UMass%20Lowell%20Computer%20Science%2091.503%20Analysis%20of%20Algorithms%20Prof.%20Karen%20Daniels%20Fall,%202004

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UMass Lowell Computer Science 91.503 Analysis
of Algorithms Prof. Karen Daniels Fall, 2004

• Lecture 1 (Part 1)
• Introduction/Overview
• Wednesday, 9/8/04

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Web Page
Web Page
http//www.cs.uml.edu/kdaniels/courses/ALG_503.h
tml
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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 s and end vertex t Problem Find
the shortest path from s 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

• 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.

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?
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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 that is a subset of V such that for
all x,y in S, (x,y) in E?
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Tools of the Trade Core Material

• Algorithm Design Patterns
• dynamic programming, linear programming, greedy
  algorithms, approximation algorithms, randomized
  algorithms, sweep algorithms, (parallel
  algorithms)
• Advanced Analysis Techniques
• amortized analysis, probabilistic analysis
• Theoretical Computer Science principles
• NP-completeness, NP-hardness

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Prerequisites

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

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Textbook

• Required
• Introduction to Algorithms
• by T.H. Corman, C.E. Leiserson, R.L. Rivest
• McGraw-Hill
• 2001
• ISBN 0-07-013151-1
• see course web site (MiscDocuments) for errata

New Edition
Ordered for UML bookstore
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Syllabus (current plan)
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Chapter Dependencies
Math Review Appendices A, B, C Summations, Proof
Techniques (e.g. Induction), Sets, Graphs,
Counting Probability Ch 1-13 Foundations
Ch 35 Approximation Algorithms
Ch 34 NP-Completeness
Ch 29 Linear Programming
Math Linear Algebra
New Edition
Ch 33 Computational Geometry
Math Geometry (High School Level)
Ch 31 Number-Theoretic Algorithms RSA
Math Number Theory
Ch 32 String Matching
Automata
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Important Dates

• Midterm Exam Wednesday, 10/20
• Final Exam to be determined

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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 W 9/8 W 9/15 91.404 review
  Chapter 15