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UMass Lowell Computer Science 91.404 Analysis of Algorithms Prof. Karen Daniels Spring, 2004 Lecture 1 Introduction/Overview Text: Chapters 1, 2 Wed. 1/28/04 Web PageWeb Page http://www.cs.uml.edu/~kdaniels/courses/ALG_404.html Nature of the Course Core course: required for all CS majors Advanced undergraduate level Graduate students take separate course (91.503) No programming required “Pencil-and-paper” exercises Lectures supplemented by: Programs Real-world examples What’s It All About? Algorithm: steps for the computer to follow to solve a problem well-defined computational procedure that transforms input into output Some of our goals: recognize structure of some common problems understand important characteristics of algorithms to solve common problems select appropriate algorithm to solve a problem tailor existing algorithms create new algorithms Some Algorithm Application Areas Robotics Bioinformatics Geographic Information Systems Design Analyze Telecommunications Apply Computer Graphics Medical Imaging Astrophysics Some Typical Problems Sorting Input: Set of items Problem: Arrange items “in order” Median finding Input: Set of numbers or keys Problem: Find item smaller than half of items and bigger than half of items Minimum Spanning Tree Input: Graph G = (V,E) with weighted edges Problem: Find subset of E of G of minimum weight which forms a tree on V Shortest Path Input: Edge-weighted graph G , with start vertex and end vertex t Problem: Find the shortest path from to t in G SOURCE: Steve Skiena’s Algorithm Design Manual (for problem descriptions, see graphics gallery at http://www.cs.sunysb.edu/~algorith) Tools of the Trade Algorithm Design Patterns such as: divide-and-conquer Data Structures such as: trees, linked lists, hash tables, graphs Algorithm Analysis Techniques such as: asymptotic analysis probabilistic analysis Summations Proofs Sets Combinations Growth of Functions Permutations MATH Probability Recurrences Logarithms Tools of the Trade: (continued) Algorithm Animation http://www.cs.brockport.edu/cs/java/apps/sorters/insertsortaniminp.html What are we measuring? Some Analysis Criteria: Scope “Dimension” Upper? Lower? Both? Type of Input Time Complexity? Space Complexity? Type of Bound The problem itself? A particular algorithm that solves the problem? Best-Case? Average-Case? Worst-Case? Type of Implementation Choice of Data Structure Prerequisites Computing I (91.101) Computing II (91.102) Discrete Math I & II (92.321, 92.322) Statistics for Scientists and Engineers (92.386) Calculus I-III (92.131-231) Summations Proofs Sets Combinations Growth of Functions Permutations MATH Probability Recurrences Logarithms Course Structure: 4 Parts Foundations Sorting Heapsort, Priority Queues, Quicksort, Sorting in Linear Time Data Structures Analyzing & Designing Algorithms, Growth of Functions, Recurrences, Probability & Randomized Algorithms Stacks and Queues, Linked Lists, Introduction to Trees, Hash Tables, Binary Search Trees, Balancing Trees: RedBlack Trees Graph Algorithms DFS, BFS, Topological Sort, MST, Shortest paths Textbook Required: Introduction to Algorithms by T.H. Corman, C.E. Leiserson, R.L. Rivest McGraw-Hill New Edition 2001 ISBN 0-07-013151-1 see course web site (MiscDocuments) for errata (1st edition) Ordered for UML bookstore Syllabus (current plan) Lecture Date Topic Reading Foundations Chapters 1-5 W 1/28 Introduction/Overview Chapter 1 F 1/30, M 2/2, W 2/4 Analyzing & Designing Algorithms Chapter 2 F 2/6, M 2/9, W 2/11 Growth of Functions Chapter 3 F 2/13, W 2/18, Th 2/19 Recurrences Chapter 4 F 2/20, M 2/23, W 2/25 Probability & Randomized Algorithms Chapter 5 Sorting Chapters 6-9 F 2/27, M 3/1 Heapsort/ Priority Queues Chapter 6 W 3/3, F 3/5 Quicksort Chapter 7 M 3/8, W 3/10 Review Chapters 1-5 F 3/12 Midterm Exam Chapters 1-5 M 3/22 Quicksort W 3/24, F 3/26, M 3/29 Sorting in Linear Time Chapter 8 Data Structures Chapters 10-13 W 3/31, F 4/1, M 4/5 Stacks, Queues, Linked Lists, Trees Chapter 10 W 4/7, F 4/9 Hash Tables Chapter 11 M 4/12, W 4/16, F 4/18 Binary Search Trees Chapter 12 W 4/21, F 4/23 Balancing Trees: Red-Black Trees Chapter 13 Graph Algorithms Chapters 22-24 M 4/26, W 4/28, M 5/3 Elementary Graph Algorithms Chapter 22 W 5/5, F 5/7 Minimum Spanning Trees Chapter 23 M 5/10 Shortest Paths Chapter 24 W 5/12, F 5/14 Review 1-13, 15-16, 22-24 To Be Determined Final Exam 1-13, 15-16, 22-24 CS Theory Math Review Sheet The Most Relevant Parts... p. 1 O, Q, W definitions Series Combinations p. 2 Recurrences & Master Method p. 3 Probability Factorial Logs Stirling’s approx p. 4 Matrices p. 5 Graph Theory p. 6 Calculus Product, Quotient rules Integration, Differentiation Logs p. 8 Finite Calculus p. 9 Series Math fact sheet (courtesy of Prof. Costello) is on our web site. Important Dates Midterm Exam (Chapters 1-5): Final Exam: Friday, 3/12 TBA Grading Homework Midterm (Chapters 1-5) Final Exam 35% 30% 35% (open book, notes ) (open book, notes )
