# Introducing Dynamic Programming with Fibonacci Numbers

In this lesson, we'll use a dynamic programming technique called memoization to reduce the time complexity of the Fibonacci function.

## We'll cover the following

We are now going to use a dynamic programming technique to reduce the time complexity to linear.

## What is dynamic programming?

Dynamic programming algorithms solve problems by combining results of subproblems, just like in the divide and conquer algorithms.

## Characteristics

Most problems that can be solved with dynamic programming can also be solved with a divide and conquer approach. The difference between the two is that the dynamic programming approach is applicable when a problem has **overlapping subproblems**; i.e. the subproblems of a given problem are not independent such as when two subproblems share a sub-subproblem. An example would be the Fibonacci code we saw in the last lesson where `fib(3)`

had to be calculated in order to calculate both `fib(4)`

and `fib(5)`

.

Furthermore, dynamic programming problems have the **optimal substructure property**, which means that the overall optimal solution of the problem can be constructed from the optimal solutions of its subproblems. For Fibonacci numbers, we know that,

```
fib(n) = fib(n-1) + fib(n-2)
```

Hence, a problem of size ‘n’ can be reduced to subproblems of size ‘n-1’ and ‘n-2’. Therefore, Fibonacci numbers have the optimal substructure property.

Dynamic programming algorithms solve every subproblem just once and save the answer in a lookup table, thereby avoiding recalculating the answer every time the subproblem is encountered.

There are two dynamic programming patterns: memoization and tabulation.

## Memoization (top-down)

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