UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Fall, 2001 Lecture 1 (Part 1) Introduction/Overview Tuesday, 9/4/01

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Nature of the Course ä Core course: required for all CS graduate students ä Advanced algorithms ä Builds on undergraduate algorithms ä No programming required ä “Pencil-and-paper” exercises ä Lectures supplemented by: ä Programs ä Real-world examples

What’s It All About? ä Algorithm: ä steps for the computer to follow to solve a problem ä Some of our goals :(at an advanced level) ä recognize structure of some common problems ä understand important characteristics of algorithms to solve common problems ä select appropriate algorithm to solve a problem ä tailor existing algorithms ä create new algorithms

Some Algorithm Application Areas Computer Graphics Geographic Information Systems Robotics Bioinformatics Astrophysics Medical Imaging Telecommunications Design Apply Analyze

Some Typical Problems Fourier Transform Input: A sequence of n real or complex values h_i, 0 <= i <= n-1, sampled at uniform intervals from a function h. Problem: Compute the discrete Fourier transform H of h Nearest Neighbor Input: A set S of n points in d dimensions; a query point q. Input: A set S of n points in d dimensions; a query point q. Problem: Which point in S is closest to q? Problem: Which point in S is closest to q? SOURCE: Steve Skiena’s Algorithm Design Manual (for problem descriptions, see graphics gallery at ) (for problem descriptions, see graphics gallery at Shortest Path Input: Edge-weighted graph G, with start vertex and end vertex t Problem: Find the shortest path from to t in G Bin Packing Input: A set of n items with sizes d_1,...,d_n. A set of m bins with capacity c_1,...,c_m. Problem: How do you store the set of items using the fewest number of bins?

Some Typical Problems Transitive Closure Input: A directed graph G=(V,E). Problem: Construct a graph G'=(V,E') with edge (i,j) \in E' iff 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 G' iff (i,j) \in E. Convex Hull Input: A set S of n points in d- dimensional space. Problem: Find the smallest convex polygon containing all the points of S. Problem: Find the smallest convex polygon containing all the points of S. Eulerian Cycle Input: A graph G=(V,E). Problem: Find the shortest tour of G visiting each edge at least once. Edge Coloring Input: A graph 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?

Some Typical Problems Hamiltonian Cycle Input: A graph 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). Problem: What is the largest S \subset V such that for all x,y \in S, (x,y) \in E?

Tools of the Trade ä Algorithm Design Patterns ä dynamic programming, greedy, approximation algorithms ä Advanced Analysis Techniques ä asymptotic analysis ä Theoretical Computer Science principles ä NP-completeness, hardness ä Advanced Data Structures ä interval trees, binomial heaps Asymptotic Growth of Functions Summations Recurrences Sets Probability MATH Proofs Calculus Combinations Logarithms Number Theory Geometry Trigonometry Complex Numbers Permutations Linear Algebra Polynomials

Prerequisites ä and or ä Co-requisitive ä Standard graduate-level prerequisites for math background apply. Asymptotic Growth of Functions Summations Recurrences Sets Probability MATH Proofs Calculus Combinations Logarithms Number Theory Geometry Trigonometry Complex Numbers Permutations Linear Algebra Polynomials

Course Structure ä Advanced Algorithmic Paradigms ä Dynamic programming, greedy algorithms ä Approximation algorithms, parallel programming ä Graph Algorithms ä Shortest paths (single source & all pairs), Maximum flow ä Theory: NP-Completeness ä Complexity classes, reductions, hardness, completeness ä Advanced Agorithms for Special Applications ä Cryptography, String/Pattern Matching ä Computational Geometry ä Advanced Data Structures ä Interval trees, binomial heaps

Detailed Topics

Detailed Topics (continued)

Chapter Dependencies Ch 1-6 Math Review: Asymptotics, Recurrences, Summations, Sets, Graphs, Counting, Probability, Calculus, Proofs Techniques (e.g. Inductive) Logarithms Ch 7-10 Sorting Ch 11-14, 15 Data Structures Ch 16, 17, 18 Advanced Design & Analysis Techniques Ch Advanced Data Structures Ch 23-25,26,27 Graph Algorithms Ch 28, 29 Parallel Comparison-Based Sorting Networks, Arithmetic Circuits Ch 30 Parallel Comparison-Based Sorting Networks Ch 31 Matrix Operations Math: Linear Algebra Ch 35 Computational Geometry Math: Geometry (High School Level) Ch 36 NP-Completeness Ch 37 Approximation Algorithms

Chapter Dependencies (continued) Math: Polynomials, Convolution, Complex Numbers, Trig Ch 32 Polynomials, FFT Ch 33 Number-Theoretic Algorithms RSA Math: Number Theory Ch 34 String Matching Automata

Textbook - - Required: ä Introduction to Algorithms ä by T.H. Corman, C.E. Leiserson, R.L. Rivest ä McGraw-Hill + MIT Press ä 1993 ä ISBN ä see course web site (MiscDocuments) for errata Available in UML bookstore

Syllabus (current plan)

Important Dates ä Midterm Exam:Tuesday, 10/23 ä Final Exam:TBA

Grading ä ä Homework35% ä ä Midterm 30% (open book, notes ) ä ä Final Exam35% (open book, notes )

Homework 1 T, 9/4 Part I: T 9/ review & Chapters Part II: T 9/ review & Chapters Part II: T 9/ review & Chapters HW# Assigned Due Content