I am trying to solve this question from google’s foobar. However, my code exceeds the time limit, even though when I tested it I got correct answers. How can I make my code more efficient?

Lovely Lucky LAMBs

Being a henchman isn’t all drudgery. Occasionally, when Commander
Lambda is feeling generous, she’ll hand out Lucky LAMBs (Lambda’s
All-purpose Money Bucks). Henchmen can use Lucky LAMBs to buy things
like a second pair of socks, a pillow for their bunks, or even a third
daily meal!

However, actually passing out LAMBs isn’t easy. Each henchman squad
has a strict seniority ranking which must be respected – or else the
henchmen will revolt and you’ll all get demoted back to minions again!

There are 4 key rules which you must follow in order to avoid a
revolt:
1. The most junior henchman (with the least seniority) gets exactly 1 LAMB. (There will always be at least 1 henchman on a team.)
2. A henchman will revolt if the person who ranks immediately above them gets more than double the number of LAMBs they do.
3. A henchman will revolt if the amount of LAMBs given to their next two subordinates combined is more than the number of LAMBs they
get. (Note that the two most junior henchmen won’t have two
subordinates, so this rule doesn’t apply to them. The 2nd most junior
henchman would require at least as many LAMBs as the most junior
henchman.)
4. You can always find more henchmen to pay – the Commander has plenty of employees. If there are enough LAMBs left over such that
another henchman could be added as the most senior while obeying the
other rules, you must always add and pay that henchman.

Note that you may not be able to hand out all the LAMBs. A single LAMB
cannot be subdivided. That is, all henchmen must get a positive
integer number of LAMBs.

Write a function called answer(total_lambs), where total_lambs is the
integer number of LAMBs in the handout you are trying to divide. It
should return an integer which represents the difference between the
minimum and maximum number of henchmen who can share the LAMBs (that
is, being as generous as possible to those you pay and as stingy as
possible, respectively) while still obeying all of the above rules to
avoid a revolt. For instance, if you had 10 LAMBs and were as
generous as possible, you could only pay 3 henchmen (1, 2, and 4
LAMBs, in order of ascending seniority), whereas if you were as stingy
as possible, you could pay 4 henchmen (1, 1, 2, and 3 LAMBs).
Therefore, answer(10) should return 4-3 = 1.

To keep things interesting, Commander Lambda varies the sizes of the
Lucky LAMB payouts: you can expect total_lambs to always be between 10
and 1 billion (10 ^ 9).

Languages

To provide a Python solution, edit solution.py To provide a Java
solution, edit solution.java

Test cases

Inputs:
(int) total_lambs = 10 Output:
(int) 1

Inputs:
(int) total_lambs = 143 Output:
(int) 3

Use verify [file] to test your solution and see how it does. When you
are finished editing your code, use submit [file] to submit your
answer. If your solution passes the test cases, it will be removed
from your home folder.

def answer(lambs):
    totals = [generous(lambs), stingy(lambs)]
    difference = max(totals) - min(totals)
    return difference

def generous(lambs):
    num = 1
    while True:
        total = 2**(num) - 1
        if total <= lambs:
            num += 1
        else:
            num -= 1
            break
    return num

def stingy(lambs):
    num = 1
    while True:
        total = (fibonacci(num + 2) - 1)
        if total <= lambs:
            num += 1
        else:
            num -= 1
            break
    return num

def fibonacci(num):
    if num > -1 and num < 3:
        return 1
    else:
        num = fibonacci(num - 1) + fibonacci(num - 2)
        return num

Short answer: don’t use recursion.

Slightly longer answer: don’t use recursion if you have more than a single recursive call of your function in the non-trivial paths without memoization.

Your fibonacci calls itself twice, except when num is in range(0,3). Those two calls will call fibonacci again twice (unless one of them is already in your base case), so we now end up with four calls. Let’s have a look at fibonacci(5) for example:

fibonacci(5) = fibonacci(5 - 1) + fibonacci(5 - 2)
             =   (fibonacci((5 - 1) - 1) + fibonacci((5 - 1) - 2))
               + (fibonacci((5 - 2) - 1) + fibonacci((5 - 2) - 2))
             =   (fibonacci(((5 - 1) - 1) - 1) + fibonacci(((5 - 1) - 1) - 2)
                  + fibonacci((5 - 1) - 2))
                 )
               + (fibonacci((5 - 2) - 1) + fibonacci((5 - 2) - 2))
             =   fibonacci(2) + fibonacci(1) + fibonacci(2)
               + fibonacci(2) + fibonacci(1)

We end up with five calls of the base case, which is what you would expect. But if we count the in-between calls, we get 9 calls for num = 5, 15 calls for num = 6, 25 calls for num = 7. The number of calls increases by a factor of $1.618$.

This is called an exponential time algorithm, because its runtime has the asymptotic behaviour of $mathcal O (1.618^{n})$. The recursive variant looks great on paper, and it’s how Fibonacci numbers are defined after all, but the run-time is ungodly.

One can speed the recursive variant up if we would remember the values; if we calculate fib(5), we can remember the result of fib(3) that we need in fib(4) = fib(3) + fib(2) for our later use in fib(5) = fib(4) + fib(3). That process is called memoization, but it’s an overkill and not easy to implement.

Instead, write fibonacci in an iterative fashion:

def fibonacci(num):
    """"Returns the num-th fibonacci number, 1-indexed.

    The Fibonacci sequence follows the following rules:

    1) The first number is 1
    2) The second number is 1
    3) Every other number is the sum of its two predecessors

    This yields the sequence

        1,1,2,3,5,8,13,21,34,...
    """
    a,b = 1,1

    for _ in range(0,num):
        a, b = b, a + b

    return a

Other than that, your code seems fine. You could use better names and documentation, but given that you write code for a code challenge, it’s not that necessary. You might want to use a binary search for your limits, though, if you need more speed.

In generous, you can speed things up if you use bitshifts instead of 2**num.

You can speed up stingy if you calculate the Fibonacci numbers there. That prevents you from calculating the same sequence over and over again. Hint: you can implement fibonacci as a generator, which makes this a lot easier.

Building upon @Zeta excellent answer, if you turn fibonacci into a generator, you only need to sum each term in stingy and stop when the sum is higher than the amount of lambs. The number of sums being the number of henchmen you can hire. To do so efficiently, you can use itertools.accumulate.

You are also reinventing the wheel in generous as what you are computing is a base 2 logarithm.

Lastly, it’s pretty clear in the question that the stingy strategy will always give you more henchmen than the generous one. So there is no need for min and max here. But even if you want to be sure, you’d rather use abs instead.

Proposed improvements:

from math import log2
from itertools import accumulate


def answer(lambs):
    return stingy(lambs) - generous(lambs)


def generous(lambs):
    return int(log2(lambs + 1))


def stingy(lambs):
    for henchmen, total_pay in enumerate(accumulate(fibonacci())):
        if total_pay > lambs:
            return henchmen


def fibonacci():
    a, b = 1, 1
    while True:
        yield a
        a, b = b, a + b