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 that can be solved using the Dynamic Programming method is the factorial computation. Traditionally, if you have to compute the factorial of N using recursion, you would solve smaller, overlapping subproblems multiple times, leading to exponential time complexity. However, by employing Dynamic Programming (as in the task mentioned above), you can store the result of each solved subproblem in a memoization table. The next time you require the result, you check the table first, thus avoiding re-computation and dramatically reducing the time complexity.

Here is what the solution looks like:

C++
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1#include <iostream>
2#include <unordered_map>
3
4// Function to compute factorial using memoization
5int factorial(int n, std::unordered_map<int, int>& memo) {
6    // Check if the result is already in the memoization table
7    if (memo.find(n) != memo.end()) {
8        return memo[n];
9    }
10    
11    // Base case: factorial of 0 or 1 is 1
12    if (n <= 1) {
13        return 1;
14    }
15    
16    // Recursively compute factorial and store it in the memoization table
17    memo[n] = n * factorial(n - 1, memo);
18    return memo[n];
19}
20
21int main() {
22    std::unordered_map<int, int> memo;
23    // Compute and print the factorial of 5
24    std::cout << factorial(5, memo) << std::endl;  // Output: 120
25    return 0;
26}

1. Checking the Memoization Table: Before performing any calculations, we check if the result for the given n is already stored in the memo table. This helps in avoiding redundant calculations.

2. Base Case Handling: The base case for factorial computation is when n is 0 or 1, and the result is 1.

3. Memoization and Recursion: If the result is not in the table, we calculate it recursively and store it in the table for future use.

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!