Welcome to our next exciting lesson, where we introduce the basics of Dynamic Programming — a powerful method for solving optimization, combinatorics, and other complex problems. Dynamic Programming provides an approach to solving problems by breaking them down into subproblems and storing the results of certain calculations that are likely to be used again. This method helps to avoid repetitive calculations, thus enhancing the efficiency of algorithms.

One of the famous introductory problems solved using the Dynamic Programming method is Fibonacci Series computation. Traditionally, if you have to compute the N-th Fibonacci number using recursion, you would solve smaller, overlapping subproblems multiple times, leading to exponential time complexity. However, by employing Dynamic Programming, you can store the result of each solved subproblem in a memoization array. The next time you require the result, you check the array first, thus avoiding recomputation and dramatically reducing the time complexity.

Here is what the solution may look like:

php
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1<?php
2
3function fibonacci($n, &$memo = []) {
4    if (array_key_exists($n, $memo)) {
5        return $memo[$n];
6    }
7    if ($n <= 1) {
8        return $n;
9    }
10    $result = fibonacci($n - 1, $memo) + fibonacci($n - 2, $memo);
11    $memo[$n] = $result;
12    return $result;
13}
14
15// Testing the fibonacci function
16echo fibonacci(10); // Output: 55
17
18?>

Remember, understanding Dynamic Programming requires a different mindset; it is all about recognizing and solving interrelated subproblems. Take a pause and let this approach sink in. Next up, we dive deep into practice problems to shape up your Dynamic Programming skills. We aim to provide you with the ability to simplify complex problems into smaller, manageable tasks to achieve efficient and effective solutions. Get ready to jump in!