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Efficiency Announcements Measuring Efficiency Recursive - PowerPoint PPT Presentation

Sep 05, 2023 •339 likes •1.56k views

Efficiency Announcements Measuring Efficiency Recursive Computation of the Fibonacci Sequence Our first example of tree recursion: 4 Recursive Computation of the Fibonacci Sequence def fib (n): Our first example of tree recursion: if n == 0 :

  1. Memoization Idea: Remember the results that have been computed before def memo(f): Keys are arguments that map to return values cache = {} def memoized(n): if n not in cache: cache[n] = f(n) return cache[n] return memoized 6

  2. Memoization Idea: Remember the results that have been computed before def memo(f): Keys are arguments that map to return values cache = {} def memoized(n): if n not in cache: cache[n] = f(n) return cache[n] return memoized Same behavior as f, 
 if f is a pure function 6

  3. Memoization Idea: Remember the results that have been computed before def memo(f): Keys are arguments that map to return values cache = {} def memoized(n): if n not in cache: cache[n] = f(n) return cache[n] return memoized Same behavior as f, 
 if f is a pure function (Demo) 6

  4. Memoized Tree Recursion fib(5) fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  5. Memoized Tree Recursion Call to fib fib(5) fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  6. Memoized Tree Recursion Call to fib fib(5) Found in cache fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  7. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  8. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  9. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  10. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  11. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  12. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  13. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  14. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  15. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  16. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  17. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  18. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  19. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  20. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  21. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  22. Memoized Tree Recursion Call to fib fib(5) Found in cache Skipped fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 7

  23. Exponentiation

  24. Exponentiation 9

  25. Exponentiation Goal: one more multiplication lets us double the problem size 9

  26. Exponentiation Goal: one more multiplication lets us double the problem size def exp(b, n): if n == 0: return 1 else: return b * exp(b, n-1) 9

  27. Exponentiation Goal: one more multiplication lets us double the problem size def exp(b, n): � if n == 0: 1 if n = 0 b n = return 1 b · b n − 1 otherwise else: return b * exp(b, n-1) 9

  28. Exponentiation Goal: one more multiplication lets us double the problem size def exp(b, n): � if n == 0: 1 if n = 0 b n = return 1 b · b n − 1 otherwise else: return b * exp(b, n-1)  1 if n = 0  b n =  1 2 n ) 2 ( b if n is even  b · b n − 1 if n is odd  9

  29. Exponentiation Goal: one more multiplication lets us double the problem size def exp(b, n): � if n == 0: 1 if n = 0 b n = return 1 b · b n − 1 otherwise else: return b * exp(b, n-1) def exp_fast(b, n): if n == 0: return 1 elif n % 2 == 0:  1 if n = 0 return square(exp_fast(b, n//2))  b n =  1 else: 2 n ) 2 ( b if n is even return b * exp_fast(b, n-1)  b · b n − 1 if n is odd  def square(x): return x * x 9

  30. Exponentiation Goal: one more multiplication lets us double the problem size def exp(b, n): � if n == 0: 1 if n = 0 b n = return 1 b · b n − 1 otherwise else: return b * exp(b, n-1) def exp_fast(b, n): if n == 0: return 1 elif n % 2 == 0:  1 if n = 0 return square(exp_fast(b, n//2))  b n =  1 else: 2 n ) 2 ( b if n is even return b * exp_fast(b, n-1)  b · b n − 1 if n is odd  def square(x): return x * x (Demo) 9

  31. Exponentiation Goal: one more multiplication lets us double the problem size def exp(b, n): Linear time: if n == 0: • Doubling the input 
 return 1 doubles the time else: • 1024x the input takes 
 return b * exp(b, n-1) 1024x as much time def exp_fast(b, n): Logarithmic time: if n == 0: • Doubling the input 
 return 1 increases the time 
 elif n % 2 == 0: by a constant C return square(exp_fast(b, n//2)) • 1024x the input 
 else: increases the time 
 return b * exp_fast(b, n-1) by only 10 times C def square(x): return x * x 10

  32. Orders of Growth

  33. Quadratic Time Functions that process all pairs of values in a sequence of length n take quadratic time 12

  34. Quadratic Time Functions that process all pairs of values in a sequence of length n take quadratic time def overlap(a, b): count = 0 for item in a: for other in b: if item == other: count += 1 return count overlap([3, 5, 7, 6], [4, 5, 6, 5]) 12

  35. Quadratic Time Functions that process all pairs of values in a sequence of length n take quadratic time 3 5 7 6 def overlap(a, b): count = 0 0 0 0 0 4 for item in a: for other in b: 0 1 0 0 5 if item == other: count += 1 0 0 0 1 return count 6 overlap([3, 5, 7, 6], [4, 5, 6, 5]) 0 1 0 0 5 12

  36. Quadratic Time Functions that process all pairs of values in a sequence of length n take quadratic time 3 5 7 6 def overlap(a, b): count = 0 0 0 0 0 4 for item in a: for other in b: 0 1 0 0 5 if item == other: count += 1 0 0 0 1 return count 6 overlap([3, 5, 7, 6], [4, 5, 6, 5]) 0 1 0 0 5 12

  37. Quadratic Time Functions that process all pairs of values in a sequence of length n take quadratic time 3 5 7 6 def overlap(a, b): count = 0 0 0 0 0 4 for item in a: for other in b: 0 1 0 0 5 if item == other: count += 1 0 0 0 1 return count 6 overlap([3, 5, 7, 6], [4, 5, 6, 5]) 0 1 0 0 5 (Demo) 12

  38. Exponential Time def fib (n): Tree-recursive functions can take exponential time if n == 0 : return 0 elif n == 1 : return 1 else : return fib(n- 2 ) + fib(n- 1 ) 13 13 http://en.wikipedia.org/wiki/File:Fibonacci.jpg

  39. Exponential Time def fib (n): Tree-recursive functions can take exponential time if n == 0 : return 0 elif n == 1 : return 1 else : return fib(n- 2 ) + fib(n- 1 ) fib(2) fib(0) fib(1) 0 1 13 13 http://en.wikipedia.org/wiki/File:Fibonacci.jpg

  40. Exponential Time def fib (n): Tree-recursive functions can take exponential time if n == 0 : return 0 elif n == 1 : return 1 else : return fib(n- 2 ) + fib(n- 1 ) fib(3) fib(1) fib(2) fib(0) fib(1) 1 0 1 13 13 http://en.wikipedia.org/wiki/File:Fibonacci.jpg

  41. Exponential Time def fib (n): Tree-recursive functions can take exponential time if n == 0 : return 0 elif n == 1 : return 1 else : return fib(n- 2 ) + fib(n- 1 ) fib(4) fib(2) fib(3) fib(0) fib(1) fib(1) fib(2) fib(0) fib(1) 0 1 1 0 1 13 13 http://en.wikipedia.org/wiki/File:Fibonacci.jpg

  42. Exponential Time def fib (n): Tree-recursive functions can take exponential time if n == 0 : return 0 elif n == 1 : fib(5) return 1 else : return fib(n- 2 ) + fib(n- 1 ) fib(3) fib(4) fib(1) fib(2) fib(2) fib(3) fib(0) fib(1) 1 fib(0) fib(1) fib(1) fib(2) 0 1 fib(0) fib(1) 0 1 1 0 1 13 13 http://en.wikipedia.org/wiki/File:Fibonacci.jpg

  43. Common Orders of Growth Exponential growth . E.g., recursive fib Quadratic growth . E.g., overlap Linear growth . E.g., slow exp Logarithmic growth . E.g., exp_fast Constant growth . Increasing n doesn't affect time 14

  44. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  45. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  46. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">ACv3icbVHbSsMwGE7rac5T1UtvgkPZGIy2CHqhMPFCLye4A6zbSLNshqVpSVJhlL2kF4JvY7p17viTwJfvkMfP2JUKtv+Ncyd3b39g9xh/uj45PTMOr9oyDAWmNRxyELR8pEkjHJSV1Qx0oEQYHPSNMfvaR684sISUP+ocYR6QRoyOmAYqQ01bN+EPRwP1TQ7ya87Ezg7RMsLjhemkPoebORn6tF7S913ZUE7olWIb/FpfrlVPaFp4JK+Esu71GNc7LUzLmVTjy0emhAt7VsGu2NOCm8DJQAFkVetZ314/xHFAuMIMSdl27Eh1EiQUxYxM8l4sSYTwCA1JW0OAiI7ybT/E3ijmT4chEJPruCUXU4kKJByHPjaGSD1Kde1lNymtWM1eOgklEexIhzPDhrEDKoQp8J+1QrNhYA4QF1XeF+BMJhJX+8rxugrP+5E3QcCuOXHe7wrV16wdOXAFrkEROAeVMEbqIE6wMaj4Rsjg5nP5tDkZjSzmkaWuQrZY7/ABqfyBM=</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  47. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">ACv3icbVHbSsMwGE7rac5T1UtvgkPZGIy2CHqhMPFCLye4A6zbSLNshqVpSVJhlL2kF4JvY7p17viTwJfvkMfP2JUKtv+Ncyd3b39g9xh/uj45PTMOr9oyDAWmNRxyELR8pEkjHJSV1Qx0oEQYHPSNMfvaR684sISUP+ocYR6QRoyOmAYqQ01bN+EPRwP1TQ7ya87Ezg7RMsLjhemkPoebORn6tF7S913ZUE7olWIb/FpfrlVPaFp4JK+Esu71GNc7LUzLmVTjy0emhAt7VsGu2NOCm8DJQAFkVetZ314/xHFAuMIMSdl27Eh1EiQUxYxM8l4sSYTwCA1JW0OAiI7ybT/E3ijmT4chEJPruCUXU4kKJByHPjaGSD1Kde1lNymtWM1eOgklEexIhzPDhrEDKoQp8J+1QrNhYA4QF1XeF+BMJhJX+8rxugrP+5E3QcCuOXHe7wrV16wdOXAFrkEROAeVMEbqIE6wMaj4Rsjg5nP5tDkZjSzmkaWuQrZY7/ABqfyBM=</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  48. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  49. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  50. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  51. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  52. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  53. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  54. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  55. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Incrementing n increases time by a constant Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  56. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">ACv3icbVHbSsMwGE7rac5T1UtvgkPZGIy2CHqhMPFCLye4A6zbSLNshqVpSVJhlL2kF4JvY7p17viTwJfvkMfP2JUKtv+Ncyd3b39g9xh/uj45PTMOr9oyDAWmNRxyELR8pEkjHJSV1Qx0oEQYHPSNMfvaR684sISUP+ocYR6QRoyOmAYqQ01bN+EPRwP1TQ7ya87Ezg7RMsLjhemkPoebORn6tF7S913ZUE7olWIb/FpfrlVPaFp4JK+Esu71GNc7LUzLmVTjy0emhAt7VsGu2NOCm8DJQAFkVetZ314/xHFAuMIMSdl27Eh1EiQUxYxM8l4sSYTwCA1JW0OAiI7ybT/E3ijmT4chEJPruCUXU4kKJByHPjaGSD1Kde1lNymtWM1eOgklEexIhzPDhrEDKoQp8J+1QrNhYA4QF1XeF+BMJhJX+8rxugrP+5E3QcCuOXHe7wrV16wdOXAFrkEROAeVMEbqIE6wMaj4Rsjg5nP5tDkZjSzmkaWuQrZY7/ABqfyBM=</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">ACv3icbVHbSsMwGE7rac5T1UtvgkPZGIy2CHqhMPFCLye4A6zbSLNshqVpSVJhlL2kF4JvY7p17viTwJfvkMfP2JUKtv+Ncyd3b39g9xh/uj45PTMOr9oyDAWmNRxyELR8pEkjHJSV1Qx0oEQYHPSNMfvaR684sISUP+ocYR6QRoyOmAYqQ01bN+EPRwP1TQ7ya87Ezg7RMsLjhemkPoebORn6tF7S913ZUE7olWIb/FpfrlVPaFp4JK+Esu71GNc7LUzLmVTjy0emhAt7VsGu2NOCm8DJQAFkVetZ314/xHFAuMIMSdl27Eh1EiQUxYxM8l4sSYTwCA1JW0OAiI7ybT/E3ijmT4chEJPruCUXU4kKJByHPjaGSD1Kde1lNymtWM1eOgklEexIhzPDhrEDKoQp8J+1QrNhYA4QF1XeF+BMJhJX+8rxugrP+5E3QcCuOXHe7wrV16wdOXAFrkEROAeVMEbqIE6wMaj4Rsjg5nP5tDkZjSzmkaWuQrZY7/ABqfyBM=</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for n+n Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Incrementing n increases time by a constant Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  57. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for n+n Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Incrementing n increases time by a constant Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  58. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> Time for n+n Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Incrementing n increases time by a constant Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Constant growth . Increasing n doesn't affect time 14

  59. <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">ACv3icbVHbSsMwGE7rac5T1UtvgkPZGIy2CHqhMPFCLye4A6zbSLNshqVpSVJhlL2kF4JvY7p17viTwJfvkMfP2JUKtv+Ncyd3b39g9xh/uj45PTMOr9oyDAWmNRxyELR8pEkjHJSV1Qx0oEQYHPSNMfvaR684sISUP+ocYR6QRoyOmAYqQ01bN+EPRwP1TQ7ya87Ezg7RMsLjhemkPoebORn6tF7S913ZUE7olWIb/FpfrlVPaFp4JK+Esu71GNc7LUzLmVTjy0emhAt7VsGu2NOCm8DJQAFkVetZ314/xHFAuMIMSdl27Eh1EiQUxYxM8l4sSYTwCA1JW0OAiI7ybT/E3ijmT4chEJPruCUXU4kKJByHPjaGSD1Kde1lNymtWM1eOgklEexIhzPDhrEDKoQp8J+1QrNhYA4QF1XeF+BMJhJX+8rxugrP+5E3QcCuOXHe7wrV16wdOXAFrkEROAeVMEbqIE6wMaj4Rsjg5nP5tDkZjSzmkaWuQrZY7/ABqfyBM=</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">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</latexit> <latexit sha1_base64="VxC4Ky3frwSgEWdLXRZ0iEtjZH4=">ACv3icbVHbSsMwGE7rac5T1UtvgkPZGIy2CHqhMPFCLye4A6zbSLNshqVpSVJhlL2kF4JvY7p17viTwJfvkMfP2JUKtv+Ncyd3b39g9xh/uj45PTMOr9oyDAWmNRxyELR8pEkjHJSV1Qx0oEQYHPSNMfvaR684sISUP+ocYR6QRoyOmAYqQ01bN+EPRwP1TQ7ya87Ezg7RMsLjhemkPoebORn6tF7S913ZUE7olWIb/FpfrlVPaFp4JK+Esu71GNc7LUzLmVTjy0emhAt7VsGu2NOCm8DJQAFkVetZ314/xHFAuMIMSdl27Eh1EiQUxYxM8l4sSYTwCA1JW0OAiI7ybT/E3ijmT4chEJPruCUXU4kKJByHPjaGSD1Kde1lNymtWM1eOgklEexIhzPDhrEDKoQp8J+1QrNhYA4QF1XeF+BMJhJX+8rxugrP+5E3QcCuOXHe7wrV16wdOXAFrkEROAeVMEbqIE6wMaj4Rsjg5nP5tDkZjSzmkaWuQrZY7/ABqfyBM=</latexit> Time for n+n Time for input n+1 Time for input n Common Orders of Growth Exponential growth . E.g., recursive fib a · b n +1 = ( a · b n ) · b Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap a · ( n + 1) 2 = ( a · n 2 ) + a · (2 n + 1) Incrementing n increases time by n times a constant Linear growth . E.g., slow exp a · ( n + 1) = ( a · n ) + a Incrementing n increases time by a constant Logarithmic growth . E.g., exp_fast a · ln(2 · n ) = ( a · ln n ) + a · ln 2 Doubling n only increments time by a constant Constant growth . Increasing n doesn't affect time 14

  60. Order of Growth Notation

  61. Big Theta and Big O Notation for Orders of Growth Exponential growth . E.g., recursive fib Incrementing n multiplies time by a constant Quadratic growth . E.g., overlap Incrementing n increases time by n times a constant Linear growth . E.g., slow exp Incrementing n increases time by a constant Logarithmic growth . E.g., exp_fast Doubling n only increments time by a constant Constant growth . Increasing n doesn't affect time 16

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sha1_base64="R68JbFuSOwmg0uCtKaVsudswboc=">AB9XicbVA9SwNBEJ2LXzF+RS1tFoMQm3Bno2XQxjJCPoTkDHubuWTJ3t6xu6eEI/DxkIRW/+Lnf/GTXKFJj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMTqPqAaBZfYMtwIvE8U0igQ2AnGNzO/84hK81g2zSRBP6JDyUPOqLHSQ685QkOrPREPiTzvlytuzZ2DrBIvJxXI0eiXv3qDmKURSsME1bruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfza+ekjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCKz/jMkNSrZYFKaCmJjMIiADrpAZMbGEMsXtrYSNqKLM2KBKNgRv+eV0r6oeW7Nu3Mr9es8jiKcwClUwYNLqMtNKAFDBQ8wyu8OU/Oi/PufCxaC04+cwx/4Hz+AHqpkdE=</latexit> Big Theta and Big O Notation for Orders of Growth Θ ( b n ) Exponential growth . E.g., recursive fib Incrementing n multiplies time by a constant Θ ( n 2 ) Quadratic growth . E.g., overlap Incrementing n increases time by n times a constant Linear growth . E.g., slow exp Θ ( n ) Incrementing n increases time by a constant Θ (log n ) Logarithmic growth . E.g., exp_fast Doubling n only increments time by a constant Θ (1) Constant growth . Increasing n doesn't affect time 16

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sha1_base64="R68JbFuSOwmg0uCtKaVsudswboc=">AB9XicbVA9SwNBEJ2LXzF+RS1tFoMQm3Bno2XQxjJCPoTkDHubuWTJ3t6xu6eEI/DxkIRW/+Lnf/GTXKFJj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMTqPqAaBZfYMtwIvE8U0igQ2AnGNzO/84hK81g2zSRBP6JDyUPOqLHSQ685QkOrPREPiTzvlytuzZ2DrBIvJxXI0eiXv3qDmKURSsME1bruYnxM6oMZwKnpV6qMaFsTIfYtVTSCLWfza+ekjOrDEgYK1vSkLn6eyKjkdaTKLCdETUjvezNxP+8bmrCKz/jMkNSrZYFKaCmJjMIiADrpAZMbGEMsXtrYSNqKLM2KBKNgRv+eV0r6oeW7Nu3Mr9es8jiKcwClUwYNLqMtNKAFDBQ8wyu8OU/Oi/PufCxaC04+cwx/4Hz+AHqpkdE=</latexit> Big Theta and Big O Notation for Orders of Growth Θ ( b n ) O ( b n ) Exponential growth . E.g., recursive fib Incrementing n multiplies time by a constant Θ ( n 2 ) O ( n 2 ) Quadratic growth . E.g., overlap Incrementing n increases time by n times a constant Linear growth . E.g., slow exp O ( n ) Θ ( n ) Incrementing n increases time by a constant Θ (log n ) O (log n ) Logarithmic growth . E.g., exp_fast Doubling n only increments time by a constant Θ (1) O (1) Constant growth . Increasing n doesn't affect time 16

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