Question

In a recent interview I was asked:

Find the middle element of a sorted List of unknown length starting from the first position.

I responded with this:

Have 2 position counters:

counter1
counter2

Increment counter1 by 1 and counter2 by 2. When counter 2 reaches the end of the list counter 1 will be in the middle. I feel that this isn’t efficient because I am revisiting nodes I have already seen. Either way, is there a more efficient algorithm?

Answer 1

Assuming a linked list, you can do it while visiting arbitrarily-close-to N of the items.

To do 5/4 N:

• Iterate over the list until you hit the end, counting the items.
• Drop an anchor at every power of 2’th element. Track the last 2 anchors.
• Iterate the before-last anchor until it reaches the middle of the list.

When you hit the end of the list, the before-last anchor is before the mid-point but at least half way there already. So N for the full iteration + at most 1/4 N for the anchor = 5/4 N.

Dropping anchors more frequently, such as at every ceiling power of 1.5^th item, gets you as close to N as needed (at the cost of tracking more anchors; but for any given X as the power step, the asymptotic memory is constant).