CS2420: Lecture 4

1
CS2420 Lecture 4

• Vladimir Kulyukin
• Computer Science Department
• Utah State University

2
Outline

• Algorithm Analysis (Chapter 2)

3
Recall Big Question 1

• I have designed an algorithm. How fast does it
  run?

4
Asymptotic Analysis

• We would like to determine time/space performance
  of programs regardless of constant factors
  (differences in compilers and operating systems
  and differences in programming languages).
• In asymptotic analysis, N is an instance
  characteristic, typically the size of the input
  to a program P that implements some algorithm A.
• F(N) is the time/space performance of P on
  instances of size N.

5
Model of Computation

• We have a standard digital computer.
• One instruction is executed at a time.
• There is a small finite set of primitive
  instructions that can represented an arbitrary
  complex instruction in any programming language.
• Every primitive instruction can be executed in
  constant time.
• We have sufficient memory to execute complex
  programs.

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Computing The Arithmetic Sum

• int sum(int n)
• 1 int rslt 0
• 2 for(int i 1 i lt n i)
• 3 rslt i
• 4 return rslt

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Computing The Arithmetic Sum

• Assignment on line 1 is executed once and takes 1
  time unit.
• Assignment inside the for-loop is executed once
  and takes 1 time unit.
• Comparison inside the for-loop is executed n1
  times and takes (n1) time units.
• Increment inside the for-loop is executed n times
  and takes n time units.
• Plus-equal on line 3 is executed n times and
  takes 2n time units.
• Return on line 4 is executed once and takes 1
  time unit.

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Computing The Arithmetic Sum

• We need to compute the sum total of the following
  time units
• 1 time unit
• 1 time unit
• n1 time units
• n time units
• 2n time units
• 1 time unit
• The sum total 4 4n.

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Computing The Arithmetic Sum

• What is the complexity of computing the
  arithmetic sum?
• T(N) 4N 4.

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Computing The Arithmetic Sum

• int sum2(int n)
• return n (n 1) / 2

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Computing The Arithmetic Sum

• What is the complexity of sum?
• T(N) 4.

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Big-Oh Notation

• F(N) O(G(N)) IFF there exist positive constants
  Cgt0 And N0 gt 0 such that F(N) CG(N) for All N
  N0
• The Objective is to find the Least (Smallest)
  Upper Bound

13
Big-Oh Tight Bounds

• The objective is to find as tight an upper bound
  as possible.
• Loose bounds are easy but not that Useful
• F(N) 4N is O(2N) is correct, but too loose to
  be useful.
• F(N) 5N2 Is O(N4) is correct, but too loose to
  be useful.

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Worst Case vs. Average Case

• Worst-Case Analysis (Big-Oh) gives an upper
  performance bound over all inputs of size N. This
  is the most useful and most frequent measure of
  performance.
• Average-Case Analysis gives an average time bound
  over all inputs of size N. Average case analysis
  is often hard to do, because it is unclear what
  is average.

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Big-Oh of Polynomials

• Theorem f(n) a0 a1n a2n2 amnm, Then
  f(n) O(nm).
• Proof f(n) ?aini nm?ain(i-m) nm?ai.
• Take C ?ai. Take n0 1.
• f(n) O(nm).

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Big-Oh of Polynomials Example 1

• Claim F(N) 3N 2 O(N).
• Since m 1, then F(N) O(N1) O(N).

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Big-Oh of Polynomials Example 2

• Claim F(N) 5N2 7N 3 O(N2).
• Since m 2, then F(N) O(N2).