Added a fibonacci algorithm which uses memoization

[JDeepD]

Describe your change:

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Documentation change?

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[ghost]

Click here to look at the relevant links ⬇️

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• Formatted string literals (f-strings)
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[ghost]

As there is no test file in this pull request nor any test function or class in the file maths/fibonacci_sequence_memoization.py, please provide doctest for the function fibo

Please provide descriptive name for the parameter: n

[cclauss]

Prefill the cache...

Suggested change

	cache: dict[int, int] = {}
	cache: dict[int, int] = {1: 1, 2: 1}

[cclauss]

Suggested change

	elif num in [1, 2]:
	value = 1
	else:
	value = fibo(num - 1) + fibo(num - 2)
	value = fibo(num - 1) + fibo(num - 2)

[andrei-polukhin]

@JDeepD btw, why is it chosen to define cache at the module level? Classical memoization technique enables creating an outer and inner function, something like:

def fibo() -> Callable:
    cache: dict[int, int] = {1: 1, 2: 1}

    def func_(n: int) -> int:
        nonlocal cache
        ...

    return func_

... and then we do something like:

fibonacci = fibo()
fibonacci(5)

The code in this PR is perfectly workable, but what about showcasing the "classical" ways of solving the problems?

[JDeepD]

@JDeepD btw, why is it chosen to define cache at the module level? Classical memoization technique enables creating an outer and inner function, something like:

def fibo() -> Callable:
    cache: dict[int, int] = {1: 1, 2: 1}

    def func_(n: int) -> int:
        nonlocal cache
        ...

    return func_

... and then we do something like:

fibonacci = fibo()
fibonacci(5)

The code in this PR is perfectly workable, but what about showcasing the "classical" ways of solving the problems?

Yeah sorry, I will change it right away.

[JDeepD]

fibonacci = fibo()
fibonacci(5)

Rather than creating another object, wouldn't it be better if we just pass the num from fibonacci to fibo directly?
Something like:

def fibo(num: int) -> int: 
       cache: dict[int, int] = {}
       def fibo(num:int) -> int:
           ...
       return fibo(num) 

[JDeepD]

Wait I messed something while pushing

[andrei-polukhin]

Hmm, quite frankly, I never saw such an implementation. My checks led me to one classical book about competitive programming (link), where we can see such an implementation of memoization in C++:

/* ready[x] indicates whether the value of solve(x) has been calculated,
and if it is, value[x] contains this value.
The constant N has been chosen so that all required values fit in the arrays. */
bool ready[N];
int value[N];

int solve(int x) {
    // the function per se
    if (x < 0) return INF;
    if (x == 0) return 0;
    if (ready[x]) return value[x];
    int best = INF;
    for (auto c : coins) {
       best = min(best, solve(x-c)+1);
    }
    value[x] = best;
    ready[x] = true;
    return best;
}

This piece of code is structurally implemented as your PR: cache is defined at the module level. However, this example shows how to write the code in the fastest way (because writing inner and outer functions takes time in programming contests).

However, if we look at some other example, we can see we create an outer and an inner function in all cases: because we re-use this code and it should be written well.

Edit: regarding returning value, not function: it just then makes little sense to write an inner and outer function. The purpose of two of them is that each cache is defined separately which is extremely convenient when you have several different functions which use the memoization technique.

[JDeepD]

Yeah, I also think putting the cache outside the function is not good practice.Nevertheless, I have somehow managed to mess up the commit(It disappeared from remote smh). I will make a new PR shortly...

[andrei-polukhin]

OK, I'll take a look at it then :)

[JDeepD]

Thnx. The new PR is here #5856