PPT-22C:19 Discrete Math Algorithms and Complexity

Fall 2011 Sukumar Ghosh What is an algorithm A finite set or sequence of precise instructions for performing a computation Example Maxima finding procedure max . Sets and Functions. Fall . 2011. Sukumar Ghosh. What is a set?. Definition. . A set is an unordered collection of objects.. . S = {2, 4, 6, 8, …}. . COLOR = {red, blue, green, yellow}. Each object is called an element or a member of the set.. Discrete Probability. Fall 2011. Sukumar Ghosh. Sample Space. DEFINITION. . The . sample space S . of an experiment is the set . of possible outcomes. An . event. . E. is a . subset. of the sample space.. Logic and Proof. Fall . 2011. Sukumar Ghosh. Predicate Logic. Propositional logic has limitations. Consider this:. Is . x. . > 3. a proposition? No, it is a . predicate. . Call it . P(x. ). . Integers and Modular Arithmetic . Fall 2010. Sukumar Ghosh. Preamble. Historically, . number theory . has been a beautiful area of . study in . pure mathematics. . However, in modern times, . number theory is very important in the . Counting. Fall . 2011. Sukumar Ghosh. The Product Rule. Example of Product Rule. Example of Product Rule. The Sum Rule. Example of Sum Rule. Example of Sum Rule. Wedding picture example. Counting subsets of a finite set. Graphs. Fall . 2011. Sukumar Ghosh. Seven Bridges of . K. ⍥. nigsberg. Is it possible to walk along a route that cross . each bridge exactly once?. Seven Bridges of . K. ⍥. nigsberg. A Graph. What is a Graph. Integers and Modular Arithmetic . Fall 2010. Sukumar Ghosh. Preamble. Historically, . number theory . has been a beautiful area of . study in . pure mathematics. . However, in modern times, . number theory is very important in the . Lecture 1: . Intro; Turing machines; . Class P and NP . . . Indian Institute of Science. About the course. Computational complexity attempts . to classify computational . problems. Structures. Introduction and Scope:. Propositions. Spring 2015. Sukumar Ghosh. The Scope. Discrete . mathematics. . studies mathematical . structures that are . fundamentally . discrete. ,. . not . Applied Discrete Mathematics Week 3: Algorithms. 1. The Growth of Functions. The growth of functions is usually described using the . big-O notation. .. Fall . 2011. Sukumar Ghosh. The Product Rule. Example of Product Rule. Example of Product Rule. The Sum Rule. Example of Sum Rule. Example of Sum Rule. Wedding picture example. Counting subsets of a finite set. Today’s class. 1) Lecture. 2) . Blackbox. presentations. 3) Guest Lecture: Jonathan Mills. O. rganized . complexity. organized complexity. study of organization. whole is more than sum of parts. Systemhood. Fall 2011. Sukumar Ghosh. Sequence. A sequence is an . ordered. list of elements. . Examples of Sequence. Examples of Sequence. Examples of Sequence. Not all sequences are arithmetic or geometric sequences.. Lijie. Chen. MIT. Today’s Topic. Background. . What is Fine-Grained Complexity?. The Methodology of Fine-Grained Complexity. Frontier: Fine-Grained Hardness for Approximation Problems. The Connection.