sorting uc berkeley fall 2004, e77 pack/e77 copyright 2005, andy packard. this work is licensed...
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SortingUC Berkeley
Fall 2004, E77http://jagger.me.berkeley.edu/~pack/e77
Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to
Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
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) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch
If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf 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
Bubble Sort (“Sink” sort here)
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch
If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf 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( steps of #
21
1
NNNi
N
i
Bubble Sort (“Sink” sort here)
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch
If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch
If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf 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)
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)?
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
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 BA nn steps"" of #
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
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
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 kkR 2222
kk RR 222 1
12
1 2log2 kk
Recursive relation
Induction hypothesis
Matlab command: sortSyntax 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.
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