Intro to Computer Algorithms Lecture 9

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Intro to Computer Algorithms Lecture 9

• Phillip G. Bradford
• Computer Science
• University of Alabama

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Announcements

• Advisory Boards Industrial Talk Series
• http//www.cs.ua.edu/9IndustrialSeries.shtm
• Coop Interview Days Oct 14th and 15th
• Signup begins on 22nd of September
• At least 40 firms looking for coop students!

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ACM announces Movie Night

• When and Where Thursday, September 25, 2003
• 630 pm in room 119 EE
• EVERYONE IS INVITED!!!!
• The Movie will be Office Space
• We will have subway platters
• No Admission fee
• If you have any questions, go to our newly
  updated website and post them on the forum
  http//www.studentacm.org

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Outline

• Integer Multiplication
• Fast Integer Multiplication
• Matrix Multiplication
• Start Strassens Algorithm

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Integer Multiplication

• Cost of multiplying two n-digit integers
• Basic Grade-School Algorithm
• 23 2101 3100 a1101 a0100
• 14 1101 4100 b1101 b0100
• (a1 a0)(b1 b0)
• Or (a1b1 a1b0) (a0b1 a0b0 )
• Consider the product of
• an10n a0100
• bn10n b0100
• What does this cost?
• O(n2) digit multiplication operations

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Integer Multiplication

• Can we do any better?
• 23 2101 3100
• 14 1101 4100
• 2314 (21)102 (3124)101 (34)100
• Four multiplications
• Powers of 10 can be done via shifting

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Integer Multiplication

• Note (31)(24) (23)(14) (21)-(34)
• Because (a1a0)(b1b0) a1b1a1b0a0b1a0b0
• Focus on a1b0a0b1 both times 101
• Subtract off a1b1 a0b0
• We must compute a1b1 a0b0 anyway!

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Integer Multiplication

• Can we generalize this?
• Let a a1101 a0100
• And b b1101 b0100
• C ab c2102 c1101 c0100
• Where c2 a1b1 and c0 a0b0
• Therefore c1 (a1a0)(b1b0) (c2 c0)

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Fast Integer Multiplication

• Divide-and-Conquer generalization!
• Conceptual Example
• Let A1 a101010 a5105
• And A0 a4104 a0100
• So, Letting A1 ? A1/105 and B1 ? B1/105
• Gives A A1105 A0
• Likewise for B B1105 B0
• Thus, AB (A1105 A0)(B1105 B0)

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Fast Integer Multiplication

• In general, giving
• AB A1B110n (A1B0A0B1)10n/2 A0B0
• For the time being, say n is a power of 2
• Do this with 3 multiplications instead of 4
• AB C210n C110n/2 C0
• Where C2 A1B1 and C0 A0B0
• And C1 (A1A0)(B1B0) (C2 C0)

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Fast Integer Multiplication

• Convince me this works!
• Recurrence
• Fundamental operation is multiplication
• What does this cost?
• T(n) 3T(n/2) for n gt 1
• T(1) 1.
• Can we use the Master Theorem?

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Fast Integer Multiplication

• Solve T(n) by backward substitution
• Let n 2k
• Then T(2k) 3iT(2k-i)
• Thus, T(n) 3k for n 2k.
• Since, k log2n
• It must be that T(n) 3log2n
• Using alogb c clogb a
• T(n) nlog23 O(n1.585)

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Fast Integer Multiplication

• How can we prove
• alogb c clogb a ?
• Brassard Bratleys simulations

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Matrix Multiplication

• Grade School Algorithm
• M and N, say n-by-n matrices
• Output P
• For i?1 to n do
• For j?1 to n do
• Pi,j ? 0
• For k? 1 to n do
• Pi,j ? Pi,j Mi,kNk,j

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Matrix Multiplication

• Cost?
• O(n3)
• Can we do better?

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Strassens Algorithm

• Cost O(n2.807)