CSE116 / CSE504
Introduction to Computer Science II
Dr. Carl Alphonce
343 Davis Hall
alphonce@buffalo.edu
Office hours:
Thursday 12:00 PM – 2:00 PM
Friday 8:30 AM – 10:30 AM
OR request appointment via e-mail
PROFESSIONALISM
Turn off and put away electronics:
cell phones
pagers
laptops
tablets
etc.
© Dr. Carl Alphonce
Today
Sorting / Merge Sort
ROADMAP
Coming up
Heaps / Priority queues
© Dr. Carl Alphonce
channel 1
Ifselectionsortispresentedwiththisinput,
12
27
16
84
72
38
10
65
whichoftheseisapossibleintermediateconfigurationof
thearray?
1) 10 27 16 84 72 38 12 65
2) 10 12 16 84 72 38 27 65
3) 10 12 16 27 38 84 72 65
A.
B.
C.
D.
E.
(1)only
(2)only
(1)and(2)only
(2)and(3)only
(1),(2)and(3)
Ifselectionsortispresentedwiththisinput,
12
27
16
84
72
38
10
65
whichoftheseisapossibleintermediateconfigurationof
thearray?
1) 10 27 16 84 72 38 12 65
2) 10 12 16 84 72 38 27 65
3) 10 12 16 27 38 84 72 65
A.
B.
C.
D.
E.
(1)only
(2)only
(1)and(2)only
(2)and(3)only
(1),(2)and(3)
Ifselectionsortispresentedwiththisinput,
12
27
16
84
72
38
10
65
whichoftheseisapossibleintermediateconfigurationof
thearray?
1) 10 27 16 84 72 38 12 65
2) 10 12 16 84 72 38 27 65
3) 10 12 16 27 38 84 72 65
A.
B.
C.
D.
E.
(1)only
(2)only
(1)and(2)only
(2)and(3)only
(1),(2)and(3)
SORTING
O(n log(n)) sorting algorithm:
MERGE SORT
© Dr. Carl Alphonce
Merge sort
Basic idea
input is presented in an array
the array is split into two partitions, of (roughly)
equal size
each partition is recursively merge sorted
partitions are sorted, they are merged to give
answer to original problem
12
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16
84
72
38
10
65
12
split
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84
72
72
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38
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10
65
65
recursively merge sort the partitions
12
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84
72
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65
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72
merge the sorted partitions
12
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72
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84
the whole process
12
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72
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10
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84
efficiency?
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84
72
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12
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10
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84
efficiency?
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10
The work of a sorting algorithm is measured by
the number of comparisons, x<y, that it makes.
65
12
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16
84
72
38
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72
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16
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16
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38
72
10
12
12
12
10
16
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16
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10
38
38
65
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72
84
Merge sort does comparisons only in the merge
phase (bottom half of diagram).
During merge, all n positions of the merged array
need to be determined. One comparison is needed
for each. So, at each level O(n) work is needed.
How many levels are there in the bottom half?
log2 n! Therefore the runtime is O(n log n).
efficiency?
12
27
16
84
72
38
10
65
12
27
16
84
72
38
10
65
12
27
16
84
72
38
10
65
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16
84
72
38
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12
27
16
84
38
72
10
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12
16
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10
12
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84
What about upper half of diagram? Splitting takes
very little time. Even if we overestimate and think
that O(n) is needed at each level, the top half
would only contribute O(n log n). Adding top and
bottom halves would yield:
O(n log n + n log n)
= O(2 n log n)
= O(n log n)
since constants are absorbed into the notation.
O(n log n)
No general purpose sorting algorithm can do
better.
How about in practice?
empirical testing
efficiency
12
27
16
84
72
38
10
65
10
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16
84
72
38
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65
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12
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84
Maybe we will get a
better bound by
computing
n+(n-1)+(n-2)+…+1
or
Σ i, for i=1…n
which is
n*(n+1)/2
which is still O(n2)
efficiency
12
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16
84
72
38
10
12
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16
84
38
72
10
65
12
16
27
84
10
38
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72
10
12
16
27
38
65
72
65
84
What about upper half of diagram? Splitting takes
very little time. Even if we overestimate and think
that O(n) is needed at each level, the top half
would only contribute O(n log n). Adding top and
bottom halves would yield:
O(n log n + n log n)
= O(2 n log n)
= O(n log n)
since constants are absorbed into the notation.