42B. An Efficient Sorting Algorithm: Merge Sort

We can do much better than insertion sort. To sort a nonempty list L of length n, break L into halves, where L₁ is the first half and L₂ is the second half. Sort lists L₁ and L₂. So now we have two sorted lists. All that is left to do is to merge those lists together into a single sorted list, taking advantage of the facts that they are already sorted. The following diagram illustrates.

This algorithm, called Merge Sort, can be shown to take time Θ(n log₂(n)) in the worst case.

Θ(n log(n)) is much better than Θ(n²). If n is 10,000, n² is 100,000,000, but nlog(n) is only about 130,000. (You should be able to show that log₂(10,000) ≈ 13. Use the fact that log₂(1000) is very close to 10.)