Depends on what you call "best." From a theoretical point of view, you cannot solve the problem in less than O(n) in a deterministic Turing machine.
The naive algorithm is too loop and update min, max. However, a recursive solution will require less comparisons than naive algorithm, if you want to get min, max simultaneously (it isn't necessarily faster due to function call overhead).
struct MinMax{
public int Min,Max;
}
MinMax FindMinMax(int[] array, int start, int end) {
if (start == end)
return new MinMax { Min = array[start], Max = array[start] };
if (start == end - 1)
return new MinMax { Min = Math.Min(array[start], array[end]), Max = Math.Max(array[start], array[end]) } ;
MinMax res1 = FindMinMax(array, start, (start + end)/2);
MinMax res2 = FindMinMax(array, (start+end)/2+1, end);
return new MinMax { Min = Math.Min(res1.Min, res2.Min), Max = Math.Max(res1.Max, res2.Max) } ;
}
The simplest solution would be to sort and get the first and last item, though it's obviously not the fastest ;)
The best solution, performance-wise, to find the minimum or maximum is the naive algorithm you written (with a single loop).