UMass Lowell Computer Science 91'503 Analysis of Algorithms Prof' Karen Daniels Fall, 2001

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UMass Lowell Computer Science 91.503 Analysis
of Algorithms Prof. Karen Daniels Fall, 2001

• Lecture 1 (Part 1)
• Introduction/Overview
• Tuesday, 9/4/01

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Web Page
Web Page
http//www.cs.uml.edu/kdaniels/courses/ALG_503_F
01.html
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Nature of the Course

• Core course required for all CS graduate
  students
• Advanced algorithms
• Builds on undergraduate algorithms 91.404
• No programming required
• Pencil-and-paper exercises
• Lectures supplemented by
• Programs
• Real-world examples

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Whats It All About?

• Algorithm
• steps for the computer to follow to solve a
  problem
• Some of our goals(at an advanced level)
• 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

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Some Algorithm Application Areas
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Some Typical Problems
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?

• Fourier Transform
• Input A sequence of n real or complex values
  h_i, 0 lt i lt 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.
• ProblemWhich point in S is closest to q?

SOURCE Steve Skienas Algorithm Design Manual
(for problem descriptions, see graphics gallery
at http//www.cs.sunysb.edu/algorith)
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Some Typical Problems
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?

• 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.

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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?
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Tools of the Trade

• Algorithm Design Patterns
• dynamic programming, greedy, approximation
  algorithms
• Advanced Analysis Techniques
• asymptotic analysis
• Theoretical Computer Science principles
• NP-completeness, hardness
• Advanced Data Structures
• interval trees, binomial heaps

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Prerequisites

• 91.500 and 91.404 or 91.583.
• Co-requisitive 91.502.
• Standard graduate-level prerequisites for math
  background apply.

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Course Structure

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

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Detailed Topics
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Detailed Topics (continued)
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Detailed Topics (continued)
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Detailed Topics (continued)
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Chapter Dependencies
Ch 28, 29 Parallel Comparison-Based Sorting
Networks, Arithmetic Circuits
Ch 1-6 Math Review Asymptotics, Recurrences,
Summations, Sets, Graphs, Counting,
Probability, Calculus, Proofs Techniques (e.g.
Inductive) Logarithms
Ch 30 Parallel Comparison-Based Sorting Networks
Ch 7-10 Sorting
Ch 16, 17, 18 Advanced Design Analysis
Techniques
Ch 23-25,26,27 Graph Algorithms
Ch 36 NP-Completeness
Ch 37 Approximation Algorithms
Ch 11-14, 15 Data Structures
Ch 19-22 Advanced Data Structures
Ch 31 Matrix Operations
Math Linear Algebra
Math Geometry (High School Level)
Ch 35 Computational Geometry
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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
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Textbook

• Required
• Introduction to Algorithms
• by T.H. Corman, C.E. Leiserson, R.L. Rivest
• McGraw-Hill MIT Press
• 1993
• ISBN 0-07-013143-0
• see course web site (MiscDocuments) for errata

Available in UML bookstore
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Syllabus (current plan)
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Important Dates

• Midterm Exam Tuesday, 10/23
• Final Exam TBA

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Grading

• Homework 35
• Midterm 30 (open book, notes )
• Final Exam 35 (open book, notes )

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Homework
HW Assigned Due Content

• 1 T, 9/4 Part I T 9/11 91.404 review
  Chapters 16-18
• Part II T 9/18 91.404 review
  Chapters 16-18