Design and Analysis of
Algorithms
4th Semester
Computer Engineering
Spring 2016
Conf.dr.ing. Ioana Ṣora
ioana.sora@cs.upt.ro
Our goal:
design
algorithms
that are
correct
and
efficient
The Need for Correctness
• Much too often, algorithms are designed in
an ad-hoc manner and “validated” through
testing on a few particular data sets …
Example of a “sorting algorithm”
The algorithm:
A:array[1..n] of
integer
The “tests”:
5, 1, 2, 7, 6
15, 2, 4, 17,16
for i=1 to n-1 do
if A[i]>A[i+1]
swap(A[i], A[i+1])
But it does not work for 3, 2, 1, 5, 4 !
The Need for Correctness
• Much too often, algorithms are designed in
an ad-hoc manner and “validated” through
testing on a few particular data sets …
• The fact that an algorithm passed a number of
tests on certain data sets DOES NOT guarantee
that the algorithm is correct
• The fact that an algorithm fails a test on ONE
data set proves that it is NOT correct
The Bubblesort Algorithm
A:array[1..n] of
integer
for lim:=n downto 2
for i:=1 to lim-1 do
if A[i]>A[i+1]
swap(A[i], A[i+1])
• How can we
know for sure
that the algorithm
is correct ? (Other
than simply believing
the programming
textbooks ;-))
– ….we will learn this
later in this course !
Now that we have a correct algorithm
…
• Implement it
• Run the program and extensively test it with arrays
of 10 numbers
• Be happy that everything is fine
• But: Run the program on an array of 10000 numbers
…
• Is it necessary to implement an algorithm to see that
it takes too long for big sizes or can we predict this ?
Analyzing Algorithms
• We need methods and metrics to analyze
algorithms for:
– Correctness
• Methods for proving correctness
– Efficiency
• Time complexity, Asymptotic analysis
Designing Algorithms
• Ok, so you will know (will learn) how to
analyze a given algorithm.
• But where do these algorithms come from ?
– Clever people already designed a plethora of
solutions (algorithms) for different problems and
we find them in textbooks, internet, etc.
– But how will you design solutions for new
problems ?
How to learn algorithms ?
http://www.neatorama.com/twaggies/2010/11/07/no-112-robcorddry/
Kruskal’s algorithm
Prim’s algorithm
Floyd’s algorithm
Dijkstra’s algorithm
Tarjan’s algorithm
Design methods:
• Greedy
• Design by Induction
• Divide and Conquer
• Dynamic Programming
What about data structures ?
• Fundamental data structures:
– Model fundamental data
• lists, queues, graphs
• Special data structures:
– Created to optimize a specific (set of)
operation(s) for a specific context
• Search trees, balanced trees
• Optimal trees
• Disjoint sets
Course Goals
• Learn to design algorithms that are correct and
efficient
– How do we know that:
• an algorithm is correct ? -> correctness proofs
• an algorithm is efficient ? -> analysis of algorithms (time
complexity)
– How to design solutions for new problems ?
• Learning general techniques for design
• Studying a set of well-known algorithms, to serve as
examples of success stories for applying general design
techniques
• Become aware that good algorithms are key
parts of software engineering practice !
Course Schedule
• Review: Analysis
• Data structures: Search Trees, Balanced trees,
Augmented data structures, Graphs
• Proving Correctness of Algorithms. Induction
• Design of Algorithms by Induction
• Dynamic Programming
• Great algorithms in real life problems:
– Data compression
– Search engines
Textbooks
[CLRS]
[Unlocked]
[Manber]
[McCormick]
Course Webpage
All official information related to the
Algorithm Design and Analysis classes:
http://staff.cs.upt.ro/~ioana/algo/
Lab Assignments
• Comprise different types of exercises:
– Questions - to be solved with pen on paper
– Simple implementation problems - implement something that
was described in the course
– Algorithm design problems
• This kind of assignments require to:
– Describe the proposed algorithm
– Analyze your algorithm (prove correctness and efficiency)
– Implement the algorithm
– Tool projects
• Complex projects containing also a part of algorithm design and
implementation
• They are optional, but: in order to get a 10 for the lab activity, you
have to do at least one tool project
Examination and Grading
• Final grade = 2/3 final written exam + 1/3 work during the
semester
• Written exam:
– Questions and Algorithm design problems (such as these the
lab description above)
• Work during the semester:
– Lab assignments
– Lab quizzes
– Tool project (for 10)
• Optional bonus points: for exceptional activity = doing
more than one optional tool project during the semester,
you can gain extra-points to the grade of your final
written exam