Sorting

1
Sorting
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)gtA(2) switch If A(2)gtA(3) switch If
A(3)gtA(4) switch If A(4)gtA(5) switch If
A(N-3)gtA(N-2) switch

• If A(1)gtA(2)
• switch
• If A(2)gtA(3)
• switch
• If A(3)gtA(4)
• switch
• If A(4)gtA(5)
• switch
• If A(N-3)gtA(N-2)
• switch
• If A(N-2)gtA(N-1)
• switch
• If A(N-1)gtA(N)
• switch

If A(1)gtA(2) switch
If A(1)gtA(2) switch If A(2)gtA(3) switch If
A(3)gtA(4) switch If A(4)gtA(5) switch If
A(N-3)gtA(N-2) switch If A(N-2)gtA(N-1) 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
A(N-2) is now 3rd largest entry A(N-1) is still
2nd largest entry A(N) is still largest enry
A(N-1) is now 2nd largest entry A(N) is still
largest enry
A(N) is now largest entry
4
Bubble Sort (Sink sort here)
If A(1)gtA(2) switch If A(2)gtA(3) switch If
A(3)gtA(4) switch If A(4)gtA(5) switch If
A(N-3)gtA(N-2) switch

• If A(1)gtA(2)
• switch
• If A(2)gtA(3)
• switch
• If A(3)gtA(4)
• switch
• If A(4)gtA(5)
• switch
• If A(N-3)gtA(N-2)
• switch
• If A(N-2)gtA(N-1)
• switch
• If A(N-1)gtA(N)
• switch

If A(1)gtA(2) switch If A(2)gtA(3) switch If
A(3)gtA(4) switch If A(4)gtA(5) switch If
A(N-3)gtA(N-2) switch If A(N-2)gtA(N-1) switch
If A(1)gtA(2) switch
1 step
N-3 steps
N-2 steps
N-1 steps
5
Bubble Sort (Sink sort here)
If A(1)gtA(2) switch If A(2)gtA(3) switch If
A(3)gtA(4) switch If A(4)gtA(5) switch If
A(N-3)gtA(N-2) switch

• If A(1)gtA(2)
• switch
• If A(2)gtA(3)
• switch
• If A(3)gtA(4)
• switch
• If A(4)gtA(5)
• switch
• If A(N-3)gtA(N-2)
• switch
• If A(N-2)gtA(N-1)
• switch
• If A(N-1)gtA(N)
• switch

If A(1)gtA(2) switch If A(2)gtA(3) switch If
A(3)gtA(4) switch If A(4)gtA(5) switch If
A(N-3)gtA(N-2) switch If A(N-2)gtA(N-1) switch
If A(1)gtA(2) switch
for lastcompareN-1-11 for i1lastcompare
if A(i)gtA(i1)
6
Matlab code for Bubble Sort

• function S bubblesort(A)
• Assume A row/column Copy A to S
• S A
• N length(S)
• for lastcompareN-1-11
• for i1lastcompare
• if S(i)gtS(i1)
• tmp S(i)
• S(i) S(i1)
• S(i1) 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 1N A(Idx) equals S
• for lastcompareN-1-11
• for i1lastcompare
• if S(i)gtS(i1)
• tmp S(i) tmpi Idx(i)
• S(i) S(i1) Idx(i) Idx(i1)
• S(i1) tmp Idx(i1) 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,nAnB)
• for iC1nAnB
• if A(iA)ltB(iB) compare smallest unused
• C(iC) A(iA) iA iA1 use A
• else
• C(iC) B(iB) iB iB1 use B
• end
• end

10
MergeSort

• function S mergeSort(A)
• n length(A)
• if n1
• S A
• else
• hn floor(n/2)
• S1 mergeSort(A(1hn))
• S2 mergeSort(A(hn1end))
• S merge(S1,S2)
• end

Base Case
Split in half
Sort 1st half
Sort 2nd half
Merge 2 sorted arrays
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 n1
• S A
• else
• hn floor(n/2)
• S1 mergeSort(A(1hn))
• S2 mergeSort(A(hn1end))
• S merge(S1,S2)
• end

R(1) 0
R(n/2) to sort array of length n/2
R(n/2) to sort array of length n/2
n steps to merge two sorted arrays of total
length n
Recursive relation R(1)0, R(n) 2R(n/2) n
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Rough Operation Count for MergeSort

• The recursive relation for R
• R(1)0, R(n) 2R(n/2) n
• Claim For n2m, it is true that R(n) n log2(n)
• Case (m0) true, since log2(1)0
• Case (mk1 from mk)

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.