Divideand
Conquer
Algorithms
COMP157
Sept212007
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Notes on Divide and Conquer Algorithms | COMP 157, Study notes of Algorithms and Programming
Material Type: Notes; Class: Design/Analysis of Algorithms; Subject: Computer Science; University: University of the Pacific; Term: Fall 2007;
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Divide
and
Conquer Algorithms
COMP
Sept
Divide
and
Conquer
The
most
‐well
known
algorithm
design
strategy:
- Divide
instance
of
problem
into
two
or
more
smaller
instances
- Solve
smaller
instances
recursively
- Obtain
solution
to
original
(larger)
instance
by
combining
these
solutions
General Divide-and-Conquer Recurrence
T
n
aT
n/b
f
n
b = number of sub-problems
(usually 2)
a = number of sub-problems that need to be solved
(usually 2)
f(n) = cost of splitting problem +
cost of merging solutions to sub-problems
Divide-and-Conquer
Parallelization
T
n
aT
n/b
f
n
If we can concurrently solve all the sub-problems
(through parallel processing) , then a = 1.
Master Theorem - Example
T
(
n
) = 2
T
( n
/2) +
^1
(recursive sum of array elements)
a = 2b = 2d = 0
Master Theorem: If
a > b
d
,^
T
(
n
)
∈ Θ
( n
log
b a
)
T
(
n
)
∈ Θ
( n
log
2 2
)
=
Θ
( n
)
Master Theorem - Example T
(
n
) = 4
T
(
n
/2) +
n
a = 4b = 2d = 1
Master Theorem: If
a > b
d
,^
T
(
n
)
∈ Θ
( n
log
b a
)
T
(
n
)
∈ Θ
( n
log
2 4
)
=
Θ
( n
2
)
Master Theorem - Example
T
(
n
) = 4
T
(
n
/2) +
n
3
a = 4b = 2d = 3
Master Theorem: If
a < b
d
,^
T
( n
)^
∈ Θ
(
n
d
)
T
(
n
)
∈ Θ
( n
3
)
Mergesort
Split
array
A
into
two
(nearly)
equal
halves
and
make
copies
of
each
half
in
arrays
B
and
C
Sort
arrays
B
and
C
recursively
Merge
sorted
arrays
B
and
C
back
into
array
A
(next
slide)
Pseudocode
of
Mergesort
Pseudocode
of
Merge
Analysis
of
Mergesort
T(n)
2T(n/2)
n
a=2,
b=2,
d=
Master
Theorem: If
a
b
d
,^
T
( n
)^
( n
d
log
n
T
n
n
log
n
Analysis
of
Mergesort
All
cases
have
same
efficiency:
n
log
n
C
worst
(n)
n
log
2
n
n
which
is
close
to
theoretical
minimum:
⎡
log
2
n
!⎤
≈
n
log
2
n
‐^
n
(chapter
Space
requirement:
n
(not in
place)
(correct?)
Can
be
implemented
without
recursion
(bottom
up)
Partitioning
Algorithm
Analysis
of
Quicksort
Best
case:
split
in
the
middle:
n
log
n
Worst
case:
sorted
array:
n
2
Average
case:
random
arrays:
n
log
n
C
avg
(n)
n
log
2
n
Improvements:^ •
better
pivot
selection:
median
of
three
partitioning
-^
switch
to
insertion
sort
on
small
subfiles
-^
elimination
of
recursion
- switch
to
iteration
These
combine
to
20
‐25%
improvement