CS 312: Algorithm Analysis

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CS 312 Algorithm Analysis

• Lecture 12 Fast Fourier Transform

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Announcements

• Project 1
• Report Due Today
• Improvement due Wednesday
• Project 2
• Guidelines available early

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Objectives

• Lecture 12 Fast Fourier Transform
• To recall how to multiply two polynomials
• To recognize polynomial multiplication as an
  instance of convolution
• To review n-th complex roots of unity
• To discover how those roots make a divide and
  conquer algorithm possible
• To get an intuition for how divide and conquer
  makes the FFT fast.

• Lecture 11 Median Matrix Multiplication
• Design a (expected) linear-time algorithm to
  select the k-th smallest element in a list
• Find the Median
• Apply divide and conquer to matrix multiplication
• Analyze using the Master Theorem

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Polynomial Multiplication

• Product of two degree d polynomials is a
  polynomial of degree 2d.
• Example

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Polynomial Multiplication

• More generally
• Then
• Where
• For igtd, use ai and bi 0
• An instance of a process called Convolution

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Questions

• Is it Correct?
• How long does it take?
• Can we do better?

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More Questions

• Why is polynomial multiplication (aka
  Convolution) interesting?
• Where is it used?

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1-D Convolution

Smoothing (noise removal)

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2-D Convolution

• A 2-D signal (an Image) is convolved with a
  second Image (the filter, or convolution
  Kernel).
• f(x,y) g(x,y)
  h(x,y)f(x,y)g(x,y)

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Value Representation

• Fact A degree-d polynomial is uniquely
  characterized by its values at any d 1 distinct
  points.
• E.g., any two points determine a line

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Mapping Between The Representations
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A New Algorithm
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Questions

• Is it Correct?
• How long does it take?
• Can we do better?

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Evaluation as Divide and Conquer
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Choose Clever Points
If we can choose wisely, we can have a DC
algorithm whose running time is captured by our
favorite recurrence relation
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N-th Complex Roots of Unity
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Review The Complex Plane
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Multiplication of Complex Numbers
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N-th Complex Roots of Unity
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For Divide and Conquer
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The FFT
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Interpolation
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Questions

• Is it Correct?
• How long does it take?
• Can we do better?

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• We got this far on 2/5/2007
• That is OK because
• You have intuition into why the FFT works
• You understand that it is a Divide and Conquer
  algorithm
• You know how to analyze this algorithm
• Other perspectives on the FFT are provided in the
  following slides.

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Matrix Reformulation
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Assignment

• HW 10 2.11, 2.8
• Read Chapter 3 on Graph Connectivity Algorithms