CSC 2300 Data Structures

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CSC 2300Data Structures Algorithms

• January 23, 2007
• Chapter 2. Algorithm Analysis

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Today

• Counting operations
• Algorithm analysis
• Order notation
• Running time calculations

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Example 1 Search

• Sequential search

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Example 2 Sort

• Insertion Sort

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Preliminary Analysis
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Algorithm Analysis Rules
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Computation Time
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Typical Growth Rates
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Order Notation
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Graph of O(f(n))
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Graph of T(f(n))
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Graph of O(f(n))
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Use of Limits
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Two Examples
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Order Notation Manipulation Rules
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Big-Oh Estimate for Summation

• Evaluate the summation using techniques we have
  learned. Then determine an O (or T) estimate from
  the resultant function.
• Place upper bounds on terms inside the summation
  to simplify the terms and eliminate the summation.

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Running Time Calculations

• We assume no particular units of time. Thus, we
  throw away leading constants.
• We also throw away low-order terms.
• Since Big-Oh is an upper bound, we must not
  underestimate the running time of the program.
• Our answer is a guarantee that the program will
  terminate within a certain time period. The
  program may stop earlier, but never later.

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Four Rules

• FOR loops
• Nested loops
• Consecutive statements
• If/Else

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FOR Loops

• The running time of a FOR loop is at most the
  running time of the statements inside the FOR
  loop (including tests) times the number of
  iterations.

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Nested Loops

• Analyze these inside out. The total running time
  of a statement inside a group of nested loops is
  the running time of the statement multiplied by
  the product of the sizes of all the loops.

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Consecutive Statements

• These just add, which means that the maximum is
  the one that counts.

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If/Else

• if condition
• S1
• else
• S2
• The running time of an if/else statement is never
  more than the running time of the test plus the
  larger of the running times of S1 and S2.

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Two More Examples