Loops, Summations, Order Reduction

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Loops, Summations, Order Reduction

• Chapter 2 Highlights

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Constant times Linear

• for x 1 to n
• operation 1
• operation 2
• operation 3

• for x 1 to n
• constant time operation
• Note Constants are not a factor in evaluating
  algorithms. Why?

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Linear-time Loop

• for x 1 to n
• constant-time operation
• Note Constant-time mean independent of the input
  size.

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Linear-time Loop

• for x 0 to n-1
• constant-time operation
• Note Dont let starting at zero throw you off.
• Brackets not necessary for one statement.

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2-Nested Loops ? Quadratic

• for x 0 to n-1
• for y 0 to n-1
• constant-time operation
• The outer loop restarts the inner loop

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3-Nested Loops ? Cubic

• for x 0 to n-1
• for y 0 to n-1
• for z 0 to n-1
• constant-time operationf(n) n3
• The number of nested loops determines the
  exponent

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4-Nested Loops ? n4

• for x 0 to n-1
• for y 0 to n-1
• for z 0 to n-1
• for w 0 to n-1
• constant-time operationf(n) n4

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Add independent loops

• for x 0 to n-1
• constant-time op
• for y 0 to n-1
• for z 0 to n-1
• constant-time op
• for w 0 to n-1
• constant-time op

• f(n) n n2 n
• n2 2n

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Non-trivial loops

• for x 1 to n
• for y 1 to x
• constant-time operation
• Note x is controlling the inner loop.

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Equivalent Loops

• for x 1 to n
• for y 1 to x
• constant-time op

• for x 1 to n
• for y x to n
• constant-time op

• Note
• These two loops behave differently, but
• They perform the same number of basic operations.

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Order reduction

• Given the following function
• Constants dont matter

• Only the leading exponent matters
• Thus

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Order reduction

• Given the following function
• Constants dont matter

• Only the leading exponent matters

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Example
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• for z 1 to n
• for y 1 to z
• for x 1 to y
• constant-op

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Example
0

• for z 1 to n
• for y 1 to z
• for x 1 to y
• constant-op

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Example
0

• for z 1 to n
• for y 1 to z
• for x 1 to y
• constant-op

• for z 1 to n
• for y 1 to z
• y operations

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Example
0

• for z 1 to n
• for y 1 to z
• y operations

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Example
0

• for z 1 to n
• for y 1 to z
• y operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Example
0

• for z 1 to n
• z(z1)/2 operations

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Proof By Induction
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• Prove the following

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Another Example

• if (n is odd)
• for x 1 to n
• for y x to n
• 1 operation
• else
• for x 1 to n/2
• for y 1 to n/2
• 1 operation

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Another Example
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• if (n is odd)
• for x 1 to n
• for y x to n
• 1 operation
• else
• for x 1 to n/2
• for y 1 to n/2
• for z 1 to n/2
• 1 operation

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Another Example
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Best Case

• if (n is odd)
• for x 1 to n
• for y x to n
• 1 operation
• else
• for x 1 to n/2
• for y 1 to n/2
• for z 1 to n/2
• 1 operation

Average Case
Worst Case
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Algorithm Analysis Overview

• Counting loops leads to summations
• Summations lead to polynomials
• N (or n) used to to characterize input size
• Constants are not a factor
• Only the leading exponent matters
• Depending on input, algorithm can have different
  running times.