COT 5407: Introduction to Algorithms

COT 5407: Introduction to Algorithms Tao Li ECS 318; Phone: x6036 taoli@cs.fiu.edu http://www.cs.fiu.edu/~taoli/class/COT5407-F09/index.html COT 5407

Why should I care about Algorithms? Cartoon from Intractability by Garey and Johnson COT 5407

More questions you should ask • Who should know about Algorithms? • Is there a future in this field? • Would I ever need it if I want to be a software engineer or work with databases? COT 5407

Why are theoretical results useful? Cartoon from Intractability by Garey and Johnson COT 5407

Self-Introduction • Ph.D. in Computer Science from University of Rochester, 2004 • Research Interests: data mining, machine learning, information retrieval, bioinformatics and more ? • Associate Professor in the School of Computer Science at Florida International University • Industry Experience: • Summer internships at Xerox Research (summer 2001, 2002) and IBM Research (Summer 2003, 2004) COT 5407

Student Self-Introduction • Name • I will try to remember your names. But if you have a Long name, please let me know how should I call you  • Major and Academic status • Programming Skills • Java, C/C++, VB, Matlab, Scripts etc. • Research Interest • Anything you want us to know COT 5407

What this course is about • Introduction to Algorithms • Analysis of Algorithms • How does one design programs and ascertain their efficiency? • Divide-and-conquer techniques, string processing, graph algorithms, mathematical algorithms. Advanced data structures such as balanced tree schemes. COT 5407

Course Logistics • Meeting Time and Location: Tuesday and Thursday 17:00pm-18:15pm, ECS134 • Office Hours:Tuesday and Thursday 14:30pm-15:30pm • TA: Yali Wu • Textbook: Introduction to Algorithms, (Third Edition) Thomas Cormen, Charles Leiserson, Ronald Rivest, and Clifford Stein. MIT Press. COT 5407

Evaluation • Class participation and Quizzes: 10% • Midterm Exam: 30% • Final Exam: 30% • Assignments:30% • You may work with one other person on homeworks, but you must each write up your solutions separately. If you work with another person, indicate who you worked with on your solution. • Please start a new page for each problem on your solutions, and include your name on each page, so the TA can choose the problems for grading. • Exams are open/closed book ?? COT 5407

Algorithm • A computational problem is a mathematical problem, specified by an input/output relation. • An algorithm is a computational procedure for solving a computational problem. problem algorithm “computer” input output COT 5407

Some Well-known Computational Problems • Sorting • Searching • Shortest paths in a graph • Minimum spanning tree • Primality testing • Traveling salesman problem • Knapsack problem • Chess • Towers of Hanoi • Program termination

History of Algorithms The great thinkers of our field: • Euclid, 300 BC • Bhaskara, 6th century • Al Khwarizmi, 9th century • Fibonacci, 13th century • Babbage, 19th century • Turing, 20th century • von Neumann, Knuth, Karp, Tarjan, … COT 5407

Euclid’s Algorithm • GCD(12,8) = 4; GCD(49,35) = 7; • GCD(210,588) = ?? • GCD(a,b) = ?? • Observation: [a and b are integers and a  b] • GCD(a,b) = GCD(a-b,b) • Euclid’sRule: [a and b are integers and a  b] • GCD(a,b) = GCD(a mod b, b) • Euclid’s GCD Algorithm: • GCD(a,b) If (b = 0) then return a; return GCD(a mod b, b) COT 5407

Basic Issues Related to Algorithms • How to design algorithms • How to express algorithms • Proving correctness • Efficiency • Theoretical analysis • Empirical analysis • Optimality

Algorithm design strategies Brute force Divide and conquer Decrease and conquer Transform and conquer Greedy approach Dynamic programming Backtracking Branch and bound Space and time tradeoffs

Analysis of Algorithms • How good is the algorithm? • Correctness • Time efficiency: amount of work done • Space efficiency: amount of space used • Simplicity, clarity • Does there exist a better algorithm? • Lower bounds • Optimality

Correctness Proving correctness is dreadful for large algorithms. A strategy that can be used is: divide the algorithm into smaller pieces, and then clarify what the preconditions and postconditions are and prove correct assuming everything else is correct. COT 5407

Amount of Work Done Rather than counting the total number of instructions executed, we'll focus on a set of key instructions and count how many times they are executed. Use asymptotic notation and pay attention only to the largest growing factor in the formula of the running time. Two major types of analysis: worst-case analysis and average-case analysis COT 5407

More • Amount of space used: The amount of space used can be measured similarly. Consideration of this efficiency is often important. • Simplicity, clarity: Sometimes, complicated and long algorithms can be simplified and shortened by the use of recursive calls. • Optimality: For some algorithms, you can argue that they are the best in terms of either amount of time used or amount of space used. There are also problems for which you cannot hope to have efficient algorithms. COT 5407

Asymptotic Growth Rates of Functions • Big O • Big Omega • Little O • Little Omega • Theta Notation COT 5407

Notations COT 5407

Other mathematical background • The ceiling function • The floor function • The exponentials and logarithms • Fibonacci number • Summations and Series COT 5407

Why study algorithms? • Theoretical importance • the core of computer science • Practical importance • A practitioner’s toolkit of known algorithms • Framework for designing and analyzing algorithms for new problems

Two main issues related to algorithms • How to design algorithms • How to analyze algorithm efficiency

Algorithm design strategies Brute force Divide and conquer Decrease and conquer Transform and conquer Greedy approach Dynamic programming Backtracking Branch and bound Space and time tradeoffs

Analysis of algorithms • How good is the algorithm? • time efficiency • space efficiency • Does there exist a better algorithm? • lower bounds • optimality

Important problem types • sorting • searching • string processing • graph problems • combinatorial problems • geometric problems • numerical problems

Fundamental data structures list array linked list string stack queue, priority queue Graph Tree set and dictionary

Search • You are asked to guess a number X that is known to be an integer lying between integers A and B. How many guesses do you need in the worst case? • Number of guesses = log2(B-A) • You are asked to guess a positive integer X. How many guesses do you need in the worst case? • NOTE: No upper bound B is known for the number. COT 5407

Search • You are asked to guess a number X that is known to be an integer lying between integers A and B. How many guesses do you need in the worst case? • Number of guesses = log2(B-A) • You are asked to guess a positive integer X. How many guesses do you need in the worst case? • NOTE: No upper bound B is known for the number. • Algorithm: • figure out B (by using Doubling Search) • perform binary search in the range B/2 through B. • Number of guesses = log2B + log2(B – B/2) • Since X is between B/2 and B, we have: log2(B/2) < log2X, • Number of guesses < 2log2X - 1 COT 5407

Polynomials • Given a polynomial • p(x) = a0 + a1 x + a2 x2 + … + an-1 xn-1 + an xn compute the value of the polynomial for a given value of x. • How many additions and multiplications are needed? • Simple solution: • Number of additions = n • Number of multiplications = 1 + 2 + … + n = n(n+1)/2 • Improved solution using Horner’s rule: COT 5407

Polynomials • Given a polynomial • p(x) = a0 + a1 x + a2 x2 + … + an-1 xn-1 + an xn compute the value of the polynomial for a given value of x. • How many additions and multiplications are needed? • Simple solution: • Number of additions = n • Number of multiplications = 1 + 2 + … + n = n(n+1)/2 • Improved solution using Horner’s rule: • p(x) =p(x) = a0 + x(a1 + x(a2 + … x(an-1 + x an))…)) • Number of additions = n • Number of multiplications = n COT 5407

COT 5407: Introduction to Algorithms