Unit #2: Complexity Theory and Asymptotic Analysis CPSC 221: - - PowerPoint PPT Presentation

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Unit #2: Complexity Theory and Asymptotic Analysis CPSC 221: - - PowerPoint PPT Presentation

Jun 11, 2023 167 likes •308 views

Unit #2: Complexity Theory and Asymptotic Analysis CPSC 221: Algorithms and Data Structures Lars Kotthoff 1 larsko@cs.ubc.ca 1 With material from Will Evans, Steve Wolfman, Alan Hu, Ed Knorr, and Kim Voll. Types of analysis if random() == 0

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Unit #2: Complexity Theory and Asymptotic Analysis

CPSC 221: Algorithms and Data Structures

Lars Kotthoff1 larsko@cs.ubc.ca

1With material from Will Evans, Steve Wolfman, Alan Hu, Ed Knorr, and

Kim Voll.

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Types of analysis

if random() == 0 return if random() == 1 for i = 1 to n do for j = 1 to n do k = 1 return else for i = 1 to n do k = 1 return

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Types of analysis

if random() == 0 return if random() == 1 for i = 1 to n do for j = 1 to n do k = 1 return else for i = 1 to n do k = 1 return Best case: T(n) = 1

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Types of analysis

if random() == 0 return if random() == 1 for i = 1 to n do for j = 1 to n do k = 1 return else for i = 1 to n do k = 1 return Best case: T(n) = 1 Worst case: T(n) = n2

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Types of analysis

if random() == 0 return if random() == 1 for i = 1 to n do for j = 1 to n do k = 1 return else for i = 1 to n do k = 1 return Best case: T(n) = 1 Worst case: T(n) = n2 Average case: T(n) = n

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Types of analysis

Best case: T(n) = 1 Worst case: T(n) = n2 Average case: T(n) = n . . . . . .

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Types of analysis

Best case: T(n) = 1 Worst case: T(n) = n2 Average case: T(n) = n . . n . T(n) . n0 . Ω . O . Θ . . . .

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Types of analysis

Best case: T(n) = 1 Worst case: T(n) = n2 Average case: T(n) = n . . n . T(n) . n0 . Ω . O . Θ . . n . T(n) . n0 . Ω . O . Θ . .

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Types of analysis

Best case: T(n) = 1 Worst case: T(n) = n2 Average case: T(n) = n . . n . T(n) . n0 . Ω . O . Θ . . n . T(n) . n0 . Ω . O . Θ . . n . T(n) . n0 . Ω . O . Θ

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“Tight” bounds

▷ informally: want to have “good” bounds ▷ no better reasonable bound which is asymptotically different ▷ rigid definition: Θ

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Runtime example #3

i = 1 while i < n do for j = 1 to i do sum = sum + 1 i += i

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Runtime example #4

int max(A, n) if (n == 1) return A[0] return larger of A[n-1] and max(A, n-1) Recursion almost always yields a recurrence relation: T(1) ≤ b T(n) ≤ c + T(n − 1) if n > 1 Solving recurrence: T(n) ≤ c + c + T(n − 2) (substitution) ≤ c + c + c + T(n − 3) (substitution) ≤ kc + T(n − k) (extrapolating k > 0) = (n − 1)c + T(1) (for k = n − 1) ≤ (n − 1)c + b T(n) ∈

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Runtime example #5: Mergesort

Mergesort algorithm: Split list in half, sort first half, sort second half, merge together Recurrence relation: T(1) ≤ b T(n) ≤ 2T(n/2) + cn if n > 1 Solving recurrence: T(n) ≤ 2T(n/2) + cn ≤ 2(2T(n/4) + cn/2) + cn (substitution) = 4T(n/4) + 2cn ≤ 4(2T(n/8) + cn/4) + 2cn (substitution) = 8T(n/8) + 3cn ≤ 2kT(n/2k) + kcn (extrapolating k > 0) = nT(1) + cn lg n (for 2k = n) T(n) ∈