CEG 221 Lesson 5: Algorithm Development II

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CEG 221Lesson 5 Algorithm Development II

• Mr. David Lippa

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Overview

• Algorithm Development II
• Review of basic algorithm development
• Advanced algorithm development
• Optimization of algorithm
• Optimization of code
• Questions

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What is an Algorithm?

• An algorithm is a high-level set of clear,
  step-by-step actions taken in order to solve a
  problem, frequently expressed in English or
  pseudo code.
• Examples of Algorithms
• Computing the remaining angles and side in an SAS
  Triangle
• Computing an integral using rectangle
  approximation method (RAM) or the Trapezoidal Rule

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Example Triangulation with SAS

• If we return to the SAS triangle, theres nothing
  really much to be done to improve speed or
  efficiency, as the computation is very
  straightforward.

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Example Trapezoidal Rule

• You notice that we compute f(x) more times than
  is necessary for all the inner values
• Lets compute SUM2 2 ( f(0) f(0.25)
  f(0.5) f(0.75) f(1.0) ) f(0) f(1.0).
• AREA2 0.5 SUM2
• 4 trapezoids ? 8 computations of f
• 8 trapezoids ? 12 computations of f
• 1024 trapezoids ? 1028 computations

If the interval here is 0, 1, then we need to
compute SUM ( f(0) f(0.25) f(0.25)
f(0.5) f(0.5) f(0.75) f(0.75) f(1.0) ).
Then, AREA 0.5 SUM. 4 trapezoids ? 9
computations of f 8 trapezoids ? 15 computations
of f 1024 trapezoids ? 2047 computations Notice
a pattern?
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Trapezoidal Rule Improvements

• For a small number of trapezoids, this method is
  slightly more work.
• For, say 1024 trapezoids, this is significantly
  more efficient in terms of number of mathematical
  calculations.
• CONCLUSION Given that greater accuracy comes
  with more trapezoids, this optimization is
  sufficient, since this algorithm will rarely be
  used with few trapezoids.

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Optimizing Implemented Code

• There are other ways to speed up code
• Sacrifice memory for improved speed (ie. Always
  try to work from memory, not from disk)
• Avoid algorithms where the ratio of work required
  to number of elements processed is n, namely an
  n2 algorithm.
• Pass by reference or pointer where appropriate to
  prevent unnecessary memory copies of large
  structures
• Use algorithm analysis to try to find the cause
  of the lack of speed

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Algorithm Analysis

• Big-Oh notation how much work is required to
  process n inputs in terms of n
• Constants are less important for Big-Oh notation
• O(1), O(log2 n), O(n), O(n log2 n), O(n2), O(n3),
  O(2n), O(n!)
• Associate algorithms with each
• Matrix, Integration, SAS, factorial
• Formal definition

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Using Algorithm Analysis

• Analyze an algorithm by computing the number of
  operations performed per unit input
• Avoid O(n2) or worse algorithms
• Convert code to pseudo code if needed, to do a
  theoretical analysis

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Algorithm Analysis Example

• Matrix Multiplication
• Pseudo code To multiply an m x n matrix A and
  an n x p matrix B, dot product each row of A
  with each column of B.
• Results in m p dot products (see previous notes
  for pseudo code details)
• Dot product
• Pseudo code to dot product two vectors,
  multiply the first element of each, the second,
  the third, and so on and add them all together
• Results in n multiplication and addition
  operations (see previous notes for pseudo code
  details)
• RESULT Matrix multiplication is an O(m n p)
  operation. With square matrices, it is O(n3)

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Next Time

• Building Libraries
• Using Libraries
• Tradeoffs
• Questions

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Questions?