Question

I’ve got an array of n elements. Those elements are numbers. What I have to do now is find a lot of them in my array. I will have to make about sqrt(n) searches.

What is a better option for me from efficiency point of view?

1. Linear search
2. Array Sorting (lets say in linear time, it is possible when we use RadixSort for example) and then searching the elements with binary-search – O(logn)

I’ve got some intuitions, but I dont know how to check and proof it. Linear search cost is (in worst case) : O(n) * O(sqrt(n)). Binary-search after sorting would be O(n) + O(logn) * O(sqrt(n)).

Answer 1

As you said, you have two options:

1. Linear search sqrt(n) times. Cost is O(n * sqrt(n)).
2. Sorting an array in O(n) and then binary search sqrt(n) times.
   Cost is O(n + log(n) * sqrt(n)). n is bigger then log(n) * sqrt(n) for all positive n. Proof.
   Thanks to this, we can write O(n + log(n) * sqrt(n)) = O(n).
   Sorting part is dominating, it’s the bottleneck.

So, second approach, for big n, should be faster – approximately linear.