Text

1

• Text
• Chapters 1, 2

2
Sorting

• Algorithm
• well-defined computational procedure that
  transforms input into output
• steps for the computer to follow to solve a
  problem

instance

• Sorting Problem
• Input A sequence of n numbers
• Output A permutation (reordering)
  of the input sequence such that

3
Insertion Sort Animation
Finding a place for item with value 5 in
position 1 Swap item in position 0 with item in
position 1.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
4
Insertion Sort Animation
Positions 0 through 1 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
5
Insertion Sort Animation
Finding a place for item with value 1 in
position 2 Swap item in position 1 with item in
position 2.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
6
Insertion Sort Animation
Finding a place for item with value 1 Swap item
in position 0 with item in position 1. Positions
0 through 2 are now in non-decreasing order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
7
Insertion Sort Animation
Finding a place for item with value 3 in
position 3 Swap item in position 2 with item in
position 3.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
8
Insertion Sort Animation
Finding a place for item with value 3 Swap item
in position 1 with item in position 2.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
9
Insertion Sort Animation
Positions 0 through 3 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
10
Insertion Sort Animation
Finding a place for item with value 2 in
position 4 Swap item in position 3 with item in
position 4.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
11
Insertion Sort Animation
Finding a place for item with value 2 Swap item
in position 2 with item in position 3.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
12
Insertion Sort Animation
Finding a place for item with value 2 Swap item
in position 1 with item in position 2.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
13
Insertion Sort Animation
Positions 0 through 4 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
14
Insertion Sort Animation
Finding a place for item with value 6 in
position 5 Swap item in position 4 with item in
position 5.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
15
Insertion Sort Animation
Positions 0 through 5 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
16
Insertion Sort Animation
Finding a place for item with value 4 in
position 6 Swap item in position 5 with item in
position 6.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
17
Insertion Sort Animation
Finding a place for item with value 4 Swap item
in position 4 with item in position 5.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
18
Insertion Sort Animation
Positions 0 through 6 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
19
Insertion Sort Animation
Finding a place for item with value 7 in
position 7 Swap item in position 6 with item in
position 7.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
20
Insertion Sort Animation
Positions 0 through 7 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
21
Insertion Sort Animation
Positions 0 through 7 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
22
Insertion Sort Animation
Positions 0 through 7 are now in non-decreasing
order.
http//www.cs.brockport.edu/cs/java/apps/sorters/i
nsertsortaniminp.html
23
Asymptotic Notation
courtesy of Prof. Costello
O(g(n)) is a set of functions, so we often say
f(n) is in O(g(n)).
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Asymptotic Notation (cont.)
courtesy of Prof. Costello
25
Asymptotic Analysis
Math fact sheet (courtesy of Prof. Costello) is
on our web site.
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Function Order of Growth
know how to order functions asymptotically (behavi
or as n becomes large)

• O( ) upper bound
• W( ) lower bound
• Q( ) upper lower bound

know how to use asymptotic complexity notation to
describe time or space complexity
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Types of Algorithmic Input

• Best-Case Input of all possible algorithm inputs
  of size n, it generates the best result
• for Time Complexity best is smallest
  running time
• Best-Case Input Produces Best-Case Running
  Time
• provides a lower bound on the
  algorithms asymptotic running time
• (subject to any implementation
  assumptions)
• for Space Complexity best is smallest
  storage
• Average-Case Input
• Worst-Case Input

these are defined similarly
Best-Case Time lt Average-Case Time lt Worst-Case
Time
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Bounding Algorithmic Time(using cases)
Using case we can discuss lower and/or upper
bounds on best-case running time or
average-case running time or worst-case running
time
T(n) W(1)
T(n) O(2n)
very loose bounds are not very useful!
Worst-Case time of T(n) O(2n) tells us that
worst-case inputs cause the algorithm to take at
most exponential time (i.e. exponential time is
sufficient). But, can the algorithm every really
take exponential time? (i.e. is exponential time
necessary?) If, for arbitrary n, we find a
worst-case input that forces the algorithm to use
exponential time, then this tightens the lower
bound on the worst-case running time. If we can
force the lower and upper bounds on the
worst-case time to match, then we can say that,
for the worst-case running time, T(n)
Q(2n ) (i.e. weve found the minimum upper bound,
so the upper bound is tight.)
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Bounding Algorithmic Time(tightening bounds)
for example...
TW (n) O(2n)
TB (n) O(n)
TW (n) W(n2)
TB(n) W(1)
1st attempt
1st attempt
1st attempt
1st attempt
TB(n) Q(n)
TW(n) Q(n2)
2nd attempt
2nd attempt
Algorithm Bounds
Here we denote best-case time by TB(n)
worst-case time by TW(n)
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Know the Difference!
Strong Bound A worst-case lower bound on a
problem holds for every algorithm that solves the
problem and abides by the problems assumptions.
Weak Bound A worst-case upper bound on a problem
comes from just considering one algorithm.
Other, less efficient algorithms that solve this
problem might exist, but we dont care about
them!
Both the upper and lower bounds could be loose
(i.e. perhaps could be tightened later on).