Mergesort

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1. Algorithm:
   Goal. Given two sorted subarrays a[lo] to a[mid] and a[mid+1] to a[hi],
   replace with sorted subarray a[lo] to a[hi].

public class Merge
{
 private static void merge(...)
 { /* as before */ }

 private static void sort(Comparable[] a, Comparable[] aux, int lo, int hi)
 {
 if (hi <= lo) return;
 int mid = lo + (hi - lo) / 2;
 sort(a, aux, lo, mid);
 sort(a, aux, mid+1, hi);
 merge(a, aux, lo, mid, hi);
 }
 public static void sort(Comparable[] a)
 {
 aux = new Comparable[a.length];
 sort(a, aux, 0, a.length - 1);
 }
}

Improvements:
1)Use insertion sort for small subarrays.
・Mergesort has too much overhead for tiny subarrays.
・Cutoff to insertion sort for ≈ 7 items.
2)Stop if already sorted.
・Is biggest item in first half ≤ smallest item in second half?
・Helps for partially-ordered arrays.
3)Eliminate the copy to the auxiliary array. Save time (but not space)
by switching the role of the input and auxiliary array in each recursive call.

2. Complexity:
   
   
   Proposition. Any compare-based sorting algorithm must use at least
   lg ( N ! ) ~ N lg N compares in the worst-case.
   
3. Bottom Up Mergesort:
   
4. Stability:
   Q. Which sorts are stable?
   A. Insertion sort and mergesort (but not selection sort or shellsort).

本文标签： Mergesort