![Example 2: Smallest Enclosing circle
Problem definition: Given n points in a plane, compute the smallest radius circle
that encloses all n point.
Applications: Facility location problem
Best deterministic algorithm : [Megiddo, 1983]
•
O(n) time complexity, too complex, uses advanced geometry
Randomized Las Vegas algorithm: [Welzl, 1991]
•
Expected O(n) time complexity, too simple, uses elementary geometry
16](https://image.slidesharecdn.com/lecture-1-cs648-131101030822-phpapp02/85/Lecture-1-cs648-16-320.jpg)
![A real Fact
[A study by Microsoft in 2008]
Compare this probability with the failure (or exceeding
the running time) probability of various randomized
algorithms mentioned earlier.
20](https://image.slidesharecdn.com/lecture-1-cs648-131101030822-phpapp02/85/Lecture-1-cs648-20-320.jpg)
Lecture 1-cs648
- 2. Overview of the lecture • What is a randomized algorithm ? • Motivation • The structure of the course 2
- 3. What is a randomized algorithm ? 3
- 4. Deterministic Algorithm Output Input Algorithm • The output as well as the running time are functions only of the input. 4
- 5. Randomized Algorithm Random bits Output Input Algorithm • The output or the running time are functions of the input and random bits chosen . 5
- 6. EXAMPLE 1 : APPROXIMATE MEDIAN 6
- 8. EXAMPLE 2 : RANDOMIZED QUICK SORT 8
- 9. 9
- 10. 10
- 11. 11
- 12. Types of Randomized Algorithms Randomized Las Vegas Algorithms: • Output is always correct • Running time is a random variable Example: Randomized Quick Sort Randomized Monte Carlo Algorithms: • Output may be incorrect with some probability • Running time is deterministic. Example: Randomized algorithm for approximate median 12
- 14. “Randomized algorithm for a problem is usually simpler and more efficient than its deterministic counterpart.” 14
- 16. Example 2: Smallest Enclosing circle Problem definition: Given n points in a plane, compute the smallest radius circle that encloses all n point. Applications: Facility location problem Best deterministic algorithm : [Megiddo, 1983] • O(n) time complexity, too complex, uses advanced geometry Randomized Las Vegas algorithm: [Welzl, 1991] • Expected O(n) time complexity, too simple, uses elementary geometry 16
- 17. Example 3: minimum Cut 17
- 18. Example 4: Primality Testing 18
- 19. UNCERTAINTY ASSOCIATED WITH RANDOMIZED ALGORITHMS IS IT REALLY A SERIOUS ISSUE ? 19
- 20. A real Fact [A study by Microsoft in 2008] Compare this probability with the failure (or exceeding the running time) probability of various randomized algorithms mentioned earlier. 20
- 22. Prerequisites • • • • • • Fundamentals of Data structures Fundamentals of the design and analysis of Algorithms Adequate programming skills (in C++) Elementary probability (12th standard) Ability to work hard Commitment to attend classes 22
- 23. Marks Breakup Assignments: 40% • Programming as well as theoretical. • To be done in groups of 2. Mid Semester Exam: 30% End Semester Exam: 30% Passing criteria: • At least 25% marks in both the exam (Total 15 out of 60) • If a student scores less than 50% marks in first mid semester exam, he/she must attend all classes for the rest of the course. 23
- 24. Contact Details Office: 307, Department of CSE Office Hours: – Every week day 12:30PM-1:00PM and 5:30PM-6:00PM Course website will be maintained at moodle.cse.iitk.ac.in 24