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CS 361, Lecture 1
Jared Saia
University of New Mexico
Todayās Outline
Administrative Info
What is an Algorithm? Data Structure?
Why study algorithms?
Todo list for next class
Why study algorithms?
House āSeven years of College down the toiletā - John Belushi in Animal
about algorithms and data structures?Q: Can I get a programming job without knowing something
entire life?A: Yes, but do you really want to be programming GUIs your
Why study algorithms? (II)
dia, Los Alamos, etc.of algorithms: Microsoft, Google, Oracle, IBM, Yahoo, San-Almost all big companies want programmers with knowledge
questions about algorithms and/or data structuresIn most programming job interviews, they will ask you several
large masses of interviewees who know only how to programYour knowledge of algorithms will set you apart from the
algorithms for solving a problem (e.g. Google, Akamai, etc.)many startups are successful because theyāve found betterIf you want to start your own company, you should know that
Why Study Algorithms? (III)
Youāll write better, faster code
Youāll learn to think more abstractly and mathematically
Itās the most challenging and interesting area of CS!
A Real Job Interview Question
dept web page for the full compilation of questions):2002 (thanks to Maksim Noy, see the career center link from the The following is a question commonly asked in job interviews in
You are given an array with integers between 1 and 1
All integers between 1 and 1
(^) 000 are in the array at least
once, and one of those integers is in the array twice
Can you do it while iterating through the array only once?Q: Can you determine which integer is in the array twice?
Solution
Ideas on how to solve this problem??
What if we allowed
multiple iterations?
Naive Algorithm
Create a new array of ints between 1 and 1
(^) 000, which
all entries to 0weāll use to count the occurences of each number. Initialize
update its count in the new arrayGo through the input array and each time a number is seen,
twice.Go through the count array and see which number occurs
Return this number
Take Away
Designing good algorithms matters!
Not always this easy to improve an algorithm
However, with some thought and work, you can
almost al-
ways
get a better algorithm than the naive approach
How to analyze an algorithm?
about:There are several resource bounds we could be concerned
time, space, communication bandwidth, logic gates,
etc.
However, we are usually most concerned about time
guages and machine typesRecall that algorithms are independent of programming lan-
Q: So how do we measure resource bounds of algorithms
Random-access machine model
We will use RAM model of computation in this class
All instructions operate in serial
etc.) take unit timeAll basic operations (e.g. add, multiply, compare, read, store,
unit spaceAll āatomicā data (chars, ints, doubles, pointers, etc.) take
Worst Case Analysis
Weāll
generally
be
pessimistic
when
we
evaluate
resource
bounds
possible input sequenceWeāll evaluate the run time of the algorithm on the worst
Amazingly,
in most cases,
weāll still be able to get pretty
good bounds
Justification:
The āaverage caseā is often about as bad as
the worst case.
Example Analysis
redundant element in an arrayConsider the problem discussed last tuesday about finding a
are 1 toLetās consider the more general problem, where the numbers
n
instead of 1 to 1
Algorithm 1
Create a new ācountā array of ints of size
n , which weāll use
to 0to count the occurences of each number. Initialize all entries
update its count in the ācountā arrayGo through the input array and each time a number is seen,
already been counted once, return this numberAs soon as a number is seen in the input array which has
Algorithm 2
letIterate through the input array, summing up all the numbers,
S
be this sum
Let
x
=
S
n (^) + 1)
n/
Return
x
Example Analysis: Time
Worst case:
Algorithm 1 does 5
n
operations (
n
inits to 0
in ācountā array,
n
reads of input array,
n
reads of ācountā
array (to see if value is 1),
n
increments, and
n
stores into
count array)
Worst case:
Algorithm 2 does 2
(^) n
(^) + 4 operations (
n
reads
of input array,
n
additions to value
S
4 computations to
determine
x
given
S
More Examples
E.g.
n , 10
n
ā
(^) 2000, and
n
O
n )
n
(^) n
,
n (^) ā
n
are
O
n )
n 2
+^
(^) n
(^) + log
(^) n
, 10
n 2 +^
(^) n
(^) ā
n
are
O
n 2 )
n (^) log
(^) n
n
is
O
n (^) log
(^) n
)
(^) log
2 n^
is
O
(log
2 n^ )
n ā
n (^) +
(^) n
(^) log
(^) n
n
is
O
n ā
n )
50
and 4 are
O
More Examples
Algorithm 1 and 2 both take time
O
n )
Algorithm 1 uses
O
n ) extra space
But, Algorithm 2 uses
O
(1) extra space
Formal Defn of Big-O
A function
f (^) ( n )
is
O
g ( n ))
if there exist positive constants
c
and
n 0
such that
f (^) ( n ) ā¤
(^) cg
n )
for all
n
ā„
n 0
Example
Letās show that
f (^) ( n ) = 10
n (^) + 100 is
O
g ( n )) where
g ( n ) =
n
We need to give constants
c
and
n 0
such that
f (^) ( n )
ā¤
cg
( n )
for all
n
ā„
n 0
In other words, we need constants
c
and
n 0
such that 10
n (^) +
cn
for all
n
ā„
n 0
Example
We can solve for appropriate constants:
n
cn
/n
c
So if
n >
1, then
c
should be greater than 110.
In other words, for all
n >
n (^) + 100
n
So 10
n (^) + 100 is
O
n )
Questions
Express the following in
O
notation
n 3 / 1000
n 2
ā
(^100)
n (^) + 3
log
(^) n
(^) log
2 n^ (^) + 100
ā
in
(^) i
Todo
Work on pretest, due next Tuesday!
Visit the class web page: www.cs.unm.edu/
saia/361/
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Read Chapter 1 in textbook
