How to Solve Any Dynamic Programming Problem

January 9,2021 - 4 min read
How to Solve Any Dynamic Programming Problem

Coding is perceived as a coveted skill that everyone should learn because of the promising career it has to offer. The year 2020 highlighted the importance of technology and the career prospects of coding. The IT industry was one of the few sectors that registered growth even when other sectors were struggling to survive because of the pandemic.

Coding is nothing but the process of communicating with the computer in a language it can understand. The first mention of programming languages was somewhere in 1950’s and since then it has only developed and formed the base of all the modern technology we use today. The definition of coding is simple but its implication is not. There are a variety of concepts that you have to master in order to become an ace coder or programmer. One such important but complex concept in the world of coding is dynamic programming. Many coders fear this topic and maybe that’s why most company ask questions related to dynamic programming in their interviews. In this blog we are going to learn How to Solve Any Dynamic Programming Problem.

Dynamic Programming is basically an optimized version of recursive methods. In a recursive solution, there are repeated calls for same inputs, this method is optimized using Dynamic Programming. In this optimized version, we simply store the results of subproblems, in order to avoid re-computation. Whenever we use dynamic programming, the time complexity is reduced from exponential to polynomial.

Dynamic programming is a very efficient way of solving any problem in programming, however, according to a majority of developers, it is also one of the most difficult concepts of programming. Interestingly, many companies recruiting for software developers ask questions related to dynamic programming in their interviews. There are some who say that these questions aren’t really effective in evaluating someone’s ability to perform in a company as a developer.

Nonetheless, Dynamic Programming continues to be either a stepping stone or a roadblock that developers find up on their way to find a job as a developer in a company they love. Even candidates with significant experience building software products, feel nervous or underconfident while solving a dynamic programming problem in a coding interview.

If you are scared, we advise you to go on reading the blog as we have some amazing dynamic programming tips for you. One of the best things about dynamic programming questions are that they are predictable and easy to pattern match. Hence, they allow us to be more prepared before appearing in any coding interview.

Dynamic programming might seem pretty complex on the outside, and give you an impression that a person who solves them is a genius at algorithms. The reality is that the one who gets the correct answer by applying dynamic programming is just more prepared than you are! In this blog, we make sure that everyone reading it is as prepared for it as their successful counterparts. We will go over all the significant steps that are important to figure out a solution to a dynamic programming problem. We would understand how to identify a Dynamic Programming problem, identify the problem variables, comprehend and express the recurrence relation and the base cases. After understanding the problem in detail, we would then decide if the implementation would be iteratively or recursively. The final step would be to add memorization and determine time complexity. Let’s get started with our guide that will help you solve any dynamic programming problem.

  • Recognizing the Problem

    The first step is to decide whether a problem can be solved using Dynamic Programming or not. It is the first and often the most difficult step in the whole process of solving the problem. Dynamic Programming is essentially just an efficient technique to solve a problem by breaking them down into a collection of simpler subproblems where you solve each of those subproblems just once and then store their solutions. The advantage of storing the solution is that the next time the same subproblem occurs, you can simply look up the solution you previously computed. Hence, this way computation time is saved and also storage space. In order to recognise a dynamic programming problem, you have to ask if the solution can be expressed as a function of solutions to similar smaller problems.

  • Identifying Variables

    After we have decided that there is a recursive structure between the subproblems, we need to now express the problem in terms of some function parameters and decide which of these parameters are changing. One popular example where there is a single changing parameter is the problem of determine the nth term of a Fibonacci series.

    One way to determine the number of changing parameters is to list the several subproblems and compare their parameters. It’s important to enumerate the changing parameters and determine the number of subproblems to solve. After identifying the subproblem and the changing and other static parameters, we now have the describe our sub-problems.

  • Expressing the Recurrence Relation

    It is very important to clearly define the recurrence relation step in order to understand the problem and make the process of solving the problem significantly easier. This step comes after you have identified that the recurrence relation exists and specified the problems in terms of static and changing parameters.

  • The Base Cases

    In dynamic programming, a base case is a subproblem that cannot be simplified further. It is basically a subproblem that doesn’t depend on any other subproblem. In order to find the base case of any dynamic programming, you should try a few examples and identify the subproblem that cannot be simplified further.

  • The Implementation

    We have come a long way in the entire process of solving a problem through dynamic programming. When it comes to implementation, we can either solve it recursively or iteratively. However, irrespective of the approach you choose, you would have to find the recurrence relation and the base cases. When it comes to deciding the approach, you should consider the trade-offs before arriving to a conclusion.

    Recursive solutions are often fast in execution and iterative approach is slow because it needs to the same work regardless of the input. Recursive solutions, unlike iterative ones, are easier to understand and code. However, it completely depends on the solution and the programmer when it comes to choosing an implementation method for a problem.

  • Memoization

    In dynamic programming, Memoization is a crucial technique which is used for storing the results of expensive function calls and returning the values whenever required. We add Memoization so that the solutions for subproblems aren’t computed repeatedly and hence we don’t end up with exponential time complexities. In short, it means that you should store the result of your function before every return statement and then look up for the result before you start doing any other computation.

  • Time complexity

    Now that we have come a long way in the entire process of going through the entire dynamic program problem, it’s time for computing and analysing the time complexity. It is a complex task to calculate the time complexity, however, there are some simple rules that can make the task much easier. Whenever, you compute the time complexity of a dynamic programming problem, make sure you count the number of states which will depend on the number of variable parameters in your problem. The work done at each state is also important and if one state has been computed then calculate and analyse how much work do you have to do to compute the last state.

With this, we have successfully analysed the dynamic programming problem and analysed each aspect of the problem and the entire process of solving the problem. The various steps that we went through provides a robust framework for systematically solving any dynamic programming problem. It is highly recommended to practice this approach on a few dynamic programming problems to ace your skills and be fully prepared the next time someone asks you a dynamic programming problem. We suggest you to solidify your understanding of various concepts of dynamic programming like base cases, recursive and iterative approach, memorization and other concepts mentioned above. You should also practice solving various dynamic programming problems in other computing languages. While you are practicing various dynamic programming problems online by following the steps we went through, keep in mind that you have to learn ideas and concepts and not just memorise problems. It’s easy to learn ideas because they are small in number compared to wide variety and number of problems present out there. We hope you make the most of this blog and learn to solve problem through the method of dynamic programming. We wish you all the best for your journey in understanding a complex programming concept.

Competitive coding is a great way of making your way in the world of technology. Be it making your way to the top IT companies or just to ace your coding skills, competitive programming is always beneficial! Talking about competitive coding, dynamic programming is the most popular concept. Companies like asking questions that revolves around dynamic programming maybe because they think it’s the easiest and the most efficient way of choosing the best from the lot. If you want to be picked, make sure dynamic programming is on your tips.

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