The document discusses sorting algorithms. It explains that keeping data in order allows for more efficient searching. Two common sorting algorithms are described: bubble sort and merge sort. Bubble sort repeatedly compares adjacent elements and swaps them if they are in the wrong order, while merge sort divides the array in half, recursively sorts each half, and then merges the sorted halves. The document provides pseudocode for implementing these algorithms and analyzes their runtime operations.

1. SortingSorting

2. Sorting
Keeping data in “order” allows it to be searched more efficiently
Example: Phone Book
– Sorted by Last Name (“lots” of work to do this)
• Easy to look someone up if you know their last name
• Tedious (but straightforward) to find by First name or
Address
Important if data will be searched many times
Two algorithms for sorting today
– Bubble Sort
– Merge Sort
Searching: next lecture

3. Bubble Sort (“Sink” sort here)
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
…
If A(N-3)>A(N-2)
switch
If A(N-2)>A(N-1)
switch
If A(N-1)>A(N)
switch
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(N-3)>A(N-2)
switch
If A(N-2)>A(N-1)
switch
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(N-3)>A(N-2)
switch
If A(1)>A(2)
switch
A(N) is now
largest entry
A(N-1) is now
2nd
largest entry
A(N) is still
largest enry
A(N-2) is now 3rd
largest entry
A(N-1) is still 2nd
largest entry
A(N) is still
largest enry
A(1) is now Nth
largest entry.
A(2) is still (N-1)th
largest entry.
A(3) is still (N-2)th
largest entry.
A(N-3) is still 4th
largest entry
A(N-2) is still 3rd
largest entry
A(N-1) is still 2nd
largest entry
A(N) is still largest
entry

4. Bubble Sort (“Sink” sort here)
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
…
If A(N-3)>A(N-2)
switch
If A(N-2)>A(N-1)
switch
If A(N-1)>A(N)
switch
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(N-3)>A(N-2)
switch
If A(N-2)>A(N-1)
switch
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(N-3)>A(N-2)
switch
If A(1)>A(2)
switch
N-1 steps
N-2 steps
N-3 steps
1 step
22
)1(
stepsof#
21
1
NNN
i
N
i
≈
−
== ∑
−
=

5. Bubble Sort (“Sink” sort here)
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
…
If A(N-3)>A(N-2)
switch
If A(N-2)>A(N-1)
switch
If A(N-1)>A(N)
switch
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(N-3)>A(N-2)
switch
If A(N-2)>A(N-1)
switch
If A(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(N-3)>A(N-2)
switch
If A(1)>A(2)
switch
for lastcompare=N-1:-1:1
for i=1:lastcompare
if A(i)>A(i+1)

6. Matlab code for Bubble Sort
function S = bubblesort(A)
% Assume A row/column; Copy A to S
S = A;
N = length(S);
for lastcompare=N-1:-1:1
for i=1:lastcompare
if S(i)>S(i+1)
tmp = S(i);
S(i) = S(i+1);
S(i+1) = tmp;
end
end
end
What about returning
an Index vector Idx,
with the property that
S = A(Idx)?

7. Matlab code for Bubble Sort
function [S,Idx] = bubblesort(A)
% Assume A row/column; Copy A to S
N = length(A);
S = A; Idx = 1:N; % A(Idx) equals S
for lastcompare=N-1:-1:1
for i=1:lastcompare
if S(i)>S(i+1)
tmp = S(i); tmpi = Idx(i);
S(i) = S(i+1); Idx(i) = Idx(i+1);
S(i+1) = tmp; Idx(i+1) = tmpi;
end
end
end
If we switch two entries of S, then exchange the same
two entries of Idx. This keeps A(Idx) equaling S

8. Merging two already sorted
arrays
Suppose A and B are two sorted arrays (different
lengths)
How do you “merge” these into a sorted array C?
Chalkboard…

9. Pseudo-code: Merging two
already sorted arrays
function C = merge(A,B)
nA = length(A); nB = length(B);
iA = 1; iB = 1; %smallest unused element
C = zeros(1,nA+nB);
for iC=1:nA+nB
if A(iA)<B(iB) %compare smallest unused
C(iC) = A(iA); iA = iA+1; %use A
else
C(iC) = B(iB); iB = iB+1; %use B
end
end
BA nn +=steps""of#

10. MergeSort
function S = mergeSort(A)
n = length(A);
if n==1
S = A;
else
hn = floor(n/2);
S1 = mergeSort(A(1:hn));
S2 = mergeSort(A(hn+1:end));
S = merge(S1,S2);
end
Base Case
Split in half
Sort 2nd
half
Merge 2 sorted arrays
Sort 1st
half

11. Rough Operation Count for MergeSort
Let R(n) denote the number of operations necessary to
sort (using mergeSort) an array of length n.
function S = mergeSort(A)
n = length(A);
if n==1
S = A;
else
hn = floor(n/2);
S1 = mergeSort(A(1:hn));
S2 = mergeSort(A(hn+1:end));
S = merge(S1,S2);
end
R(1) = 0
R(n/2) to sort array of length
n/2
n steps to merge two sorted arrays of total
length n
R(n/2) to sort array of length
n/2
Recursive relation: R(1)=0, R(n) = 2*R(n/2) + n

12. Rough Operation Count for MergeSort
The recursive relation for R
R(1)=0, R(n) = 2*R(n/2) + n
Claim: For n=2m
, it is true that R(n) ≤ n log2(n)
Case (m=0): true, since log2(1)=0
Case (m=k+1 from m=k)
( )12 1
+⋅= +
kk
( ) kkk
222log22 2 ⋅+⋅≤
( ) kk
R 2222 ⋅+⋅=( ) ( )kk
RR 222 1
⋅=+
( )1
2
1
2log2 ++
⋅= kk
Recursive relation
Induction
hypothesis

13. Matlab command: sort
Syntax is
[S] = sort(A)
If A is a vector, then S is a vector in ascending
order
The indices which rearrange A into S are also
available.
[S,Idx] = sort(A)
S is the sorted values of A, and A(Idx) equals S.