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503_lecture1a_S11 - UMass Lowell Computer Science 91.503...

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Unformatted text preview: UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, Spring 2011 Lecture 1 (Part 1) Introduction/Overview I t d ti /O i Tuesday, 1/25/11 Web Page Page Web http://www.cs.uml.edu/~kdaniels/courses/ALG_503_S11.html Nature of the Course Core course: Required for all CS Masters students Required as part of doctoral qualifying structure q p f q fy g Advanced algorithms Builds on undergraduate algorithms 91.404 No programming required "Pencil-and-paper" exercises "Pencil-and Lectures supplemented by: Programs g Real-world examples Real- What's It All About? Algorithm: Algorithm Al ith : steps for the computer to follow to solve a problem recognize structure of some common g problems understand important characteristics of algorithms to solve common problems select appropriate algorithm to solve a problem tailor existing algorithms create new algorithms Some of our goals:(at an advanced level) Some Algorithm Application Areas Robotics Bioinformatics Geographic Information Systems Analyze Design Telecommunications Apply Computer Graphics Voice Recognition Medical Imaging Some Typical Problems Nearest Neighbor Input: A set S of n points in d dimensions; a query point q. Problem:Which point in S is closest to q? Shortest Path Input: Edge-weighted graph G, with Edgestart vertex s and end vertex t Problem: Find the shortest path from s to t in G Convex Hull C H ll Input: A set S of n points in ddimensional space. Bin Packing g Input: A set of n items with sizes d_1,...,d_n. d_1,...,d_n. A set of m bins with capacity c_1,...,c_m. c_1,...,c_m. Problem: Find the smallest convex polygon containing all the points of S. Problem: How do you store the set of items using the fewest number of bins? SOURCE: SOURCE: Steve Skiena's Algorithm Design Manual (for problem descriptions, see graphics gallery at http://www.cs.sunysb.edu/~algorith) ) Some Typical Problems Transitive Closure Input: A directed graph G=(V,E). G=(V,E). Problem: Construct a graph G'=(V,E') with edge (i,j) in E' iff G (V,E ) E there is a directed path from i to j in G. For transitive reduction, construct a small graph G'=(V,E') with a directed path from i to j in p G' iff (i,j) in E. Edge Coloring Input: A graph G=(V,E). G=(V,E). Problem: What is the smallest set of colors needed to color the edges of E such that no two edges with the same color share a vertex in common? Hamiltonian Cycle y Input: A graph G=(V,E). G=(V,E). Problem: Find an ordering of the vertices such that each vertex is visited exactly once. Clique Input: A graph G=(V,E). G=(V,E). Problem: What is the largest S that is a subset of V such that for all x,y in S, (x,y) in E? Tools of the Trade: Core Material Algorithm Design Patterns dynamic programming, linear p g y p g g, programming, g, greedy algorithms, approximation algorithms, randomized algorithms, sweep algorithms, (parallel algorithms) amortized analysis, probabilistic analysis NPNP-completeness, NP-h d l NP-hardness Permutations Number Theory Probability Geometry Advanced Analysis Techniques Theoretical Computer Science principles MATH Logarithms Proofs Calculus Trigonometry Recurrences Linear Algebra Sets Combinations Polynomials P l i l Summations Asymptotic Growth of Functions Prerequisites 91.404 or 94.404. Standard graduate-level prerequisites for g graduatep q math background apply. MATH Combinations Polynomials Summations Permutations Logarithms Calculus Linear Algebra Sets Probability Proofs Recurrences Asymptotic Growth of Functions Geometry Trigonometry Number Theory Textbooks Required: q Introduction to Algorithms by T.H. Corman, C E Leiserson, R L Rivest T H Corman, C.E. Leiserson, R.L. McGraw/Hill and MIT Press 3rd Edition 2009 ISBN 0716710455 see course web site for book s web site containing errata book's see p. 20-22 for pseudocode conventions (different from 2nd edition) 20- Recommended: Garey & Johnson Ordered for UML bookstore Syllabus (current plan) y ( p ) math quiz th i midterm exam final exam Chapter Dependencies Math Review Appendices A, B, C, D Summations, Proof Techniques (e.g. Induction), Sets Induction) Sets, Graphs, Counting & Probability, Matrices Ch 1-13 Foundations Ch 29 Linear Programming Math: Linear Algebra (Appendix D) Ch 33 Computational Geometry Ch 15, 16, 17 15 16 Advanced Design & Analysis Techniques C Ch 22-24,25,26 , 5, 6 Graph Algorithms Ch 34 NP-Completeness Ch 35 Approximation Algorithms Math: Geometry (High School Level) Math: Number Theory Foundations: Automata 3rd Edition Ch 31 Number-Theoretic Algorithms RSA Ch 32 String Matching Important Dates Math Quiz: Tuesday, 2/1 Tuesday, 3/8 y, to be determined In l I class Closed book, no calculators In class Open book, open notes (5/10 is a possibility) Open book, open notes Midterm Exam: Final E am: Exam: Exam am: Grading Homework Midterm Final Exam Instructor's Discretion 30% 30% 35% 5% (open book, notes ) (open book, notes ) Homework HW# Assigned 1 T 1/25 Due T 2/1 Content 91.404 Review (for q ) ( quiz & Chapter 15 ...
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