quartz/content/notes/08-mergesort-2.md

title	tags
08-mergesort-2	cosc201 lecture

recall definition of merge sort

• pre ⇒ split
• rec ⇒ sort pieces
• post ⇒ merge

1 Complexity

no counters

pre and post pahses are constant and ϴ(n)

so M(n) = ϴ(n) + 2 * M(n/2)

does this even help. what if n is odd

pretend ϴ(n) is C \times n

\begin{align*}
M(n) &= C \times n+2 \times M(n/2) \\
&= C \times n+2 \times (C \times (n/2) + 2 \times M(n/4))\\
&= C \times (2n) + 4 \times M(n/4) \\
&= C \times (2n) + 4 \times (C \times (n/4)) + 2 \times M(n/8))\\
&= C \times (3n) + 8 \times M(n/8)\\ \\
&= C \times (kn) + 2^k \times M(n/2^k)
\end{align*}

ends when we find base case when we get to $n/2^k = 1$ we could split earlier. the work done base case is (bounded by) some constatn D so if k is large enough that n/2^k is a base case, we get

M(n) = C \times (kn) + 2^k \times D

how big is k

k <=lg(n)

so:

M(n) ≤ C \times (n lg(n)) + D(n) = ϴ(n lg(n))

which is true

In a divide and consiwer algo wher pre and pst processign work are Ο(n) and the division is into parts of size at least n for some contatn c > 0 tge total time complexity is Ο(n lg n) and generally ϴ(n log n)

2 Variations of mergesort

unite and conquer

5 | 8 | 2 | 3 | 4 | 1 | 7 | 6

5 8 | 2 3 | 1 4 | 6 7

2 3 5 8 | 1 4 6 7

1 2 3 4 5 6 7 8

	public static void mergeSort(int[] a) {
		int blockSize = 1;
		while(blockSize < a.length) {
			int lo = 0;
			while (lo + blockSize < a.length) {
				int hi = lo + 2*blockSize;
				if (hi > a.length) hi = a.length;
				merge(a, lo, lo + blockSize, hi);
				lo = hi;			
			}
			blockSize *=2;		
		}	
	}

outer loop is executed lg n times, where n is the length of a

inner loop proceeds until we find a block that "runs out of elements"

inner loop is having 2 x blocksize added each time, to runs most n/2 x blocksize

inside inner is call to merge which is ϴ(blocksize)

2.1 complexity from bottom up

• n is the numbe of elemetns in a
• outer loop is executed

!Pasted image 20220329114859.png#invert

2.2 improvments

some arrays have sections that are already sorted

you canm

2.3 timsort

used by python java rust etc