## Three Interview Questions

### May 13, 2014

One of the sites that I watch from time to time is Career Cup, which publishes coding questions that have been asked in actual job interviews. These questions came through this morning:

1) Consider a sorted singly-linked list having the following nodes: 10 -> 30 -> 50 -> 70 -> NULL. You are given a pointer to node 50 and a new node having the value 40. Can you insert node 40 correctly in the list maintaining the ascending order?

2) Given a linked list 5 -> 4 -> 3 -> 2 -> 1 produce a linked list 4 -> 2 -> 0 -> 2 -> 1 by subtracting the last node of the list from the first, the next-to-last node from the second, and so on, stopping at the midpoint of the list.

3) Write a program to output the number of consecutive trailing zeros in the factorial of a number. For example, if the number is 5, then 5! = 120, and there is one trailing zero.

Your task is to write programs to answer the three interview questions posed above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

The first can be a double trick question… it depends on if the nodes are modifiable. If so, you change the value in 50 to 40, and insert a new 40 node.

I mean, a new 50 node.

Ceci n’est pas une Python.

Traditional Lisp exercises. Much fun.

The above and the below produce the expected results in this form:

Test result 1: (10, (30, (40, (50, (70, ())))))

Test result 2: (4, (2, (0, (2, (1, ())))))

Oops. Yes, I did miss an initial condition in the first question.

This is correct, as long as the list is is originally in the ascending

order, and the node given to insert-at-node is effectively the node

containing the smallest item greater than the new data.

For any n>=2, in the decomposition in prime factors of n!, the

exponent of 2 is greater than the exponent of 5,

since (log n 2)/(log n 5) = (log 5 2) ~= 2.321928 > 1

Since only the multiples of an exponent of 10 will add a 0 to the

factorial, and there are more multiples of 2 than multiples of 5, we

can just sum the exponents of 5 factor by factor.

For the last problem, we can just repeatedly divide by 5 and add:

And for the second problem, a little Haskell function:

@informatimago –

Your factorial algorithm was interesting – building on top of it- if we just divide n by 5 – that seems to work although I did not test for very large numbers…

factorial(4) – will have 0 zeros (4/5)

factorial(10) – will have 2 zeros (10/5)

factorial(24) – will have 4 zeros (24/5)

factorial(25) – will have 5 zeros (25/5) and so on…

For number 3, counting factors in each iteration of a factorial is asymptotically less efficient than counting representation of progressively larger powers of the factor in n. Implemented in Python the faster solution looks as follows:

Even at small n, the difference in growth is significant:

(n was randomly chosen across range [1,10240], 1000 samples for each n were taken and averaged.)

n: Faster (s) Slower (s)

538: 3.74e-07 8.91e-05

1418: 4.32e-07 2.34e-04

3915: 4.85e-07 6.43e-04

4071: 4.86e-07 6.72e-04

6333: 4.85e-07 1.04e-03

6364: 4.85e-07 1.05e-03

8262: 5.03e-07 1.36e-03

8970: 5.00e-07 1.48e-03

9401: 4.99e-07 1.55e-03

9497: 5.01e-07 1.57e-03

After n=10240, the slower algorithm becomes unwieldy, while the faster method still runs in under 4e-04 seconds with n above 480 digits.

Vivek is correct. For the second problem, all we have to do is divide by 5. The reasoning is that the prime factors of 10 are 2 and 5. For n!, every 5th value of n will give us a factor of 5. Every second value of n will give us a factor of 2, so we’ll have plenty of factors of 2 (i.e. in the prime factorization of n!, we’ll always have at least as many factors of 2 as we do factors of 5). So every 5th value of n!, we’ll add another 0 to the end of the number.