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How to turn this memoized recursive solution into a 'bottom-up' iterative one?

I have a solution to a problem that uses dynamic programming. I need help turning this from a recursive solution into an iterative one.
The function takes in a number and follows the three rules:
it may divide the number in half
it may subtract one
it may add one
until the number is 1. My goal is to find the minimum number of steps it takes to do this.
Here is my solution:
def solution(n):
n = int(n)
memo = {}
return memoized_fuel_injection_perfection(n, memo)
def memoized_fuel_injection_perfection(n, memo):
if n == 1:
return 0
if n == 2:
return 1
if n in memo:
return memo[n]
if n % 2 == 0:
if n not in memo:
memo[n] = memoized_fuel_injection_perfection(n//2, memo) + 1
return memo[n]
return min(memoized_fuel_injection_perfection(n-1, memo), memoized_fuel_injection_perfection(n+1, memo)) + 1
But when I input numbers larger than 300 digits long, I am getting a recursive error. How can I turn this into an iterative solution? Any help or guidance is appreciated.
Here is an iterative solution I created, but I am getting MemoryError with very large inputs. Is there some way I can optimize storing the variables so I don't have to compute them for every number?
def solution(n):
memo = {}
memo[0] = 0
memo[1] = 0
memo[2] = 1
n = int(n)
for i in range(3, n+1):
if i % 2 == 0:
memo[i] = memo[i//2] + 1
else:
memo[i] = min(memo[i//2], memo[i//2 + 1]) + 2
return memo[n]

The problem you said you're having with writing an iterative solution is the use of memoized_fuel_injection_perfection(n+1, memo), which makes it tricky to determine what order to compute results in. The key is that you cannot repeatedly go down this code path indefinitely. If you could, even your recursive solution would be invalid.
Immediately after a +1 or -1 operation, you always perform a divide-by-2. We can fuse the +1 or -1 with the divide-by-2, producing an operation that cannot increase the number. The core of an iterative solution would then look like this:
if n % 2 == 0:
table[n] = table[n//2] + 1
else:
table[n] = min(table[n//2], table[n//2+1]) + 2
Can you complete things from there? (You'll need a way to avoid computing results for every positive integer less than n.)

Here's my attempt:
def solution(n):
def is_even(n): # helper function
return n % 2 == 0
possible_nodes = {1} # 1 is the destination
consider = [n] # stack: numbers to be considered
while consider: # as long as non-empty
x = consider.pop() # now think about where can we move from x
if x in possible_nodes: # if it is already handled before
continue
if is_even(x): # if even, we just halve it
consider.append(x//2)
else: # otherwise, -1 or +1
consider += [x-1, x+1]
possible_nodes.add(x) # mark x as 'considered'
steps = {1: 0} # dict to store min steps
for x in filter(is_even, sorted(possible_nodes)): # odds calculated only when needed
if x//2 not in steps: # if x//2 was not computed, x//2 must be odd
steps[x//2] = min(steps[x//2 - 1], steps[x//2 + 1]) + 1
steps[x] = steps[x//2] + 1
return steps[n] if is_even(n) or n == 1 else min(steps[n-1], steps[n+1]) + 1
n = 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003333000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003483983333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333
print(solution(n)) # 2467
print(*(solution(i) for i in range(1, 21)))
# 0 1 2 2 3 3 4 3 4 4 5 4 5 5 5 4 5 5 6 5
The code basically consists of two steps. In the first step, it enumerates all the possible steps from n, under the rule that for x,
if x is even, we only consider x//2 as the next step; this strategy is adopted also in your code;
if x is odd, we consider x-1 and x+1 as the next step.
I did this because calculating the minimum steps for all values up to n is wasteful. (Actually at first I tried to do so by initiating something like [None] * n, but it seems like python cannot handle such a long list. Even if it can, I guess that would be extremely slow.)
The next step is calculating the minimum steps, starting from the smallest number. I address the problem of accessingsteps[x+1] by "not calculating steps[x] for odd x eagerly." We only calculate steps[x] for even x eagerly, but lazily for odd x.
The rationale is the following; by the time odd x is needed, it must be the case that x is either k//2 - 1 or k//2 + 1 for some even k, which must be larger than x + 1. Since x + 1 is even, step[x+1] must have been already calculated, by the construction of the for loop.

The incrementing here continues to return a 0 value

I was writing the solution to this codewars problem however I've ran into a bit of an issue.
Problem statement:
Write a function, persistence, that takes in a positive parameter num and returns its multiplicative persistence, which is the number of times you must multiply the digits in num until you reach a single digit, e.g.:
persistence(39) # returns 3, because 39=27, 27=14, 1*4=4 and 4 has only one digit
def persistence(n, t=1, x=0):
if len(str(n)) > 1:
number = [int(i) for i in str(n)]
for i in number:
t = t * i
if len(str(t)) > 1:
x += 1
return(persistence(t,x))
else:
return(x)
else:
return 0
I can't quite figure out what the error is in this code. My hunch is that it's either a parameter error or the way the return() value is placed.
In essence, the code for distilling an integer to it's multiples is correct, so I just added an extra parameter to persistence; setting x = 0 and making it so that each time the if condition was fulfilled it would increment that exact x value. Once the number was distilled, simply output x. Yet it continues to simply output 0 as the final answer. What's the problem here?
Edit: Solution was in the comments, didn't realise how the parameters were passing. Correct version is:
return(persistence(t,1,x))
Also had to set x = 1 for the logic to work on codewars.

There are 2 flaws in Your code:
return(persistence(t,x))
should be
return(persistence(t,1,x))
otherwise the value of x will be assigned to t and x will be defaulted to 0.
Then you must increment x directly after the first test, otherwise You will miss one iteration.
Another way to calculate this is not to switch over to strings, but to do it numerically:
def persistence(n):
iterations = 0; # no iterations yet
while n > 9: # while n has more than 1 digit:
product = 1 # neutrum for result product
while n > 0: # while there a digit to process:
digit = n % 10 # retrieve the right most digit
product *= digit # do the multiplication
n = n // 10 # cut off the processed digit
iterations += 1 # increment iterations
n = product # let n be the newly calculated product
return iterations # return the result

I think you your function's parameters work not as you expect them to do.
When you call function persistence(t, x), the first argument n should become t, and second argument x, should become new x. But in your function, x becomes new t because of their position.
It is quite useful to have bunch of print statements to reveal the bug.
def persistence(n, x=1, t=1):
print('x:', x)
if len(str(n)) > 1:
number = [int(i) for i in str(n)]
for i in number:
t = t * i
print('t:', t)
if len(str(t)) > 1:
x += 1
print('x has changed:', x)
return persistence(t, x)
else:
return x
else:
return 0
print(persistence(39))
print('---------------')
print(persistence(999))
print('---------------')
print(persistence(4))

Passes all test cases with two changes:
You were not updating your n with the new t everytime
Your x was being set to 0 every time. That should be set to 1 in the beginning (default value)
def persistence(n, t=1, x=1):
if len(str(n)) > 1:
number = [int(i) for i in str(n)]
for i in number:
t = t * i
if len(str(t)) > 1:
x += 1
return (persistence(n=t, x=x))
else:
return (x)
else:
return 0
Actually, you can write it without needing both parameters t and n. Just one n is fine as shown below:
def persistence(n, x=1):
if len(str(n)) > 1:
number = [int(i) for i in str(n)]
t = 1
for i in number:
t = t * i
if len(str(t)) > 1:
return x + (persistence(n=t, x=x))
else:
return (x)
else:
return 0

Is there a way to make this reverse factorial code run more efficiently

I am just starting to learn python and made a program where it calculates the factorial number based on the factorial.
For example if I give the program the number 120 it will tell me it's factorial is 5
anyways my question is how can I make this code more efficient and faster.
Num = int(input())
i=0
for i in range(0,Num):
i = i + 1
x = Num/i
Num = x
if (x==1):
print(i)

Multiplications are much faster than divisions. You should try to reach the number with a factorial instead of dividing it iteratively:
def unfactorial(n):
f,i = 1,1
while f < n:
i += 1
f *= i
return i if f == n else None
unfactorial(120) # 5

A few things you can do:
Num = int(input())
i=0 # your for loop will initialize i, you don't need to do this here
for i in range(0,Num):
i = i + 1 # your for loop will increment i, no need to do this either
x = Num/i # you don't need the extra variable 'x' here
Num = x
if (x==1):
print(i)
You can rewrite this to look something like:
for index in range(1, number): # start range at 1
number /= index # this means; number = number / index
if number==1:
return index

Compute the factorials in ascending order until you reach (or exceed) the factorial you are looking for, using the previous factorial to efficiently compute the next.

def reverse_factorial(num):
i = 1
while num > 1:
i += 1
num /= i
return i
print(reverse_factorial(int(input())))

Why is this code not running fully? It doesn't run line 53

I made myself an exercise with python since I am new. I wanted to make a rever LMC calculator ( Least common multiple ) but for some reason, something as simple as a print in a loop doesn't seem o work for me. I would appreciate some help since I am stuck on this weird issue for 20 minutes now. Here is the code:
import random
import sys
def print_list():
count_4_print = 0
while count_4_print < len(values):
print(values[count_4_print])
count_4_print += 1
def lcm(x, y):
if x > y:
greater = x
else:
greater = y
while True:
if (greater % x == 0) and (greater % y == 0):
lcm1 = greater
break
greater += 1
return lcm1
def guess(index, first_guess, second_guess):
num = 1
while lcm(first_guess, second_guess) != values[num - 1]:
first_guess = random.randrange(1, 1000000)
second_guess = random.randrange(1, 1000000)
num += 1
num = 1
if lcm(first_guess, second_guess) == values[num - 1]:
return first_guess, second_guess
num += 1
lineN = int(input())
values = []
count_4_add = 0
count_4_guess = 0
for x in range(lineN):
values.append(int(input()))
count_4_add += 1
if count_4_add >= lineN:
break
print_list()
for x in range(lineN + 1):
first, second = guess(count_4_guess, 1, 1)
count_4_guess += 1
print(first + second)
# this ^^^ doesn't work for some reason
Line 57 is in the while loop with count_4_guess. Right above this text, it says print(first_guess + second_guess)
Edit: The code is supposed to take in an int x and then prompt for x values. The outputs are the inputs without x and LMC(output1, output2) where the "LMC" is one of the values. This is done for each of the values, x times. What it actually does is just the first part. It takes the x and prompts for x outputs and then prints them but doesn't process the data (or it just doesn't print it)

Note: From looking at your comments and edits it seems that you are lacking some basic knowledge and/or understanding of things. I strongly encourage you to study more programming, computer science and python before attempting to create entire programs like this.
It is tough to answer your question properly since many aspects are unclear, so I will update my answer to reflect any relevant changes in your post.
Now, onto my answer. First, I will go over some of your code and attempt to give feedback on what could improved. Then, I will present two ways to compute the least common multiple (LCM) in python.
Code review
Code:
def print_list():
count_4_print = 0
while count_4_print < len(values):
print(values[count_4_print])
count_4_print += 1
Notes:
Where are the parameters? It was already mentioned in a few comments, but the importance of this cannot be stressed enough! (see the note at the beginning of my comment)
It appears that you are trying to print each element of a list on a new line. You can do that with print(*my_list, sep='\n').
That while loop is not how you should iterate over the elements of a list. Instead, use a for loop: for element in (my_list):.
Code:
def lcm(x, y):
if x > y:
greater = x
else:
greater = y
while True:
if (greater % x == 0) and (greater % y == 0):
lcm1 = greater
break
greater += 1
return lcm1
Notes:
This is not a correct algorithm for the LCM, since it crashes when both numbers are 0.
The comparison of a and b can be replaced with greater = max(x, y).
See the solution I posted below for a different way of writing this same algorithm.
Code:
def guess(index, first_guess, second_guess):
num = 1
while lcm(first_guess, second_guess) != values[num - 1]:
first_guess = random.randrange(1, 1000000)
second_guess = random.randrange(1, 1000000)
num += 1
num = 1
if lcm(first_guess, second_guess) == values[num - 1]:
return first_guess, second_guess
num += 1
Notes:
The line num += 1 comes immediately after return first_guess, second_guess, which means it is never executed. Somehow the mistakes cancel each other out since, as far as I can tell, it wouldn't do anything anyway if it were executed.
if lcm(first_guess, second_guess) == values[num - 1]: is completely redundant, since the while loop above checks the exact same condition.
In fact, not only is it redundant it is also fundamentally broken, as mentioned in this comment by user b_c.
Unfortunately I cannot say much more on this function since it is too difficult for me to understand its purpose.
Code:
lineN = int(input())
values = []
count_4_add = 0
count_4_guess = 0
for x in range(lineN):
values.append(int(input()))
count_4_add += 1
if count_4_add >= lineN:
break
print_list()
Notes:
As explained previously, print_list() should not be a thing.
lineN should be changed to line_n, or even better, something like num_in_vals.
count_4_add will always be equal to lineN at the end of your for loop.
Building on the previous point, the check if count_4_add >= lineN is useless.
In conclusion, count_4_add and count_4_guess are completely unnecessary and detrimental to the program.
The for loop produces values in the variable x which is never used. You can replace an unused variable with _: for _ in range(10):.
Since your input code is simple you could probably get away with something like in_vals = [int(input(f'Enter value number {i}: ')) for i in range(1, num_in_vals+1)]. Again, this depends on what it is you're actually trying to do.
LCM Implementations
According to the Wikipedia article referenced earlier, the best way to calculate the LCM is using the greatest common denominator.
import math
def lcm(a: int, b: int) -> int:
if a == b:
res = a
else:
res = abs(a * b) // math.gcd(a, b)
return res
This second method is one possible brute force solution, which is similar to how the one you are currently using should be written.
def lcm(a, b):
if a == b:
res = a
else:
max_mult = a * b
res = max_mult
great = max(a, b)
small = min(a, b)
for i in range(great, max_mult, great):
if i % small == 0:
res = i
break
return res
This final method works for any number of inputs.
import math
import functools
def lcm_simp(a: int, b: int) -> int:
if a == b:
res = a
else:
res = abs(a * b) // math.gcd(a, b)
return res
def lcm(*args: int) -> int:
return functools.reduce(lcm_simp, args)
Oof, that ended up being way longer than I expected. Anyway, let me know if anything is unclear, if I've made a mistake, or if you have any further questions! :)

While loop vs a for loop in finding sum of multiples of 3 and 5 under 1000

not really familiar with while loops but I thought it was an alternative so this may be an elementary mistake.
I need to look for the sum of natural numbers under 1000 that are multiples of 3 and 5. E.g for under 10
multiples of 3 and 5 < 10 = 3,5,6,9
sum = 23
My code using a for loop works and is as follows (this was my initial solution):
def multiple():
lst = []
for x in range(334): #didn't know how else to use a for loop but to find the largest value of x using a calculator
if 3*x < limit:
lst.append(3*x)
if 5*x< 1000:
lst.append(5*x)
if (3*x > 1000) and (5*x > 1000): #unnecessary in a forloop with a range but this is just to maintain symmetry with while loop
break
lst2 = list(set(lst)) #remove duplicates
print(sum(lst2))
multiple()
My code using a while loop(this solution doesn't even come out in te console --> maybe this is were the error lies):
def multiple():
lst = []
while True:
x = 1
if 3*x < 1000:
lst.append(3*x)
if 5*x< 1000:
lst.append(5*x)
if (3*x > 1000) and (5*x > 1000):
break
x += 1
lst2 = list(set(lst)) #remove duplicates
print(sum(lst2))
multiple()
Desired output:
233168
In addition to how to rectify the while loop, any improvements on my for loop or while loop would also be welcome. Thanks

Critically Debugging
Since you're new, let's take this opportunity to analyze the bug before we solve it. Note first that you did not notice any printouts at all. Therefore, your print() statement was either not running, or was printing only spaces. We can rule out the latter since sum() will return an integer.
Therefore, the print() is never run. The function is defined and called correctly, so that's not the issue. Now notice while True:; this is an early warning sign. If the while loop never ends, the print() will never run. We do notice there are multiple break statements that should stop the loop, but it's likely there's an issue with them.
Now we check how the loop updates. First, note the i+=1. That seems right. However, i=1 is also within the while loop. This cannot be correct, since every iteration i will be reset. This would cause the loop to run forever.
This type of critical analysis of code is only built through practice, but hopefully this answer offers some insight into how you could have fixed this issue yourself (and how I looked through your code).
Also note that adding print statements into the while loop to test would have allowed you to notice that i was always 1.
Working Code
def multiple():
lst = []
x = 1 # moved from line below
while True:
# x = 1 should not go here
if 3*x < 1000:
lst.append(3*x)
if 5*x< 1000:
lst.append(5*x)
if (3*x > 1000) and (5*x > 1000):
break
x += 1
lst2 = list(set(lst)) #remove duplicates
print(sum(lst2))
multiple()

A fairly straight forward approach is to iterate over every number from 1 to 1000 and check if it is divisible by 3 or 5, and then sum them all up.
total = sum(x for x in range(1, 1001) if x%3 == 0 or x%5 == 0)
total
# returns:
234168

Another solution:
Using for loop:
def multiple():
sum = 0
for _ in xrange(1, 1001):
if _ % 3 == 0 or _ % 5 == 0:
sum += _
return sum
print(multiple())
Using while loop:
def multiple():
sum = 0
cnt = 1
while cnt <= 1000:
if cnt % 3 == 0 or cnt % 5 == 0:
sum += cnt
cnt += 1
return sum
print(multiple())
output:
234168

#!/usr/bin/env python
def multiple():
lst = []
x=1
while (3*x < 1000) or (5*x < 1000):
if 3*x < 1000:
lst.append(3*x)
if 5*x < 1000:
lst.append(5*x)
x += 1
lst2 = list(set(lst)) #remove duplicates
print(sum(lst2))
multiple()