Lecture 16

Mergesort

MCS 275 Spring 2021
David Dumas

Course bulletins:

Starting with Quiz 6, you will have 48 hours for quizzes (Noon Sunday to Noon Tuesday).
Project 2 description updated with sample data and modules policy.
Project 2 due 6pm CST Friday, February 26.
Check out the recursion sample code.
Worksheet 7 will explore recursive maze solver / generator in more depth.

A strategy that often involves recursion.

Not always possible or a good idea. It only works if merging partial solutions is easier than solving the entire problem.

Suppose you have a list of objects that can be compared with ==, >, <.

You'd like to reorder them in increasing order.

This problem is called comparison sort. There are many solutions.

A divide-and-conquer solution to comparison sort.

It is a fast solution, often used in practice.

Key: It is pretty easy to take two sorted lists and merge them into a single sorted list.

So, let's divide our list into halves, sort each one (recursively), then merge them.

Now we'll formalize this.

Algorithm mergesort:

Input: list L whose elements support comparison.

Goal: reorder the elements of L in place to achieve sorted order.

If L has 0 or 1 elements, it is already sorted. Do nothing.
Otherwise, copy the first half of L into a new list L1, and the rest into L2.
Use recursive calls to sort L1 and L2 (in place).
Use merge_sorted_lists to merge L1 and L2 into L.

This algorithm depends on having a function merge_sorted_lists that can merge to sorted lists into a single sorted list.

Algorithm merge_sorted_lists:

Input: sorted lists L1 and L2, and a list L of the proper length in which to store the results.

Goal: copy all elements of L1 and L2 into L in increasing order.

Make three integer variables i1,i2,i to keep track of current position in L1,L2,L respectively. Set all to zero.
While i1 < len(L1) and i2 < len(L2), do the following:
- Check which of L1[i1] and L2[i2] is smaller.
- Store the smaller one in L[i].
- Increment whichever one of i1,i2 was used.
- Increment i
Copy any remaining portion of L1 into L.
Copy any remaining portion of L2 into L.

Let's implement mergesort in Python.

No changes to the references from Lecture 13

Algorithms by Jeff Erickson, Chapter 1. Mergesort is example 1.4.
Lutz discusses recursive functions in Chapter 19 (pages 555-559 in the print edition).
Intro to Python for Computer Science and Data Science by Deitel and Deitel, Chapter 11.
Think Python, 2ed, by Allen B. Downey, Sections 5.8 to 5.10.
Computer Science: An Overview by Brookshear and Brylow, Section 5.5.