1. CSE310 Lecture01: Algorithms, Counting and Data Structures
Guoliang (Larry) Xue
Computer Science and Engineering
CIDSE
Arizona State University
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2. Topics of this lecture Discussion of Syllabus Algorithms
definition
time complexity
space complexity
Counting and Induction Proof
number of operations
space requirement
Data Structures
stacks and queues
advanced data structures

3. Algorithms
An algorithm is a well-defined step by step computational procedure that
is deterministic
takes some input
produces some output
stops on any input
Examples illustrating what is an algorithm and what is not an algorithm.

4. Example of what is an algorithm
Given a sequence of n numbers, compute the sum. Can we design an algorithm for this?
Given a sequence of n numbers, order them in non-decreasing order, Can we design an algorithms for this?Introduce insertion sort.

5. Example of what is not an algorithm
Print out all positive integers.
for (n=1; n>=1; n++) {
print n; }
The above is NOT an algorithm. Why?
Print our the first 10,000,000 positive integers.
for (n=1; n<= ; n++){print n;}
The above IS an algorithm. Why?

6. A more complex example The 3n+1 problem.
Given a positive integer n, if n=1 output n and stop; if n is even, output n=n/2 and continue; if n is odd, output n=3n+1 and continue;
Is the above an algorithm?

7. Another view of algorithms
An algorithm is a tool to solve a well-defined computational problem.
An instance of a problem.
Example of a problem: Given a sequence of numbers, order them into non-decreasing order.
An instance of the above problem is given by the input sequence 5, 4, 3, 2, 1.
Another instance of the above problem is given by the input sequence 7, 5, 1, 3.

8. Time complexity of an algorithm
Given an algorithm A for solving a particular problem, what is the time complexity of the algorithm?
Time complexity is measured using the number of operations required.
We also need to note the input size of the problem instance.
Examples: (1) summation, (2) matrix multiplication.

9. Space complexity of an algorithm
Given an algorithm A for solving a particular problem, what is the space complexity of the algorithm?
Space complexity is measured using the number of memory cells required.
We also need to note the input size of the problem instance.
Examples: (1) summation, (2) matrix multiplication.

10. Complexity of an algorithm
Best case analysis
Average case analysis
Worst case analysis
Amortized analysis

11. Counting techniques and induction proof

12. Growth of functions
lg (n)
sqrt(n) n
n log(n) n2 n3 2n n!

13. Data structures What is an Abstract Data Type? Why data structures?
A collection of data elements
A collection of functions on the data elements
Why data structures?
to store data efficiently
to process data efficiently
Examples of data structures:
stacks and queues
binary search trees

14. Using the right data structures
How to represent a polynomial of order n?
use an array of size n
use a linked list
How to perform addition, subtraction and multiplication using the array representation?
How to perform addition, subtraction and multiplication using the linked list representation?
Which data structure is better?

15. Using the right algorithm
If the first 6 terms in a sequence are 1, 1, 2, 3, 5, 8, what is the 7th term? What is the nth term?
This sequence is called the Fibonacci sequence.
Let us write two algorithms for computing the nth Fibonacci number.
Which algorithm is better?

16. Summary What is an algorithm? What is an ADT? Counting.
Proof using mathematical induction.
Problems and instances.
Comparison of algorithms.

17. Readings and Exercises
Most materials covered in this lecture can be found in Section 1.1, Section 1.2, Section 2.1 and 2.2.
You should work on Exercises through (p.11) and Problems 1-1 (pp.14-15).
Lecture 02 will be on asymptotic notations. To preview, read Sections 3.1 and 3.2.