 'e', 'i', 'o', 'u'];
    $count = 0;
    $str = strtolower($str);

    for ($i = 0; $i < strlen($str); $i++) {
        if (in_array($str[$i], $vowels)) {
            $count++;
        }
    }

    return $count;
}

// Test cases
echo countVowels('Hello World') . "\n"; // 3
echo countVowels('PHP') . "\n"; // 0
echo countVowels('aeiou') . "\n"; // 5

Wrap-up

Congratulations! You've completed the introduction to algorithms. Here's what you've accomplished:

✓ Learned what algorithms are and why they matter for PHP developers
✓ Explored common algorithm categories including searching, sorting, data structures, tree traversal, and string processing
✓ Understood the difference between algorithms and data structures
✓ Developed a systematic problem-solving framework with five clear steps
✓ Learned when optimization is appropriate and when it's premature
✓ Applied algorithmic thinking to solve a real-world duplicate detection problem
✓ Set up your development environment for algorithm practice and testing
✓ Completed three practice exercises to reinforce your understanding

Key Concepts Learned

Algorithms are step-by-step procedures for solving problems
Good algorithms make your PHP applications faster and more scalable
Algorithmic thinking is a systematic approach to problem-solving
Efficiency matters, especially as your data grows
Practice is essential—you'll improve with each algorithm you implement

What's Next

In the next chapter, we'll dive deep into Algorithm Complexity and Big O Notation. You'll learn to:

Analyze algorithm efficiency mathematically
Understand time and space complexity
Recognize common complexity patterns
Make informed decisions about algorithm choices

This foundation will help you evaluate every algorithm we study in this series.

💻 Code Samples

All code examples from this chapter are available in the GitHub repository:

View Chapter 00 Code Samples

Files included:

01-quick-start-examples.php - Collection of essential algorithm patterns ready to use
02-common-patterns.php - Fundamental algorithm patterns (two pointers, sliding window, hash maps)
03-performance-tips.php - Practical optimization techniques with benchmarks
README.md - Complete documentation and usage guide

Clone the repository to run the examples locally:

bash

git clone https://github.com/dalehurley/codewithphp.git
cd codewithphp/code/php-algorithms/chapter-00
php 01-quick-start-examples.php

You've mastered the fundamentals of algorithms and algorithmic thinking!Check the box when you've finished reading, or scroll to the bottom to auto-complete.

Ready to analyze algorithm efficiency? Continue to Chapter 01: Algorithm Complexity & Big O Notation.

Chapter 00: Introduction to Algorithms ​

Overview ​

Prerequisites ​

What You'll Build ​

Objectives ​

Step 1: Understanding What Algorithms Are (~5 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 2: Understanding Why Algorithms Matter for PHP Developers (~10 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 3: Understanding Algorithms vs Data Structures (~5 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 4: Exploring Types of Problems Algorithms Solve (~15 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 5: Learning Algorithmic Thinking Framework (~10 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 6: Applying Algorithmic Thinking to a Real Problem (~10 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 7: Understanding When NOT to Optimize (~5 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Step 8: Setting Up Your Development Environment (~5 min) ​

Goal ​

Actions ​

Expected Result ​

Why It Works ​

Troubleshooting ​

Exercises ​

Exercise 1: Reverse a String ​

Exercise 2: Find the Second Largest Number ​

Exercise 3: Count Vowels ​

Wrap-up ​

Key Concepts Learned ​

What's Next ​

Further Reading ​

💻 Code Samples ​

Chapter 00: Introduction to Algorithms

Overview

Prerequisites

What You'll Build

Objectives

Step 1: Understanding What Algorithms Are (~5 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 2: Understanding Why Algorithms Matter for PHP Developers (~10 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 3: Understanding Algorithms vs Data Structures (~5 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 4: Exploring Types of Problems Algorithms Solve (~15 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 5: Learning Algorithmic Thinking Framework (~10 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 6: Applying Algorithmic Thinking to a Real Problem (~10 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 7: Understanding When NOT to Optimize (~5 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Step 8: Setting Up Your Development Environment (~5 min)

Goal

Actions

Expected Result

Why It Works

Troubleshooting

Exercises

Exercise 1: Reverse a String

Exercise 2: Find the Second Largest Number

Exercise 3: Count Vowels

Wrap-up

Key Concepts Learned

What's Next

Further Reading

💻 Code Samples