Merge Sort Summary

Nothing special here. It’s just a blog post for summarising my algorithm learning course. Although this was already taught in the University, it’s still good to summarise here

Merge Sort

  • Divide array into two halves.
  • Sort each half.
    • For each half, continue dividing into 2 halves and do merge sort.
  • Merge two halves.

Here is the sample implementation from the Coursera course. Actually, I’m more familiar with the while version.

public class Merge {
    private static void merge(Comparable[] a, Comparable[] aux, int lo, int mid, int hi) {
        for (int k = lo; k <= hi; k++)
            aux[k] = a[k];

        int i = lo, j = mid + 1;
        for (int k = lo; k <= hi; k++) {
            if (i > mid) a[k] = aux[j++];
            else if (j > hi) a[k] = aux[i++];
            else if (less(aux[j], aux[i])) a[k] = aux[j++];
            else a[k] = aux[i++];

    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);

Running time estimates:

  • Assume that Laptop executes 108 compares/second.
  • Assume that Supercomputer executes 1012 compares/second.
  • Insertion sort: N2
  • Merge sort N logN
  insertionsort insertionsort insertionsort mergesort mergesort mergesort
  thousand million billion thousand million billion
laptop instant 2.8 hours 317 years instant 1 second 18 min
super instant 1 second 1 week instant instant instant

Bottom-up Merge Sort

  • The reversed way of top-down merge sort
  • Pass through array, merging subarrays of size 1.
  • Repeat for subarrays of size 2, 4, 8, 16, ….
public class MergeBU
    private static void merge(...) {
        // the same

    public static void sort(Comparable[] a)
        int N = a.length;
        Comparable[] aux = new Comparable[N];
        for (int sz = 1; sz < N; sz = sz+sz)
            for (int lo = 0; lo < N-sz; lo += sz+sz)
                merge(a, aux, lo, lo+sz-1, Math.min(lo+sz+sz-1, N-1));
  • Bottom-up merge sort is about 10% slower than recursive, top-down mergesort on typical systems
  • Recursive mergesort requires O(logN) space for the recursion stack
  • The bottom-up version lets you do better (no recursion stack, just a few integers keeping track of your position in the input)
  • If you come across some language that doesn’t support recursion and provides you with only limited memory for a stack (perhaps an embedded system?), the bottom-up version will be your only choice.