What are the key steps to solve a dynamic programming problem?

The key steps to solve a dynamic programming problem are:

1. Understanding the problem and defining the subproblems: Before starting to solve a dynamic programming problem, it is important to understand the problem statement and define its subproblems. Subproblems are smaller versions of the original problem that can be solved recursively or iteratively. For example, in the Fibonacci sequence problem, the subproblems are calculating the values of Fibonacci series until n-1 and n-2.

2. Deciding on the state representation: The state representation is a way of representing subproblems. The state can be identified by variables such as an index, a length, or a subset of items. For example, in the knapsack problem, the state is represented by weight and the remaining capacity of the knapsack.

3. Formulating the recursive relation: The recursive relation defines the solution of a subproblem in terms of its smaller subproblems. This is the most important step in dynamic programming. It involves identifying how the solution to the subproblem can be expressed in terms of the solution to smaller subproblems. For example, in the Fibonacci sequence problem, the recursive relation can be defined as F(n) = F(n-1) + F(n-2).

4. Identifying the base cases: Base cases are the smallest subproblems that can be directly solved. These are necessary to stop the recursion and provide a solution to the original problem. For example, in the Fibonacci sequence problem, the base cases are F(0) = 0 and F(1) = 1.

5. Constructing the solution: After solving the subproblems, we can construct the solution to the original problem using the results of the subproblems. For example, in the Fibonacci sequence problem, the solution is F(n).

6. Adding memorization or tabulation: To improve the efficiency of dynamic programming solutions, memoization or tabulation techniques can be used to reduce redundant computations. Memoization involves storing the results of solved subproblems in a table or array to avoid recomputing them, while tabulation involves solving all subproblems in a bottom-up approach and storing the results in a table or array.

Here is an example code in Java that implements the above steps to solve the Fibonacci sequence problem:

public int fibonacci(int n) {
    int[] cache = new int[n+1];
    return fibonacciSub(n, cache);
}

public int fibonacciSub(int n, int[] cache) {
    if(n < 2) {
        return n;
    }
    if(cache[n] != 0) {
        return cache[n];
    }
    int result = fibonacciSub(n-1, cache) + fibonacciSub(n-2, cache);
    cache[n] = result;
    return result;
}

In this code, we first define a memoization cache to store the results of subproblems. Then we define the recursive function ‘fibonacciSub‘ that takes the input ‘n‘ and the memoization cache ‘cache‘. The base case is when ‘n‘ is less than 2, where we simply return ‘n‘. Next, we check whether the solution to the subproblem has already been computed and stored in the cache. If it is present in the cache, we simply return the result. Otherwise, we recursively solve the subproblems and store the result in the cache before returning it. Finally, we define ‘fibonacci‘ function that calls ‘fibonacciSub‘ and returns its result.