AlgorithmsinaNutshell Session1 Introduction 9:109:40 Outline - PowerPoint PPT Presentation

Algorithms
in
a
Nutshell Session
1 Introduction 9:10
–
9:40

Outline • What
is
an
algorithm? • An
example:
INSERTION
SORT • Pattern
format
description • Mathematical
notations Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 2

What
is
an
Algorithm? • A
deterministic
sequence
of
operations
for solving
a
problem
given
a
specific
input
set • Deterministic
–
this
means
it
always
works, given
the
known
constraints
on
the
problem – Produces
solution
for
all
possible
inputs • Input
Set
–
the
(often
arbitrary)
way
in
which a
problem
instance
is
represented Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 3

Problem :
Sort
a
collection
of
strings eagle • Input cat ant – Collection
of
String
S dog ball • Output – Ordered
result • Assumptions ant ball – Complete
ordering
between cat dog any
two
elements
of
S eagle Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 4

Problem
Instance
Representations • Array
of
fixed
structured
content e ¬ c ¬ a e a g l a t n t ¬ ¬ ¬ ¬ ¬ • Array
of
pointers
to
content eagle cat ant • Compact
representation e ¬ c ¬ a e a g l a t n t ¬ Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 5

Common
Data
Structures • Array Data
structures
provide
various – N
dimensional ways
to
representation • Linked
List information – Doubly‐linked Note
the
glyphs
used throughout
the
book
will
need • Stack to
be
consistent
here
as
well • Queue Key
operations
for
each
data – Double‐ended
queue structure
to
be
discussed
as – Priority
queue needed • Binary
Tree • Binary
Heap Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 6

63 INSERTION
SORT • Frame
the
problem – Insert
 each
new
element
into
proper
location remaining
elements in
the
collection
to
 h g a f m be
processed sorted
part
of
 element
being
considered
next the
collection Algorithm
iteratively
applies
this
key
operation Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 7

INSERTION
SORT • Occasionally
all
elements
must
be
moved remaining
elements in
the
collection
to
 g h a f m be
processed sorted
part
of
 element
being
considered
next the
collection Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 8

INSERTION
SORT • Only
need
to
move
elements
“higher”
than the
one
being
inserted remaining
elements in
the
collection
to
 a g h f m be
processed sorted
part
of
 element
being
considered
next the
collection Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 9

INSERTION
SORT • Sometimes
the
element
to
be
inserted
is greater
than
all
existing
elements
–
no
swaps! remaining

elements in
the
collection
to
 a f g h m be
processed sorted
part
of
 element

being
considered
next the
collection Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 10

INSERTION
SORT • Sometimes
the
element
to
be
inserted
is greater
than
all
existing
elements
–
no
swaps! DONE a f g h m sorted
part
of
 the
collection Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 11

64 Algorithm
Fact
Sheet Name
of the Concepts Algorithm INSERTION
SORT Array insert
(A,
6,
“7”)
 sort 
(A) Small Pseudocod 1. for 
i=1
 to
 n–1
 do Example e 1 4 8 9 11 15 7 12 13 6 2. insert
(A,
i,
A[i]) description end Already
sorted 7 value insert
 (A,
pos,
value) 1. i
=
pos–1 Elements Insert
into 2. while 
(i
 ≥ 
0
 and 
A[i]
>
value)
 then compared proper and
bumped
up 3. A[i+1]
=
A[i] spot 4. i
=
i–1 1 4 7 8 9 11 15 12 13 6 5. A[i+1]=value Sorted
region
extended
by
one end Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 12

sort 
(A) Array 64 INSERTION 1. for 
i=1
 to
 n–1
 do 2. insert
(A,
i,
A[i]) SORT end insert
 (A,
pos,
value) 1. i
=
pos–1 2. while 
(i
 ≥ 
0
 and 
A[i]
>
value)
 then insert
(A,
6,
“7”)
 3. A[i+1]
=
A[i] 4. i
=
i–1 1 4 8 9 11 15 7 12 13 6 5. A[i+1]=value end Already
sorted value 7 Elements Insert
into compared proper and
bumped spot up 1 4 7 8 9 11151213 6 Sorted
region
extended
by
one Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 13

Code
Check • Show
actual
running
code – Handout – Debug
example Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 14

Performance
of
INSERTION
SORT • As
sorting
algorithms
go,
how
efficient
is INSERTION
SORT? – Is
it
the
fastest
algorithm?
[NO] – How
does
it
compare
with
other
algorithms? • Difficult
to
answer
without
a
theoretic
model – independent
of
programming
language – Independent
of
computer
hardware Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 15

Performance
of
Algorithms • Use
standard
“Big
O”
notation – Let
T( n )
be
time
for
algorithm
to
perform
on
an average
problem
instance
of
size
 n – How
does
T( n )
grow
in
proportion
to
increasing
n? • Example
Problem – Given
 n 
integers,
find
the
largest
integer – Expect
that
T(2n)
 ≅ 
2*
T(n) • Find
performance
family
that
most
closely matches
behavior
of
algorithm Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 16

Performance
Analysis • Linear
or
O(n) – As
problem
size
is
multiplied
by
2,
the
time
to complete
the
problem
“should
be”
multiplied
by
2 – Holds
true
for
any
constant
multiplicative
factor • How
to
capture
this
concept? – Once
n
is
“sufficiently
large”,
there
is
some
constant
 c such
that
 t ( n )
≤
 c * n – Not
an
estimate
but
a
firm
upper
bound • In
practice,
“c”
is
never
computed,
though
its existence
is
guaranteed – Depends
upon
hardware
platform,
language,
etc… Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 17

Functional
Families • You
already
know
this
concept
from
algebra – x 2 
and
4x 2 –3*x+170
are
both
“quadratic” formulae – Distinctive
shape – Highest
exponent is
most
important – In
“long
run”
they behave
similarly Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 18

66 Performance
of
INSERTION
SORT • Average
Case – Given
random
permutation
of
 n 
elements,
each element
is
(on
average)
 n /3
positions
from
proper location 3 1 2 2 0 Average
=
8/5
=1.6
 ≅ 
5/3 h g a f m a f g h m Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 19

66 Performance
of
INSERTION
SORT • insert(A,
pos,
value)
is
invoked
 n–1 
times – On
average,
 while 
loop
invoked
 n /3
times • Quick
estimate
=
( n –1
)*( n /3)
=
 ⅓ n 2 
–
n/3 – The
critical
factor
is
the
highest
exponent – ⅓ n 2 
–
n/3
is
O(n 2 ) sort 
(A) • INSERTION
SORT
is
not
linear 1. for 
i=1
 to
 n–1
 do 2. insert
(A,
i,
A[i]) – It
is
Quadratic end – As
problem
size
is
multiplied
by insert
 (A,
pos,
value) 1. 
i
=
pos–1 k =2,
the
time
to
complete
the 2. 
while 
(i
 ≥ 
0
 and 
A[i]
>
value)
 then problem
is
multiplied
by
 k 2 =4 3. A[i+1]
=
A[i] 4. i
=
i–1 5. 
A[i+1]=value Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 20 end

18 Rating
Performance
of
Algorithm • Best
Case – Problem
instances
for
which
algorithm
computes answer
most
efficiently • Average
Case – Typical
random
problem
instance.
Identifies
the expected
performance
of
the
algorithm • Worst
Case – Unusual
problem
instances
that
force
algorithm
to work
harder
and
be
less
efficient Algorithms
in
a
Nutshell (c)
2009,
George
T.
Heineman 21