Efficient way to compute sum of k largest numbers in a list?

Question

I was reading some practice interview questions and I have a question about this one. Assume a list of random integers each between 1 & 100, compute the sum of k largest integers? Discuss space and time complexity and whether the approach changes if each integer is between 1 & m where m varies?

My first thought is to sort the array and compute the sum of largest k numbers. Then, I thought if I use a binary tree structure where I can look starting from bottom right tree. I am not sure if my approach would change whether numbers are 1 to 100 or 1 to m? Any thoughts of most efficient approach?

Answer 1

The most efficient way might be to use something like randomized quickselect. It doesn't do the sorting step to completion and instead does just the partition step from quicksort. If you don't want the k largest integers in some particular order, this would be the way I'd go with. It takes linear time but the analysis is not very straightforward. m would have little impact on this. Also, you can write code in such a way that the sum is computed as you partition the array.

Time: O(n)
Space: O(1)

---

The alternative is sorting using something like counting sort which has a linear time guarantee. As you say the values are integers in a fixed range, it would work quite well. As m increases the space requirement goes up, but computing the sum is quite efficient within the buckets.

Time: O(m) in the worst case (see comments for the argument)
Space: O(m)