## Dictionaries

Read this for more on dictionaries.

### Dictionaries

#### 11.6 Memos

If you played with the fibonacci function from Section 6.7, you might have noticed that the bigger the argument you provide, the longer the function takes to run. Furthermore, the run time increases quickly.

To understand why, consider Figure 11.2, which shows the

Count how many times

One solution is to keep track of values that have already been computed by storing them in a dictionary. A previously computed value that is stored for later use is called a

Whenever

If you run this version of

To understand why, consider Figure 11.2, which shows the

**call graph**for`fibonacci`

with `n=4`

:

`fibonacci`

with `n=4`

calls `fibonacci`

with `n=3`

and `n=2`

. In turn, `fibonacci`

with `n=3`

calls `fibonacci`

with `n=2`

and `n=1`

. And so on.Count how many times

`fibonacci(0)`

and `fibonacci(1)`

are called. This is an inefficient solution to the problem, and it gets worse as the argument gets bigger.One solution is to keep track of values that have already been computed by storing them in a dictionary. A previously computed value that is stored for later use is called a

**memo**. Here is a "memoized" version of`fibonacci`

:known = {0:0, 1:1} def fibonacci(n): if n in known: return known[n] res = fibonacci(n-1) + fibonacci(n-2) known[n] = res return res

`known`

is a dictionary that keeps track of the Fibonacci numbers we already know. It starts with two items: 0 maps to 0 and 1 maps to 1.Whenever

`fibonacci`

is called, it checks `known`

. If the result is already there, it can return immediately. Otherwise it has to compute the new value, add it to the dictionary, and return it.If you run this version of

`fibonacci`

and compare it with the original, you will find that it is much faster.