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Dynamic programming is a technique that divides the problems into sub-problems and saves results for future use so that we cannot re-evaluate the results. The subproblems will optimize as ideal sub-properties to optimize the overall solution. The primary purpose of dynamic programming is to solve optimization problems. At AlgoMonster, optimization problems mean that the problem is minimal or maximum when we try to find out. The dynamic programming ensures the optimal resolution of an issue in the event of a solution.

Examples of dynamic programming

  1. Sydney – Perth minimum cost
  2. Study for Economic Feasibility
  3. 0/1 problem with a knapsack
  4. Alignment sequence problem


How does the programming dynamic work?

The following steps follow the dynamic programming:

  • The complex problem will break into simpler subproblems.
  • We will find the best solution for these under-problems.
  • Save results of subproblems (memoization). The process for storing the results of subproblems is called storage.
  • You use them again to calculate the same sub-problem more than once.
  • Calculate the outcome of the complex problem.

The five steps above are fundamental steps for dynamic programming.

The problems that overlap sub-problems and optimum substructures, In this case, an optimal substructuring means that can solve the optimization problem by combining all subproblems with an optimum solution.

When programming is dynamic, spatial complexity will increase as we save the intermediate results and reduce time complexity.

Take the example of the Fibonacci figure. As we all know, the numbers of Fibonacci are a series of numbers, each of which is the total of the two earlier numbers. The first few numbers of Fibonacci are 0, 1, 2, 3, 5, and 8, and continue.

If the nth Fibonacci number do ask to calculate, we can do this in the following equation,

Fib(n) = fib(n-1) + Fib(n-2), n > 1 Fib(n-2)

As can be seen here, we broke it up into two more minor problems, Fib(n-1) and Fib(n-2, to solve the overall problem (i.e., Fib(n)). It indicates that we can use DP to resolve it.

Dynamic Programming Characteristics.

Look at a problem’s features that tell us that we can use the DP to solve it.

Sub-problems overlapping.

The smaller versions of the original problem are sub-problems. If the solution to any problem involves solving the same subsection multiply, it overlaps sub-problems.

Optimum Property Substructure.

Any problem with an optimal substructure property can construct the entire optimal solution with the optimal solutions to his subproblems. As we know, for the number of Fibonacci,

Fib(n) = Fib(n-1) + Fib(n-2)

It shows clearly that an ‘n’ size problem will reduce to ‘n-1’ and ‘n-2’ subproblems. There is, therefore, the optimal substructure property for Fibonacci numbers.

Dynamical programming approaches.

Dynamic programming will base on two approaches:

  • Top-down approach
  • Bottom-up approach

 Top-down approach.

The top-down follows the technique of storage, while the bottom-up approach follows the method of tabulation. In this case, storage amounts to resource sum and caching sum. Recursion means the function itself being called up, while caching means the intermediate results being stored.


  • It is straightforward to comprehend and implement.
  • Only when needed, can it solve the underlying problems.
  • You can debug easily.


  • The recurrence technique used in the call stack occupies more memory. Sometimes, the overflow of the pile is too deep when the recursion is too deep.
  • It takes up more memory which degrades performance overall.

Approach from the bottom-up.

The bottom-up way is also one of the methods for implementing dynamic planning. The tabulation technique is applied to implement the dynamic programming approach. It can solve the same problem but can remove the recursion. There is no stack overflow problem and no recurring function overhead when the recursion will delete. We solve the issues and store the results in the matrix using this tabulation procedure.

Dynamic programming can apply in two ways:

  • Top-Down
  • Bottom-Up

The bottom-up approach is the recurrence method, which saves the memory space. The down-up is a starting algorithm, while the recursive algorithm begins at the end and is reverse. We start with the base case to find the solution in the bottom-up approach. In the Fibonacci series, the basis cases are 0 and 1, as we know. As the path to the bottom begins with the primary issues, we start from 0 and 1.

How can a DP problem be identified?

Dynamic Programming does usually use for problems with optimization. Can find many possible solutions to such issues. Each key has a value, but we would like to find a solution with an optimum value (at least or at most).

The overlapping subproblem, i.e., you would repeatedly need the solutions to the subproblems to find a final solution, satisfies all dynamic programming problems.

Get to know dynamic programming.

Dynamic programs are reputed to be challenging to handle, but many programs teach their algorithms without explaining how they will find them. Dynamic programming issues help you to make your solution the shortest. Learning to find these solutions in a bottom-up method could ignite your programming chops rather than the traditional top-down problem solution.

Dynamic programming courses and certificates.

edX provides courses in dynamic programming aspects that can help you understand the principles and practice of this specific methodology. Within the machine learning framework with UC San Diego, you can learn things like dynamic programming in partnership with leading institutions. This course teaches you to define your problem, develop data infrastructure or start solutions that facilitate complicated machine learning. UC San Diego also offers an extensive Algorithmic Design and Technics course, including introducing DP problems and possible solutions.

Dynamic Programming Ignite Your Career.

Only a great deal offers recursive solutions once the total number of algorithm resources will use to solve short problems. You will need dynamic programming to help you avoid the issue of the algorithm and reduce the maximum number of decision trees your program needs to create to solve. The reputation of dynamic programming is a difficult one. Still, if you understand the basics of finding suitable algorithms and how this stylish solution can create efficient solutions, you are welcome to add to every programming team.