2.2 KiB
Introduction
Merge sort is a comparison type sort that has an effective use of
recursion and the divide and conquer algorithm.
Merge sort sorts in
\(`O(N \log N)`\) time and uses \(`O(N \log N)`\) space.
We will explore
the pros and cons of this sort the proofs on its time and space
complexity and the implementation and algorithm of this sorting
method.
Abstract: Given two sorted arrays \(`a_{lo}`\) to \(`a_{mid}`\) and \(`a_{mid+1}`\) to \(`a_{hi}`\), replace with sorted subarray \(`a_{lo}`\) to \(`a_{hi}`\).
Basic Algorithm
- Divide array into 2 halves
- Recursively sort each half
- Merge the two halfs.
def mergesort(left, right, array[]):
if left <= right:
mid = (left + right) / 2
mergesort(left, mid, array[])
mergesort(mid+1, right, array[])
mergehalves(left, mid, right, array[])Running Time Analysis
- Latop computer can run close to \(`10^8`\) per second.
- Supercomputer can execute \(`10^{12}`\) compares per second.
Insertion Sort \(`(N^2)`\)
| computer | thousand \(`(10^3)`\) | million \(`(10^6)`\) | billion \(`(10^9)`\) |
|---|---|---|---|
| home | instant | 2.8 hours | 317 years |
| super | instant | 1 second | 1 week |
Merge Sort \(`(N \log N)`\)
| computer | thousand \(`(10^3)`\) | million \(`(10^6)`\) | billion \(`(10^9)`\) |
|---|---|---|---|
| home | instant | 1 second | 18 min |
| super | instant | instant | instant |
Proof Sketch
Proposition Merge sort uses \(`\le N \log N`\) compares to sort an array of length \(`N`\).
Proof Sketch The number of compares \(`C(N)`\) to mergesort an array of length \(`N`\) satisfies the recurrence:
C(N) \le C(\lceil N/2 \rceil) + C(\lfloor N/2 \rfloor) + N \text{ for } N \gt 1, \text{ with } C(1) = 0.Divide and conquer recurrence induction proof
Proposition. If \(`D(N)`\) satisfies \(`D(N)=2D(N/2) + N`\) for \(`N \gt 1`\), with \(`D(1) = 0,`\) then \(`D(N) = N \log N`\).
Proof [assuming \(`N`\) is a power of \(`2`\)] - Base case: \(`N = 1`\) - Inductive hypothesis: \(`D(N) = N \log N`\). - Goal: show that \(`D(2N) = (2N) = \log (2N)`\)
\(`D(2N) = 2D(N) + 2N`\)
\(`\quad = 2 N \log N + 2N`\)
\(`\quad = 2 N (log (2N) - 1) + 2N`\)
\(`\quad = 2 N \log (2N)`\)
Resources
- Princeton University: https://algs4.cs.princeton.edu/lectures/22Mergesort.pdf