43C. An Efficient Sorting Algorithm: Merge Sort

We can do much better than insertion sort. Merge Sort is based on the fact that it is easy to combine two linked lists that are already sorted.

Merging sorted lists

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Supposes that linked lists A and B are already in nondescending order. merge(A, B) should rearrange the list cells of A and B into a combined list that is in nondescending order, and return a pointer to the first cell in the sorted list. For example:

Merge is easy to write. If either list is empty, the merged list is the other one. If both lists are nonempty, suppose that list A begins with h_A and list B begins with h_B. Then the combined list begins with the smaller of h_A and h_B. Remove that from its list and continue with what is left.

  List merge(List A, List B)
  {
    if(A == NULL)
    {
      return B;
    }
    else if(B == NULL)
    {
      return A;
    }
    else if(A->head < B->head)
    {
      A->tail = merge(A->tail, B);
      return A;
    }
    else
    {
      B->tail = merge(A, B->tail);
      return B;
    }
  }

Merge Sort

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Empty lists are easy to sort. To sort a nonempty list L:

1. Break L into halves L₁ and L₂, where L₁ is the first half and L₂ is the second half.

2. Sort lists L₁ and L₂ using Merge Sort.

3. Merge sorted lists L₁ and L₂ together into a single sorted list.

The following diagram illustrates.

We show on the next page that Merge Sort takes time Θ(n log₂(n)) in the worst case. That is much better than Θ(n²), as shown in the following brief table. The values of nlog(n) are approximate.