This is really similar to Fibonacci Sequence problem. I understand the DP optimization with Fibonacci function using the for loop, but I'm having hard time to connect to this problem.
The recursion function I want to optimize is:
def cal(n):
if n <= 0:
return 1
else:
return cal(n-25)+cal(n-26)
Something like this may help:
(It's inspired by previous post)
from functools import cache
#cache
def cal(n):
if n <= 0:
return 1
else:
return cal(n-25) + cal(n-26)
print(cal(100))
The idea of a "for loop optimization" is that we calculate cal(0), cal(1), cal(2), ... consecutively. And when we want cal(n), we already have cal(n-25) and cal(n-26) stored in an array.
So, the following solution has linear complexity for non-negative n:
def cal(n):
mem = [1] # cal(0) is 1
for i in range(1, n + 1):
num = 1 if i - 25 < 0 else mem[i - 25]
num += 1 if i - 26 < 0 else mem[i - 26]
mem.append (num)
return mem[-1]
One can further optimize to make all the values cal(1), cal(2), ..., cal(n) globally available after calculating the last of them.
I'm trying to wrap my head around functional programming using F#. I'm sticking to purely mathematical problems for now.
My current problem is simple enough: to write a function that takes an integer N and outputs a list of all the factors of N
Because of the similarities between sequences and C# IEnumerables formed by yield return I got this solution:
let seqFactorsOf n =
seq { for i in 2 .. (n / 2) do if n % i = 0 then yield i }
I don't think lists can be generated that way, though, so I turned to List.unfold:
let listFactorsOf n =
2 |> List.unfold (fun state ->
if state <= n / 2 then
if state % 2 = 0 then
Some (state, state + 1)
else
//need something here to appease the compiler. But what?
else
None)
My other attempt uses the concept of matching, with which I'm almost totally unfamiliar:
let listFactorsOf_2 n =
2 |> List.unfold(fun state ->
match state with
| x when x > n / 2 -> None
| x when n % x = 0 -> Some(x, x + 1)
//I need a match for the general case or I get a runtime error
)
Is there a way to create such list using List.unfold? Please notice that I'm a beginner (I started F# 3 days ago) and the documentation is not very kind to newbies, so if you'd try to be as didactic as possible I would appreciate it a lot.
First - yes, of course lists can be generated using that for..in syntax (it's called "list comprehensions" by the way). Just put the whole thing in square brackets instead of seq { }:
let seqFactorsOf n =
[ for i in 2 .. (n / 2) do if n % i = 0 then yield i ]
As for unfold - every iteration is required to either produce an element of the resulting list (by returning Some) or to signal end of iteration (by returning None). There is nothing you can return from the body of unfold to indicate "skipping" the element.
Instead, what you have to do is to somehow "skip" the unwanted elements yourself, and only ever return the next divisor (or None).
One way to do that is with a helper function:
let rec nextDivisor n i =
if n % i = 0 then Some i
else if i >= n/2 then None
else nextDivisor n (i+1)
Let's test it out:
nextDivisor 16 3
> Some 4
nextDivisor 16 5
> Some 8
nextDivisor 16 10
> None
Now we can use that in the body of unfold:
let listFactorsOf n =
2 |> List.unfold (fun state ->
match nextDivisor n state with
| Some d -> Some (d, d + 1)
| None -> None
)
As a bonus, the construct match x with Some a -> f a | None -> None is a well-known and widely used concept usually called "map". In this particular case - it's Option.map. So the above can be rewritten like this:
let listFactorsOf n =
2 |> List.unfold (fun state ->
nextDivisor n state
|> Option.map (fun d -> d, d+1)
)
I decided to incorporate recursions into my code and for that I want to train using it. I have a recursive function which finds the max element in the given array.
def maxx(lst):
if len(lst) == 1:
return lst[0]
else:
m = maxx(lst[1:])
return m if m > lst[0] else lst[0]
maxx([1,3,2])
> 3
I tried to draw a sequence of events and I can't get why the code above works?
Because according to what I see it should be 2 not 3. Where is my mistake? Is there a method that correctly expands the recursion and thus helps you to understand its flow?
Depth 0:
lst = [1,3,2]
m = maxx([3,2])
Depth 1:
lst = [3,2]
m = maxx([2])
Depth 2:
lst = [2]
returning 2
Back to Depth 1:
lst = [3,2]
m = 2
return m (2) if m > lst[0] (3) else return lst[0] (3)
returning 3
Back to Depth 0:
lst = [1,3,2]
m = 3
return m (3) if m > lst[0] (1) else return lst[0] (1)
returning 3
The reply by Alfabravo is correct, I think that's where you lost track of lst[0] as you went back up the tree.
I'm new to Julia and learning use of Map, reduce, filter.
It is becoming very hard for me to comprehend how it can replace for and while loops.
For ex for below code, I would like to replace for loop
function addMultiplesOf3And5(N::Int)
sumOfMultiples = 0
if(N == 3)
return sumOfMultiples + N
end
for i = 3:N-1
if(i % 3 == 0 && i % 5 == 0)
continue
elseif(i % 3 == 0)
sumOfMultiples += i
elseif(i % 5 == 0)
sumOfMultiples += i
end
end
return sumOfMultiples
end
I would really appreciate the help.
Update :
This is what I did after going through tutorials
function addMultiplesOf3And5(N::Int)
array = range(1,N-1)
return reduce(+, map(x -> multiples_of_3_Or_5(x), array))
end
function multiples_of_3_Or_5(n)
if(n % 3 == 0 && n % 5 == 0)
return 0
elseif(n % 3 == 0)
return n
elseif(n % 5 == 0)
return n
else
return 0
end
end
Final:
function addMultiplesOf3And5(N::Int)
array = range(1,N-1)
return reduce(+, filter(x -> ((x%3==0)$(x%5==0)), array))
end
To understand how you can replace your 'for loop + if block' code with 'map / reduce / filter' you need to know exactly how they work and why they might be chosen instead.
1. The map function
map is a function that takes a function variable and a list as arguments, and returns a new list, where each element is the result of applying the function to each element of the old list. So for example if your variable f refers to a function f(x) = x + 5 you defined earlier, and you have a list L=[1,2,3,4,5], then map(f, L) will return [f(L[1]), f(L[2]), f(L[3]), f(L[4]), f(L[5])]
So if you have code like:
f(x) = x + 5;
L = [1,2,3,4,5];
A = zeros(5);
for i in L
A[i] = f(i);
end
You could rewrite this as a mapping operation like so:
A = map(x -> x + 5, [1,2,3,4,5]);
2. The reduce function
reduce takes a binary function variable (i.e. a function that takes two arguments) and a list as arguments. What it does is probably best explained by an example. Calling reduce with the + operator, and list [1,2,3,4,5] will do the following:
Step 1: [1, 2, 3, 4, 5] % : 5 elements
Step 2: [1+2, 3, 4, 5] % [3,3,4,5] : 4 elements
Step 3: [3+3, 4, 5] % [6, 4, 5] : 3 elements
Step 4: [6+4, 5] % [10, 5] : 2 elements
Step 5: [10+5] % [15] : 1 elements
result: 15
i.e. we have reduced the list to a single result by successively applying the binary function to the first pair of elements, consuming the list little by little.
So if you have code like:
f(x,y) = x + y
L = [1,2,3,4,5];
A = L[1];
for i in 2:length(L)
A = f(A, L[i])
end
you could rewrite this as a reduce operation like so:
A = reduce(x,y -> x+y, [1,2,3,4,5])
3. The filter function
filter takes a predicate function (e.g. iseven, isnull, ==, or anything that takes an argument and performs a test on it, resulting in true or false) and a list, tests each element of the list with the function and returns a new list that only contains the elements that pass that test. e.g.
filter(iseven, [1,2,3,4,5]) # returns [2,4]
The answer to your problem
If I understand correctly, addMultiplesOf3And5 takes a number N (e.g. 20), and does the following:
filter out all the elements that can be divided by either 3 or 5 from the list [1,2,3,...,20]
successively add all elements of the resulting list together using a reduce function.
You should be able to use the above to figure out the exact code :)
Not sure what the function in the question is supposed to calculate, but:
addMult3or5(N) = N==3 ? 3 : sum(filter(x->((x%3==0)$(x%5==0)),3:N-1))
calculates the same thing.
sum is a a reduce-like function for the + operation.
Hope this helps clarify.
Also, $ is the exclusive-or operation in Julia.
I need to write a recursive function that returns the remainder of two numbers. Here's what I wrote:
def remainder(a,b):
if a==b:
return 0
else:
k=a-b
return a-b + remainder(b,a-k)
If we test remainder(5,3) the function will return 2 and it's correct but if we test remainder(15,3),
we'll get 12 and its false. I just don't know how to debug it.
You are missing a case: (when a < b)
def remainder(a,b):
if a<b: #trivial: remainder = a - b*0 = a
return a
else: #reduce the problem to a simple one
return remainder(a-b, b)
Test:
print remainder(15,3)
Output:
0
Here if you are lazy and don't want to write more than 2 lines:
def remainder(a,b):
return a if a < b else remainder(a-b, b)
It should be something like this :
def remainder(a,b):
if a<b:
return a
else:
return remainder(a-b,b)
You can do:
def remainder(a, b):
if a < b:
return a
return remainder(a - b, b)
Examples:
>>> remainder(15, 3)
0
>>> remainder(14, 3)
2
>>> remainder(13, 3)
1
If a < b then it means we're done: we know the result of the computation and we can return a. Otherwise we need to subtract b from a and we call remainder again recursively. This can then repeatedly continue until a is smaller than b. Once that happens the recursion stops (i.e., it won't call remainder again) but we're able to return the result.