troubleshooting process dynamic programming optimal substructure
In general the problem can be using a three step process:
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Separate the problem into smaller subproblems.
Solving these problems optimally using this process three steps recursively. Use
optimal solutions to construct an optimal solution of original problem.
- The subproblems are themselves determined by dividing these into sub-problems and so on until it reaches the simplest case that is easy to solve.
- say that a problem has overlapping subproblems, meaning that one subproblem is used to solve many different problems larger.
- For example, the Fibonacci sequence F3 = F1 + F2 and F4 = F2 + F3, the calculation of each issue involves the calculation of F2. Since both F3 and F4 F5 necessary to calculate a naive method to calculate F5 may end up making F2 calculating two or more times.
overlapping subproblems When a method can waste time recalculating optimal solutions of subproblems already solved. 
To avoid repetition of calculations, you can save the solutions to the problems addressed in a way that if required the same problem then can be reused calculations made earlier.
- memoización This approach is called (no memorization).
- When you are sure that a particular solution and no longer needed can then be deleted, freeing up space.
- In some cases you can make computations that are known are required in later.
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