CSCI 6212
Design and Analysis of
Algorithms
Dr. Juman Byun
The George Washington University
•
•
Please drop this course if you have not taken the
following prerequisite. Sometimes enthusiasm
alone is not enough.
CSci 1311: Discrete Structures I (3)
CSci 1112: Algorithms and Data Structures (3)
Official Course Description
•
http://www.cs.gwu.edu/academics/courses/graduate/csci6212
•
Design and analysis of algorithms. Turing machines; NPComplete theory. Algorithmic techniques: divide-and-conquer,
greedy, dynamic programming, graph traversal, backtracking,
and branch-and-bound. Applications include sorting and
searching, graph algorithms, and optimization. Prerequisite:
CSci 1311, 1112. (Fall and
spring)
•
Textbook: Introduction to Algorithms, Cormen, Leiserson,
Rivest, and Stein.
Basic Principles of
Algorithm Design and
Analysis
Algorithm Design
How to Start the Design ?
•
Understand the problem. What are we trying to
achieve ?
•
Get a rough idea how to solve the problem using
your intuition and imagination
Example: Sorting an Array of
integers
5
2
4
6
1
3
Example: Sorting an Array of
integers
2
5
4
6
1
3
Example: Sorting an Array of
integers
4
2
5
6
1
3
Example: Sorting an Array of
integers
4
2
5
6
1
3
Example: Sorting an Array of
integers
6
4
2
5
1
3
Example: Sorting an Array of
integers
6
2
4
5
1
3
Example: Sorting an Array of
integers
1
6
2
4
5
3
Example: Sorting an Array of
integers
1
2
4
5
6
3
Example: Sorting an Array of
integers
3
1
2
4
5
6
Example: Sorting an Array of
integers
3
2
4
5
1
6
Example: Sorting an Array of
integers
3
2
4
5
1
6
Example: Sorting an Array of
integers
3
2
4
5
1
6
Example: Sorting an Array of
integers
3
2
4
5
1
6
Example: Sorting an Array of
integers
3
1
2
4
5
6
Example: Sorting an Array of
integers
3
1
2
4
5
6
Example: Sorting an Array of
integers
1
2
4
5
3
6
Example: Sorting an Array of
integers
1
2
4
5
3
6
Example: Sorting an Array of
integers
1
2
4
5
3
6
Example: Sorting an Array of
integers
1
2
3
4
5
6
Accept Concise Notation
•
The previous colorful animation was pretty but it
took me a some time to create it.
•
In order for a human being to perceive the entire
problem, you need to compress the problem or
commit it into the long-term storage of your brain.
•
Algorithmic notation is an excellent way of
compressing a problem so that it is easier to
remember and handle.
Accept Concise Notation
5
2
4
6
1
3
Accept Concise Notation
5
2
4
6
1
3
Disambiguate it
5 , 2 , 4 , 6 , 1, 3
Abstract it
A = { 5 , 2 , 4 , 6 , 1, 3 }
Denote it
Insertion Sort (A)
1 for j = 2 to A.length
2
key = A[j]
3
// Insert A[j] into the sorted sequence A[1..j-1]
4
i=j-1
5
while i > 0 and A[i] > key
6
A[i +1] = A[i]
7
i=i-1
8
A[i + 1] = key
Pascal Notation
Insertion Sort (A)
1 for j = 2 to A.length
2 begin
3
key = A[j]
4
// Insert A[j] into the sorted sequence A[1..j-1]
5
i=j-1
6
while i > 0 and A[i] > key
7
begin
8
A[i +1] = A[i]
9
i=i-1
10
end
11
A[i + 1] = key
12 end
C Notation
Insertion Sort (A)
1 for j = 2 to A.length
2{
3
key = A[j]
4
// Insert A[j] into the sorted sequence A[1..j-1]
5
i=j-1
6
while i > 0 and A[i] > key
7
{
8
A[i +1] = A[i]
9
i=i-1
10
}
11
A[i + 1] = key
12 }
Algorithm Analysis
Example: Running
(Execution) Time Analysis
Insertion Sort (A)
1 for j = 2 to A.length
2
key = A[j]
3
// Insert A[j] into the sorted sequence A[1..j-1]
4
i=j-1
5
while i > 0 and A[i] > key
6
A[i +1] = A[i]
7
i=i-1
8
A[i + 1] = key
How many times does each
statement execute ?
Insertion Sort (A)
1 for j = 2 to A.length
2
key = A[j]
3
// Insert A[j] into the sorted sequence A[1..j-1]
4
i=j-1
5
while i > 0 and A[i] > key
6
A[i +1] = A[i]
7
i=i-1
8
A[i + 1] = key
Factors to Consider
•
Input Size
•
Running Time
•
•
Worst-Case
•
Best-Case
•
Average
Rate of Growth
Assumptions
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Technologies
•
Memory Model: RAM (Random Access
Machine/Memory)
•
Processor: Single Processor