UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Fall, 2002 Review Lecture Tuesday, 12/10/02

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 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

Course Structure ä Advanced Algorithmic Paradigms ä Dynamic programming ä Greedy algorithms ä Simplex optimization ä Approximation algorithms & schemes ä Randomized algorithms ä Sweep-line algorithms ä Graph Algorithms ä Shortest paths (single source & all pairs) ä Maximum flow ä Theory: NP-Completeness ä Complexity classes, reductions, hardness, completeness ä Advanced Algorithms for Special Applications ä Cryptography ä String/Pattern Matching ä Computational Geometry

Chapter Dependencies Math Review: Asymptotics, Recurrences, Summations, Sets, Graphs, Counting, Probability, Calculus, Proofs Techniques (e.g. Inductive) Logarithms Ch 6-9 Sorting Ch Data Structures Math: Linear Algebra Ch 33 Computational Geometry Math: Geometry (High School Level) Ch Advanced Design & Analysis Techniques Ch Graph Algorithms Ch 34 NP-Completeness Ch 35 Approximation Algorithms Ch 31 Number-Theoretic Algorithms: RSA Math: Number Theory Ch 32 String Matching Automata Ch 29 Linear Programming

Important Dates ä Final Exam: ä Tuesday, 17 December, Ball 328, 5:30 pm

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