Question

EDIT:

This isn’t as trivial as
you think. Consider the fact that each
addition of a new number pushes out an
old number from the linkedlist.

The solution doesn’t seem to be as
simple as keeping track of a min
number with a variable. What if the
minimum gets pushed out of the
linkedlist? Then what? How do you know
what the new min is?

I heard this interview question:

You have a fixed length linked list.

• At time t=0, the linkedlist is filled
  with random numbers.

• At each increment in time, one new
  number is fed into the head of the
  linkedlist, and one number is pushed
  out from the tail.

• You are allowed only ONE traversal before the first time interval.

  • This means one traversal ever. Not once at every time t.
    .

• O(1) storage.

Your task is to be able to return the min of the linkedlist at every time interation.

What is an algorithm that would be able to do this?

---

Interesting note:

Since there’s no information regarding time complexity, you are allowed to use sort operations. The only problem then is: sorting takes more than one iteration.

Answer 1

First,

O(1) storage is not the same as an a single register. It means constant space usage.

Second,

I am going to call your LL a constant size queue (CSQ).

When initializing your queue, also initialize a min-heap where all elements of the queue keep a reference (pointer) to the heap node corresponding to them.

1 op on CSQ

• Pop 1 element out in O(1) time of the CSQ.
• Remove corresponding node from the min-heap in O(lg n) time.
• Add corresponding element to the min-heap in O(lg n) time.
• Push 1 element into the CSQ in O(1) time and mark a reference to the heap node added above.

The above operations guarentee that the heap size will always remain in sync with the queue size –hence O(1). The heap can be constructed in a single travesal.

Finding min

Clearly O(1). Just return the head of the min heap.