Sorting
UC Berkeley
Fall 2004, E77
http://jagger.me.berkeley.edu/~pack/e77
Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike
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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
– BubbleSort
– MergeSort
Searching: next lecture
Bubble Sort (“Sink” sort here)
If A(1)>A(2)
If A(1)>A(2)
If A(1)>A(2)
switch
switch
switch
If A(2)>A(3)
If A(2)>A(3)
If A(2)>A(3)
switch
switch
switch
If A(3)>A(4)
If A(3)>A(4)
If A(3)>A(4)
switch
switch
switch
If A(4)>A(5)
If A(4)>A(5)
If A(4)>A(5)
switch
switch
switch
…
If A(N-3)>A(N-2) If A(N-3)>A(N-2) If A(N-3)>A(N-2)
switch
switch
switch
If A(N-2)>A(N-1) If A(N-2)>A(N-1) A(N-2) is now 3rd
switch
switch
largest entry
If A(N-1)>A(N)
A(N-1) is now
A(N-1) is still 2nd
switch
2nd largest entry
largest entry
A(N) is now
largest entry
A(N) is still
largest enry
A(N) is still
largest enry
If A(1)>A(2)
switch
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
Bubble Sort (“Sink” sort here)
If A(1)>A(2)
If A(1)>A(2)
If A(1)>A(2)
If A(1)>A(2)
switch
switch
switch
switch
If A(2)>A(3)
If A(2)>A(3)
If A(2)>A(3)
1 step
switch
switch
switch
If A(3)>A(4)
If A(3)>A(4)
If A(3)>A(4)
switch
switch
switch
If A(4)>A(5)
If A(4)>A(5)
If A(4)>A(5)
switch
switch
switch
…
If A(N-3)>A(N-2) If A(N-3)>A(N-2) If A(N-3)>A(N-2)
switch
switch
switch
If A(N-2)>A(N-1) If A(N-2)>A(N-1) N-3 steps
switch
switch
If A(N-1)>A(N)
N-2 steps
N 1
switch
N ( N  1) N 2
N-1 steps
# of steps   i 
i 1
2

2
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(1)>A(2)
switch
If A(2)>A(3)
switch
If A(3)>A(4)
switch
If A(4)>A(5)
switch
If A(1)>A(2)
switch
If A(N-3)>A(N-2) If A(N-3)>A(N-2)
switch
switch
If A(N-2)>A(N-1)
switch
for lastcompare=N-1:-1:1
for i=1:lastcompare
if A(i)>A(i+1)
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);
What about returning
S(i+1) = tmp;
an Index vector Idx,
end
with the property that
end
S = A(Idx)?
end
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
If we switch two entries of S, then exchange the same
end
two entries of Idx. This keeps A(Idx) equaling S
end
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…
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
# of "steps"  n  n
A
B
MergeSort
function S = mergeSort(A)
n = length(A);
Base Case
if n==1
S = A;
Split in half
else
hn = floor(n/2);
Sort 1st half
S1 = mergeSort(A(1:hn));
S2 = mergeSort(A(hn+1:end));
Sort 2nd half
S = merge(S1,S2);
end
Merge 2 sorted arrays
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
R(n/2) to sort array
R(1) = 0
S = A;
of length n/2
else
hn = floor(n/2);
R(n/2) to sort array
S1 = mergeSort(A(1:hn));
of length n/2
S2 = mergeSort(A(hn+1:end));
S = merge(S1,S2);
n steps to merge two sorted
end
arrays of total length n
Recursive relation: R(1)=0, R(n) = 2*R(n/2) + n
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)
   R2  2 
 2  R2   2  2
 2  2 log 2   2  2
R2
k 1
k
Recursive relation
k
k
k
k
2
 2k 1  k  1
2
k 1
 
 log 2 2
k 1
k
Induction
hypothesis
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.