I am trying to find the most efficient way to sort the t smallest integers of an unsorted array of length n.

I am trying to have O(n) runtime but, keep getting stuck.

The best I can think of is just sorting the entire array and taking the first t. In all other cases, I keep hitting the chance that the smallest is left behind, and if I check them all then, it has the same time complexity of sorting the entire array.

Can anyone give me some ideas?

If `t`

is considerably smaller than `n`

you can find those `t`

elements in one traverse over the array, always saving the `t`

smallest items and getting rid of bigger integers - many data structures are available for this, BST for example.

Then the run time will be `min(O(n), O(t log(t)))`

Run something like quickselect to find the t-th element and then partition the data to extract the t smallest elements. This can be done in O(n) time (average case).

Quickselect is:

An algorithm, similar on quicksort, which repeatedly picks a 'pivot' and partitions the data according to this pivot (leaving the pivot in the middle, with smaller elements on the left, and larger elements on the right). It then recurses to the side which contains the target element (which it can easily determined by just counting the number of elements on either side).

Then you'll still need to sort the t elements, which can be done with, for example, quicksort or mergesort, giving a running time of O(t log t).

The total running time will be O(n + t log t) - you probably can't do much better than that.

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