efficiency announcements measuring efficiency recursive
play

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=">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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

  47. <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> <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=">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 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=">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 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=">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 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=">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> <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=">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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

  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=">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 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=">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 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=">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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 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

  62. <latexit sha1_base64="8g+c8Fm4V71cuVu27a2oG6AMhA=">AB8nicbVDLSsNAFL3xWeur6tJNaBEqQknc6LoxmWFviCJZTKdtEMnkzBzI5TQz3Djwgdu/Rp3/o3Tx0JbD1w4nHMv94TpoJrdJxva219Y3Nru7BT3N3bPzgsHR23dZIpylo0EYnqhkQzwSVrIUfBuqliJA4F64Sj26nfeWRK80Q2cZyICYDySNOCRrJ85tDhqQaPsjzXqni1JwZ7FXiLkilXvYv3gCg0St9+f2EZjGTSAXR2nOdFIOcKORUsEnRzRLCR2RAfMlSRmOshnJ0/sM6P07ShRpiTaM/X3RE5ircdxaDpjgkO97E3F/zwvw+g6yLlM2SzhdFmbAxsaf/232uGEUxNoRQxc2tNh0SRSialIomBHf5VXSvqy5Ts29N2ncwBwFOIUyVMGFK6jDHTSgBRQSeIXeLXQerberY9565q1mDmBP7A+fwCQI5I+</latexit> <latexit sha1_base64="1j5ev6ilyFSc2YovtpREqyQAMPs=">AB8nicbVDLSsNAFJ3UV62vqks3Q4tQEUriRpdFNy4r9AVJLJPpB06mYSZGyGE/oVuXCji1q9x179x+lho64ELh3Pu5d57gkRwDbY9tQobm1vbO8Xd0t7+weFR+fiko+NUdamsYhVLyCaCS5ZGzgI1ksUI1EgWDcY38387hNTmseyBVnC/IgMJQ85JWAk12uNGJBa8Cgv+uWqXbfnwOvEWZJqo+JdPk8bWbNf/vYGMU0jJoEKorXr2An4OVHAqWCTkpdqlhA6JkPmGipJxLSfz0+e4HOjDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlEomBGf15XSuao7dt15MGncogWK6AxVUA056Bo10D1qojaiKEYv6A29W2C9Wh/W56K1YC1nTtEfWF8/mY2TxA=</latexit> <latexit sha1_base64="1j5ev6ilyFSc2YovtpREqyQAMPs=">AB8nicbVDLSsNAFJ3UV62vqks3Q4tQEUriRpdFNy4r9AVJLJPpB06mYSZGyGE/oVuXCji1q9x179x+lho64ELh3Pu5d57gkRwDbY9tQobm1vbO8Xd0t7+weFR+fiko+NUdamsYhVLyCaCS5ZGzgI1ksUI1EgWDcY38387hNTmseyBVnC/IgMJQ85JWAk12uNGJBa8Cgv+uWqXbfnwOvEWZJqo+JdPk8bWbNf/vYGMU0jJoEKorXr2An4OVHAqWCTkpdqlhA6JkPmGipJxLSfz0+e4HOjDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlEomBGf15XSuao7dt15MGncogWK6AxVUA056Bo10D1qojaiKEYv6A29W2C9Wh/W56K1YC1nTtEfWF8/mY2TxA=</latexit> <latexit sha1_base64="a8UYW0Su6kS0T6+bYsH29pm5wpg=">AB8nicbVA9SwNBEJ2LXzF+RS1tFoMQm3Bno2XQxjJCviA5w95mL1myt3fszgnhyM+wsVDE1l9j579xk1yhiQ8GHu/NMDMvSKQw6LrfTmFjc2t7p7hb2ts/ODwqH5+0TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTu7nfeLaiFg1cZpwP6IjJULBKFqp12+OdJq8KguB+WKW3MXIOvEy0kFcjQG5a/+MGZpxBUySY3peW6CfkY1Cib5rNRPDU8om9AR71mqaMSNny1OnpELqwxJGtbCslC/T2R0ciYaRTYzoji2Kx6c/E/r5dieONnQiUpcsWi8JUEozJ/H8yFJozlFNLKNPC3krYmGrK0KZUsiF4qy+vk/ZVzXNr3oNbqd/mcRThDM6hCh5cQx3uoQEtYBDM7zCm4POi/PufCxbC04+cwp/4Hz+AH2kLU=</latexit> <latexit sha1_base64="cxg26Q7isxqJFXNUApBw2Z0XPYE=">AB8HicbVDLSgNBEOz1GeMr6tHLkCBEhLDrRY9BLx4j5CXZJcxOsmQmd1lZlYIS7CiwdFvfo53vwbJ4+DJhY0FXdHeFieDauO63s7a+sbm1ndvJ7+7tHxwWjo6bOk4VwaLRazaIdUoeIQNw43AdqKQylBgKxzdTv3WIyrN46huxgkGkg4i3ueMGis9+PUhGlr2zruFkltxZyCrxFuQUrXoX7wDQK1b+PJ7MUslRoYJqnXHcxMTZFQZzgRO8n6qMaFsRAfYsTSiEnWQzQ6ekDOr9Eg/VrYiQ2bq74mMSq3HMrSdkpqhXvam4n9eJzX96yDjUZIajNh8UT8VxMRk+j3pcYXMiLElClubyVsSBVlxmaUtyF4y+vkuZlxXMr3r1N4wbmyMEpFKEMHlxBFe6gBg1gIOEJXuDVUc6z8+Z8zFvXnMXMCfyB8/kDv5SRLQ=</latexit> <latexit sha1_base64="zZPfn/VjLEdIKtCk3Law2gI2I=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF2MpsMmZldZnqFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7wkRwA647dXIbm1vbO/ndwt7+weFR8fikZeJU9aksYh1JySGCa5YEzgI1k0IzIUrB2O7mZ+4lpw2PVgHCAkGikecErDSo98YMiAV76JXLtVdw68TrwlKdK/uXLtDau94rfj+mqWQKqCDGdD03gSAjGjgVbFLwU8MSQkdkwLqWKiKZCbL5wRN8bpU+jmJtSwGeq78nMiKNGcvQdkoCQ7PqzcT/vG4K0U2QcZWkwBRdLIpSgSHGs+9xn2tGQYwtIVRzeyumQ6IJBZtRwYbgrb68TlpXVc+teg82jVu0QB6doRKqIA9doxq6R3XURBRJ9Ize0LujnVfnw/lctOac5cwp+gPn6wfI/pKz</latexit> <latexit sha1_base64="zZPfn/VjLEdIKtCk3Law2gI2I=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF2MpsMmZldZnqFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7wkRwA647dXIbm1vbO/ndwt7+weFR8fikZeJU9aksYh1JySGCa5YEzgI1k0IzIUrB2O7mZ+4lpw2PVgHCAkGikecErDSo98YMiAV76JXLtVdw68TrwlKdK/uXLtDau94rfj+mqWQKqCDGdD03gSAjGjgVbFLwU8MSQkdkwLqWKiKZCbL5wRN8bpU+jmJtSwGeq78nMiKNGcvQdkoCQ7PqzcT/vG4K0U2QcZWkwBRdLIpSgSHGs+9xn2tGQYwtIVRzeyumQ6IJBZtRwYbgrb68TlpXVc+teg82jVu0QB6doRKqIA9doxq6R3XURBRJ9Ize0LujnVfnw/lctOac5cwp+gPn6wfI/pKz</latexit> <latexit sha1_base64="uecGOMSGgnxr7TgkwN0KdrGvsZQ=">AB8HicbVA9SwNBEJ2LXzF+RS1tDoMQm3Bno2XQxjJCviQ5wt5mL1myu3fszgnhyK+wsVDE1p9j579xk1yhiQ8GHu/NMDMvTAQ36HnfTmFjc2t7p7hb2ts/ODwqH5+0TZxqylo0FrHuhsQwRVrIUfBuolmRIaCdcLJ3dzvPDFteKyaOE1YIMlI8YhTglZ67DfHDEnVvxyUK17NW8BdJ35OKpCjMSh/9YcxTSVTSAUxpud7CQYZ0cipYLNSPzUsIXRCRqxnqSKSmSBbHDxzL6wydKNY21LoLtTfExmRxkxlaDslwbFZ9ebif14vxegmyLhKUmSKLhdFqXAxdufu0OuGUxtYRQze2tLh0TSjajEo2BH/15XSvqr5Xs1/8Cr12zyOIpzBOVTBh2uowz0oAUJDzDK7w52nlx3p2PZWvByWdO4Q+czx+tFY+k</latexit> <latexit sha1_base64="2Nu7S0mxhr8c+CikqDKY8Bf0+5M=">AB8nicbVDJSgNBEK2JW4xb1KOXIUGICGEmFz0GvXiMkA0mY+jp9CRNenqG7hohDPkMLx5c8OrXePNv7CwHTXxQ8Hiviqp6QSK4Rsf5tnIbm1vbO/ndwt7+weFR8fikreNUdaisYhVNyCaCS5ZCzkK1k0UI1EgWCcY3878ziNTmseyiZOE+REZSh5yStBIXq85Ykgq8qF20S+Wnaozh71O3CUp10u9yzcAaPSLX71BTNOISaSCaO25ToJ+RhRyKti0Es1SwgdkyHzDJUkYtrP5idP7XOjDOwVqYk2nP190RGIq0nUWA6I4IjverNxP8L8Xw2s+4TFJki4WhamwMbZn/9sDrhFMTGEUMXNrTYdEUompQKJgR39eV10q5VXafq3ps0bmCBPJxBCSrgwhXU4Q4a0AIKMTzBC7xaD1b79bHojVnLWdO4Q+szx9HS5IO</latexit> <latexit sha1_base64="JtNKk2OifBCJsu/nv8Q9CeO7Jno=">AB8nicbVDLSsNAFJ34rPVdelmaBEqQkm60WXRjcsKfUESy2Q6aYdOJmHmRgihf6EbF4q49Wvc9W+cPhbaeuDC4Zx7ufeIBFcg21PrY3Nre2d3cJecf/g8Oi4dHLa0XGqKGvTWMSqFxDNBJesDRwE6yWKkSgQrBuM72Z+94kpzWPZgixhfkSGkoecEjCS67VGDEhVPtYv+6WKXbPnwOvEWZJKo+xdPU8bWbNf+vYGMU0jJoEKorXr2An4OVHAqWCTopdqlhA6JkPmGipJxLSfz0+e4AujDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlIomBGf15XSqdcu+Y8mDRu0QIFdI7KqIocdI0a6B41URtRFKMX9IbeLbBerQ/rc9G6YS1nztAfWF8/ULWTlA=</latexit> <latexit sha1_base64="JtNKk2OifBCJsu/nv8Q9CeO7Jno=">AB8nicbVDLSsNAFJ34rPVdelmaBEqQkm60WXRjcsKfUESy2Q6aYdOJmHmRgihf6EbF4q49Wvc9W+cPhbaeuDC4Zx7ufeIBFcg21PrY3Nre2d3cJecf/g8Oi4dHLa0XGqKGvTWMSqFxDNBJesDRwE6yWKkSgQrBuM72Z+94kpzWPZgixhfkSGkoecEjCS67VGDEhVPtYv+6WKXbPnwOvEWZJKo+xdPU8bWbNf+vYGMU0jJoEKorXr2An4OVHAqWCTopdqlhA6JkPmGipJxLSfz0+e4AujDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlIomBGf15XSqdcu+Y8mDRu0QIFdI7KqIocdI0a6B41URtRFKMX9IbeLbBerQ/rc9G6YS1nztAfWF8/ULWTlA=</latexit> <latexit sha1_base64="WALeiKi4PWgAumKikdV7B6ZkfQ=">AB8nicbVA9SwNBEJ3zM8avqKXNYRBiE+7SaBm0sYyQL7icYW+zSZbs7R67c0I48jNsLBSx9dfY+W/cJFdo4oOBx3szMyLEsENet63s7G5tb2zW9gr7h8cHh2XTk7bRqWashZVQuluRAwTXLIWchSsm2hG4kiwTjS5m/udJ6YNV7KJ04SFMRlJPuSUoJWCXnPMkFTkY+2qXyp7VW8Bd534OSlDjka/9NUbKJrGTCIVxJjA9xIM6KRU8FmxV5qWELohIxYKkMTNhtjh5l5aZeAOlbYl0V2ovycyEhszjSPbGRMcm1VvLv7nBSkOb8KMyRFJuly0TAVLip3/r874JpRFNLCNXc3urSMdGEok2paEPwV19eJ+1a1feq/oNXrt/mcRTgHC6gAj5cQx3uoQEtoKDgGV7hzUHnxXl3PpatG04+cwZ/4Hz+ADTMkIU=</latexit> <latexit sha1_base64="jMSL2OINSwlIU1DFc5CHfF+CiMs=">AB8HicbVDLSgNBEOz1GeMr6tHLkiBEhLDrRY9BLx4j5CXZJcxOsmQmdlZlYIS7CiwdFvfo53vwbJ4+DJhY0FXdHdFCWfaeN63s7a+sbm1ndvJ7+7tHxwWjo6bOk4VxQaNeazaEdHImcSGYZjO1FIRMSxFY1up37rEZVmsaybcYKhIAPJ+owSY6WHoD5EQ8ryvFsoeRVvBneV+AtSqhaDi3cAqHULX0EvpqlAaSgnWnd8LzFhRpRhlOMkH6QaE0JHZIAdSyURqMNsdvDEPbNKz+3HypY07kz9PZERofVYRLZTEDPUy95U/M/rpKZ/HWZMJqlBSeL+il3TexOv3d7TCE1fGwJoYrZW106JIpQYzPK2xD85ZdXSfOy4nsV/96mcQNz5OAUilAGH6gCndQgwZQEPAEL/DqKOfZeXM+5q1rzmLmBP7A+fwBHFSRag=</latexit> <latexit sha1_base64="gdLfQI6euH6g8XZvgkCxRKZJspI=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF20psMmZldZmaFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7woQzbVx36uQ2Nre2d/K7hb39g8Oj4vFJS8epotCkMY9VJyQaOJPQNMxw6CQKiAg5tMPR3cxvP4HSLJYNM04gEGQgWcQoMVZ69BtDMKQiL3rFslt158DrxFuScq3kX75Ma+N6r/jt92OaCpCGcqJ13MTE2REGUY5TAp+qiEhdEQG0LVUEgE6yOYHT/C5Vfo4ipUtafBc/T2REaH1WIS2UxAz1KveTPzP6YmugkyJpPUgKSLRVHKsYnx7HvcZwqo4WNLCFXM3orpkChCjc2oYEPwVl9eJ62rqudWvQebxi1aI/OUAlVkIeuUQ3dozpqIoEekZv6N1Rzqvz4XwuWnPOcuYU/YHz9QMlvpLw</latexit> <latexit sha1_base64="gdLfQI6euH6g8XZvgkCxRKZJspI=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF20psMmZldZmaFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7woQzbVx36uQ2Nre2d/K7hb39g8Oj4vFJS8epotCkMY9VJyQaOJPQNMxw6CQKiAg5tMPR3cxvP4HSLJYNM04gEGQgWcQoMVZ69BtDMKQiL3rFslt158DrxFuScq3kX75Ma+N6r/jt92OaCpCGcqJ13MTE2REGUY5TAp+qiEhdEQG0LVUEgE6yOYHT/C5Vfo4ipUtafBc/T2REaH1WIS2UxAz1KveTPzP6YmugkyJpPUgKSLRVHKsYnx7HvcZwqo4WNLCFXM3orpkChCjc2oYEPwVl9eJ62rqudWvQebxi1aI/OUAlVkIeuUQ3dozpqIoEekZv6N1Rzqvz4XwuWnPOcuYU/YHz9QMlvpLw</latexit> <latexit sha1_base64="oB6jlXGIkO9IKqkRvd0Cu68+Q6c=">AB8HicbVA9SwNBEJ2LXzF+RS1tDoMQm3Bno2XQxjJCviQ5wt5mL1myu3fszgnhyK+wsVDE1p9j579xk1yhiQ8GHu/NMDMvTAQ36HnfTmFjc2t7p7hb2ts/ODwqH5+0TZxqylo0FrHuhsQwRVrIUfBuolmRIaCdcLJ3dzvPDFteKyaOE1YIMlI8YhTglZ67DfHDElVXQ7KFa/mLeCuEz8nFcjRGJS/+sOYpIpIY0/O9BIOMaORUsFmpnxqWEDohI9azVBHJTJAtDp65F1YZulGsbSl0F+rviYxIY6YytJ2S4NisenPxP6+XYnQTZFwlKTJFl4uiVLgYu/Pv3SHXjKYWkKo5vZWl46JhRtRiUbgr/68jpX9V8r+Y/eJX6bR5HEc7gHKrgwzXU4R4a0AIKEp7hFd4c7bw4787HsrXg5DOn8AfO5w8J1Y/h</latexit> <latexit sha1_base64="DK/wbEiUiPA2hr2keU+G1aLr5M=">AB9XicbVDLSgNBEOyNrxhfUY9ehgQhIoRdL3oMevEYIS/IrmF2MpsMmZ1ZmaVsOQ/vAgq4tV/8ebfOHkcNLGgoajqprsrTDjTxnW/ndza+sbmVn67sLO7t39QPDxqaZkqQptEcqk6IdaUM0GbhlO4miOA45bYejm6nfqBKMykaZpzQIMYDwSJGsLHSvd8YUoMrPpcDJM56xbJbdWdAq8RbkHKt5J+/AEC9V/zy+5KkMRWGcKx13MTE2RYGUY4nRT8VNMEkxEe0K6lAsdUB9ns6gk6tUofRVLZEgbN1N8TGY61Hseh7YyxGeplbyr+53VTE10FGRNJaqg80VRypGRaBoB6jNFieFjSzBRzN6KyBArTIwNqmBD8JZfXiWti6rnVr07m8Y1zJGHEyhBTy4hBrcQh2aQEDBE7zCm/PoPDvzse8NecsZo7hD5zPH40ok1o=</latexit> <latexit sha1_base64="RHa1m03WbZ7W9E8ltOzuGdhNgY=">AB9XicbVDLSgNBEOz1GeMr6tHLkCBEhLDrRY9BLx4j5AXZNcxOZpMhszPLzKyhPyFBy8eFPHqv3jL3zh5HDSxoKGo6qa7K0w408Z1J87a+sbm1nZuJ7+7t39wWDg6bmqZKkIbRHKp2iHWlDNBG4YZTtuJojgOW2Fw9up3qkSjMp6iZLaBDjvmARI9hY6cGvD6jBZ/LPhLn3ULJrbgzoFXiLUipWvQvnifVrNYtfPs9SdKYCkM41rjuYkJRlgZRjgd5/1U0wSTIe7TjqUCx1QHo9nVY3RmlR6KpLIlDJqpvydGONY6i0PbGWMz0MveVPzP6Qmug5GTCSpoYLMF0UpR0aiaQSoxQlhmeWYKYvRWRAVaYGBtU3obgLb+8SpqXFc+tePc2jRuYIwenUIQyeHAFVbiDGjSAgIXeIN358l5dT6cz3nrmrOYOYE/cL5+AJaSlOA=</latexit> <latexit sha1_base64="RHa1m03WbZ7W9E8ltOzuGdhNgY=">AB9XicbVDLSgNBEOz1GeMr6tHLkCBEhLDrRY9BLx4j5AXZNcxOZpMhszPLzKyhPyFBy8eFPHqv3jL3zh5HDSxoKGo6qa7K0w408Z1J87a+sbm1nZuJ7+7t39wWDg6bmqZKkIbRHKp2iHWlDNBG4YZTtuJojgOW2Fw9up3qkSjMp6iZLaBDjvmARI9hY6cGvD6jBZ/LPhLn3ULJrbgzoFXiLUipWvQvnifVrNYtfPs9SdKYCkM41rjuYkJRlgZRjgd5/1U0wSTIe7TjqUCx1QHo9nVY3RmlR6KpLIlDJqpvydGONY6i0PbGWMz0MveVPzP6Qmug5GTCSpoYLMF0UpR0aiaQSoxQlhmeWYKYvRWRAVaYGBtU3obgLb+8SpqXFc+tePc2jRuYIwenUIQyeHAFVbiDGjSAgIXeIN358l5dT6cz3nrmrOYOYE/cL5+AJaSlOA=</latexit> <latexit 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

  63. <latexit sha1_base64="KVPMTpnH9s6vUR/4ASm+0cLiDGE=">AB63icbVA9SwNBEJ2LXzF+RS1tlgQhIoQ7Gy2DNnZGMB+QHGFvs5cs2d07dveEcOQv2FgoYmHjH7Lz37h3SaGJDwYe780wMy+IOdPGdb+dwtr6xuZWcbu0s7u3f1A+PGrKFGEtkjEI9UNsKacSdoyzHDajRXFIuC0E0xuMr/zSJVmkXw05j6Ao8kCxnBJpPuat7ZoFx1624OtEq8Bak2Kv3zDwBoDspf/WFEkGlIRxr3fPc2PgpVoYRTmelfqJpjMkEj2jPUokF1X6a3zpDp1YZojBStqRBufp7IsVC6kIbKfAZqyXvUz8z+slJrzyUybjxFBJ5ovChCMToexNGSKEsOnlmCimL0VkTFWmBgbT8mG4C2/vEraF3XPrXv3No1rmKMIJ1CBGnhwCQ24hSa0gMAYnuAFXh3hPDtvzvu8teAsZo7hD5zPH+6Ljvg=</latexit> <latexit sha1_base64="k2Gtb5Vy1XCfuk+6GicE2CxElfQ=">AB63icbVBNSwMxEJ31s9avqkcvoUWoCGXix6LXrxZwX5Au5Rsm1Dk+ySZIVl6V/woqCIV/+Qt/4bs20P2vpg4PHeDPzgpgzbVx36qytb2xubRd2irt7+weHpaPjlo4SRWiTRDxSnQBrypmkTcMp51YUSwCTtvB+Db3209UaRbJR5PG1Bd4KFnICDa5dF/1zvuliltzZ0CrxFuQSr3cu3id1tNGv/TdG0QkEVQawrHWXc+NjZ9hZRjhdFLsJZrGmIzxkHYtlVhQ7WezWyfozCoDFEbKljRopv6eyLDQOhWB7RTYjPSyl4v/ed3EhNd+xmScGCrJfFGYcGQilD+OBkxRYnhqCSaK2VsRGWGFibHxFG0I3vLq6R1WfPcmvdg07iBOQpwCmWogdXUIc7aEATCIzgGd7g3RHOi/PhfM5b15zFzAn8gfP1A/f1kH4=</latexit> <latexit sha1_base64="k2Gtb5Vy1XCfuk+6GicE2CxElfQ=">AB63icbVBNSwMxEJ31s9avqkcvoUWoCGXix6LXrxZwX5Au5Rsm1Dk+ySZIVl6V/woqCIV/+Qt/4bs20P2vpg4PHeDPzgpgzbVx36qytb2xubRd2irt7+weHpaPjlo4SRWiTRDxSnQBrypmkTcMp51YUSwCTtvB+Db3209UaRbJR5PG1Bd4KFnICDa5dF/1zvuliltzZ0CrxFuQSr3cu3id1tNGv/TdG0QkEVQawrHWXc+NjZ9hZRjhdFLsJZrGmIzxkHYtlVhQ7WezWyfozCoDFEbKljRopv6eyLDQOhWB7RTYjPSyl4v/ed3EhNd+xmScGCrJfFGYcGQilD+OBkxRYnhqCSaK2VsRGWGFibHxFG0I3vLq6R1WfPcmvdg07iBOQpwCmWogdXUIc7aEATCIzgGd7g3RHOi/PhfM5b15zFzAn8gfP1A/f1kH4=</latexit> <latexit sha1_base64="qw+XdynGm6NTsSDR/q8RhDC1eTU=">AB63icbVBNSwMxEJ31s9avqkcvwSLUS8l60WPRizcr2A9ol5JNs21okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemAhuLMbf3tr6xubWdmnvLu3f3BYOTpumzjVlLVoLGLdDYlhgivWstwK1k0IzIUrBNObnO/8S04bF6tNOEBZKMFI84JTaX7mv+xaBSxXU8B1olfkGqUKA5qHz1hzFNJVOWCmJMz8eJDTKiLaeCzcr91LCE0AkZsZ6jikhmgmx+6wydO2WIoli7UhbN1d8TGZHGTGXoOiWxY7Ps5eJ/Xi+10XWQcZWklim6WBSlAtkY5Y+jIdeMWjF1hFDN3a2Ijokm1Lp4yi4Ef/nlVdK+rPu47j/gauOmiKMEp3AGNfDhChpwB01oAYUxPMrvHnSe/HevY9F65pXzJzAH3ifP9wMjW8=</latexit> <latexit sha1_base64="cxg26Q7isxqJFXNUApBw2Z0XPYE=">AB8HicbVDLSgNBEOz1GeMr6tHLkCBEhLDrRY9BLx4j5CXZJcxOsmQmd1lZlYIS7CiwdFvfo53vwbJ4+DJhY0FXdHeFieDauO63s7a+sbm1ndvJ7+7tHxwWjo6bOk4VwaLRazaIdUoeIQNw43AdqKQylBgKxzdTv3WIyrN46huxgkGkg4i3ueMGis9+PUhGlr2zruFkltxZyCrxFuQUrXoX7wDQK1b+PJ7MUslRoYJqnXHcxMTZFQZzgRO8n6qMaFsRAfYsTSiEnWQzQ6ekDOr9Eg/VrYiQ2bq74mMSq3HMrSdkpqhXvam4n9eJzX96yDjUZIajNh8UT8VxMRk+j3pcYXMiLElClubyVsSBVlxmaUtyF4y+vkuZlxXMr3r1N4wbmyMEpFKEMHlxBFe6gBg1gIOEJXuDVUc6z8+Z8zFvXnMXMCfyB8/kDv5SRLQ=</latexit> <latexit sha1_base64="zZPfn/VjLEdIKtCk3Law2gI2I=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF2MpsMmZldZnqFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7wkRwA647dXIbm1vbO/ndwt7+weFR8fikZeJU9aksYh1JySGCa5YEzgI1k0IzIUrB2O7mZ+4lpw2PVgHCAkGikecErDSo98YMiAV76JXLtVdw68TrwlKdK/uXLtDau94rfj+mqWQKqCDGdD03gSAjGjgVbFLwU8MSQkdkwLqWKiKZCbL5wRN8bpU+jmJtSwGeq78nMiKNGcvQdkoCQ7PqzcT/vG4K0U2QcZWkwBRdLIpSgSHGs+9xn2tGQYwtIVRzeyumQ6IJBZtRwYbgrb68TlpXVc+teg82jVu0QB6doRKqIA9doxq6R3XURBRJ9Ize0LujnVfnw/lctOac5cwp+gPn6wfI/pKz</latexit> <latexit sha1_base64="zZPfn/VjLEdIKtCk3Law2gI2I=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF2MpsMmZldZnqFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7wkRwA647dXIbm1vbO/ndwt7+weFR8fikZeJU9aksYh1JySGCa5YEzgI1k0IzIUrB2O7mZ+4lpw2PVgHCAkGikecErDSo98YMiAV76JXLtVdw68TrwlKdK/uXLtDau94rfj+mqWQKqCDGdD03gSAjGjgVbFLwU8MSQkdkwLqWKiKZCbL5wRN8bpU+jmJtSwGeq78nMiKNGcvQdkoCQ7PqzcT/vG4K0U2QcZWkwBRdLIpSgSHGs+9xn2tGQYwtIVRzeyumQ6IJBZtRwYbgrb68TlpXVc+teg82jVu0QB6doRKqIA9doxq6R3XURBRJ9Ize0LujnVfnw/lctOac5cwp+gPn6wfI/pKz</latexit> <latexit sha1_base64="uecGOMSGgnxr7TgkwN0KdrGvsZQ=">AB8HicbVA9SwNBEJ2LXzF+RS1tDoMQm3Bno2XQxjJCviQ5wt5mL1myu3fszgnhyK+wsVDE1p9j579xk1yhiQ8GHu/NMDMvTAQ36HnfTmFjc2t7p7hb2ts/ODwqH5+0TZxqylo0FrHuhsQwRVrIUfBuolmRIaCdcLJ3dzvPDFteKyaOE1YIMlI8YhTglZ67DfHDEnVvxyUK17NW8BdJ35OKpCjMSh/9YcxTSVTSAUxpud7CQYZ0cipYLNSPzUsIXRCRqxnqSKSmSBbHDxzL6wydKNY21LoLtTfExmRxkxlaDslwbFZ9ebif14vxegmyLhKUmSKLhdFqXAxdufu0OuGUxtYRQze2tLh0TSjajEo2BH/15XSvqr5Xs1/8Cr12zyOIpzBOVTBh2uowz0oAUJDzDK7w52nlx3p2PZWvByWdO4Q+czx+tFY+k</latexit> <latexit sha1_base64="0DLF3Gz1ia/tOnvj4Xajp0cA=">AB8HicbVDJSgNBEK1xjXGLevTSJAgRIcx40WPQizcjmEUyQ+jpdJImvQzdPUIY8hVePCjq1c/x5t/YWQ6a+KDg8V4VfXihDNjf/bW1ldW9/YzG3lt3d29/YLB4cNo1JNaJ0ornQrxoZyJmndMstpK9EUi5jTZjy8nvjNR6oNU/LejhIaCdyXrMcItk56uC2HXPWRPO0USn7FnwItk2BOStViePYOALVO4SvsKpIKi3h2Jh24Cc2yrC2jHA6zoepoQkmQ9ynbUclFtRE2fTgMTpxShf1lHYlLZqvycyLIwZidh1CmwHZtGbiP957dT2LqOMyS1VJLZol7KkVo8j3qMk2J5SNHMNHM3YrIAGtMrMso70IFl9eJo3zSuBXgjuXxhXMkINjKEIZAriAKtxADepAQMATvMCrp71n7837mLWuePOZI/gD7/MHsd6RJQ=</latexit> <latexit sha1_base64="A9AjNuW7D5Ucq28MGn7dBUBMLmI=">AB8HicbVDLSgNBEOyNrxhfUY9ehgQhIoRdL3oMevFmBPOQ7BJmJ5NkyMzsMjMrLEu+Qg8eFPHq53jL3zh5HDSxoKGo6qa7K4w508Z1J05ubX1jcyu/XdjZ3ds/KB4eNXWUKEIbJOKRaodYU84kbRhmOG3HimIRctoKRzdTv/VElWaRfDBpTAOB5L1GcHGSo93FZ9HAyTPusWyW3VnQKvEW5ByreSfv0xqab1b/PZ7EUkElYZwrHXHc2MTZFgZRjgdF/xE0xiTER7QjqUSC6qDbHbwGJ1apYf6kbIlDZqpvycyLRORWg7BTZDvexNxf+8TmL6V0HGZJwYKsl8UT/hyERo+j3qMUWJ4aklmChmb0VkiBUmxmZUsCF4y+vkuZF1XOr3r1N4xrmyMJlKACHlxCDW6hDg0gIOAZ3uDdUc6r8+F8zltzmLmGP7A+foBu0iSqw=</latexit> <latexit sha1_base64="A9AjNuW7D5Ucq28MGn7dBUBMLmI=">AB8HicbVDLSgNBEOyNrxhfUY9ehgQhIoRdL3oMevFmBPOQ7BJmJ5NkyMzsMjMrLEu+Qg8eFPHq53jL3zh5HDSxoKGo6qa7K4w508Z1J05ubX1jcyu/XdjZ3ds/KB4eNXWUKEIbJOKRaodYU84kbRhmOG3HimIRctoKRzdTv/VElWaRfDBpTAOB5L1GcHGSo93FZ9HAyTPusWyW3VnQKvEW5ByreSfv0xqab1b/PZ7EUkElYZwrHXHc2MTZFgZRjgdF/xE0xiTER7QjqUSC6qDbHbwGJ1apYf6kbIlDZqpvycyLRORWg7BTZDvexNxf+8TmL6V0HGZJwYKsl8UT/hyERo+j3qMUWJ4aklmChmb0VkiBUmxmZUsCF4y+vkuZF1XOr3r1N4xrmyMJlKACHlxCDW6hDg0gIOAZ3uDdUc6r8+F8zltzmLmGP7A+foBu0iSqw=</latexit> <latexit sha1_base64="yULcGfPqvH4yvZXAswBK59K6KU=">AB8HicbVDLSgNBEOyNrxhfUY9eBoMQL2HXix6DXrwZwTwkWcLsZDYZMo9lZlYIS7CiwdFvPo53vwbJ8keNLGgoajqprsrSjgz1ve/vcLa+sbmVnG7tLO7t39QPjxqGZVqQptEcaU7ETaUM0mblO4mWESctqPxzcxvP1FtmJIPdpLQUOChZDEj2Drp8a7a42qI5Hm/XPFr/hxolQ5qUCORr/81RsokgoqLeHYmG7gJzbMsLaMcDot9VJDE0zGeEi7jkosqAmz+cFTdOaUAYqVdiUtmqu/JzIsjJmIyHUKbEdm2ZuJ/3nd1MZXYcZkloqyWJRnHJkFZp9jwZMU2L5xBFMNHO3IjLCGhPrMiq5EILl1dJ6IW+LXg3q/Ur/M4inACp1CFAC6hDrfQgCYQEPAMr/Dmae/Fe/c+Fq0FL585hj/wPn8An1+PnA=</latexit> <latexit sha1_base64="thSTtFLDoPeMBhXVDBCZsdC5kIE=">AB63icbVA9SwNBEJ2LXzF+RS1tlgQhIoQ7Gy2DNnZGMB+QHGFvs5cs2d07dveEcOQv2FgoYmHjH7Lz37h3SaGJDwYe780wMy+IOdPGdb+dwtr6xuZWcbu0s7u3f1A+PGrKFGEtkjEI9UNsKacSdoyzHDajRXFIuC0E0xuMr/zSJVmkXw05j6Ao8kCxnBJpPuavJsUK6dTcHWiXeglQblf75BwA0B+Wv/jAiaDSEI617nlubPwUK8MIp7NSP9E0xmSCR7RnqcSCaj/Nb52hU6sMURgpW9KgXP09kWKh9VQEtlNgM9bLXib+5/USE175KZNxYqgk80VhwpGJUPY4GjJFieFTSzBRzN6KyBgrTIyNp2RD8JZfXiXti7rn1r17m8Y1zFGE6hADTy4hAbcQhNaQGAMT/ACr45wnp0353eWnAWM8fwB87nD0tLjzU=</latexit> <latexit sha1_base64="m19+DXNT7/N3ErQr0l/iSNjr9GI=">AB63icbVBNSwMxEJ31s9avqkcvoUWoCGXix6LXrxZwX5Au5Rsm1Dk+ySZIVl6V/woqCIV/+Qt/4bs20P2vpg4PHeDPzgpgzbVx36qytb2xubRd2irt7+weHpaPjlo4SRWiTRDxSnQBrypmkTcMp51YUSwCTtvB+Db3209UaRbJR5PG1Bd4KFnICDa5dF+V5/1Sxa25M6BV4i1IpV7uXbxO62mjX/ruDSKSCoN4VjrufGxs+wMoxwOin2Ek1jTMZ4SLuWSiyo9rPZrRN0ZpUBCiNlSxo0U39PZFhonYrAdgpsRnrZy8X/vG5iwms/YzJODJVkvihMODIRyh9HA6YoMTy1BPF7K2IjLDCxNh4ijYEb/nlVdK6rHluzXuwadzAHAU4hTJUwYMrqMdNKAJBEbwDG/w7gjnxflwPueta85i5gT+wPn6AVS1kLs=</latexit> <latexit sha1_base64="m19+DXNT7/N3ErQr0l/iSNjr9GI=">AB63icbVBNSwMxEJ31s9avqkcvoUWoCGXix6LXrxZwX5Au5Rsm1Dk+ySZIVl6V/woqCIV/+Qt/4bs20P2vpg4PHeDPzgpgzbVx36qytb2xubRd2irt7+weHpaPjlo4SRWiTRDxSnQBrypmkTcMp51YUSwCTtvB+Db3209UaRbJR5PG1Bd4KFnICDa5dF+V5/1Sxa25M6BV4i1IpV7uXbxO62mjX/ruDSKSCoN4VjrufGxs+wMoxwOin2Ek1jTMZ4SLuWSiyo9rPZrRN0ZpUBCiNlSxo0U39PZFhonYrAdgpsRnrZy8X/vG5iwms/YzJODJVkvihMODIRyh9HA6YoMTy1BPF7K2IjLDCxNh4ijYEb/nlVdK6rHluzXuwadzAHAU4hTJUwYMrqMdNKAJBEbwDG/w7gjnxflwPueta85i5gT+wPn6AVS1kLs=</latexit> <latexit sha1_base64="tndW4QtEoTbzjT1XjQapIifJ9wA=">AB63icbVBNSwMxEJ31s9avqkcvwSLUS8l60WPRizcr2A9ol5JNs21okl2SrFCW/gUvHhTx6h/y5r8x2+5BWx8MPN6bYWZemAhuLMbf3tr6xubWdmnvLu3f3BYOTpumzjVlLVoLGLdDYlhgivWstwK1k0IzIUrBNObnO/8S04bF6tNOEBZKMFI84JTaX7mvqYlCp4jqeA60SvyBVKNAcVL76w5imkilLBTGm5+PEBhnRlPBZuV+alhC6ISMWM9RSQzQTa/dYbOnTJEUaxdKYvm6u+JjEhjpjJ0nZLYsVn2cvE/r5fa6DrIuEpSyxRdLIpSgWyM8sfRkGtGrZg6Qqjm7lZEx0QTal08ZReCv/zyKmlf1n1c9x9wtXFTxFGCUziDGvhwBQ24gya0gMIYnuEV3jzpvXjv3seidc0rZk7gD7zPHzjMjaw=</latexit> <latexit sha1_base64="fxdqK8mpvWTQC6DjJIhxecHR+Nc=">AB7XicbVDLSgNBEOyNrxhfUY9ehgQhIoTdXOIx6MWbEcwDkjXMTmaTMbMzy8ysEJb8gxcPiog3/8ebf+PkcdDEgoaiqpvuriDmTBvX/XYya+sbm1vZ7dzO7t7+Qf7wqKloghtEMmlagdYU84EbRhmOG3HiuIo4LQVjK6mfuRKs2kuDPjmPoRHgWMoKNlZo3JXFfOevli27ZnQGtEm9BirVC9/wDAOq9/Fe3L0kSUWEIx1p3PDc2foqVYTSa6baBpjMsID2rFU4IhqP51dO0GnVumjUCpbwqCZ+nsixZHW4yiwnRE2Q73sTcX/vE5iwgs/ZSJODBVkvihMODISTV9HfaYoMXxsCSaK2VsRGWKFibEB5WwI3vLq6RZKXtu2bu1aVzCHFk4gQKUwIMq1OAa6tAg/wBC/w6kjn2Xlz3uetGWcxcwx/4Hz+AHIoj9k=</latexit> <latexit sha1_base64="fUWbN3kzsxJRSRE8/MVNcuo0i0=">AB7XicbVDLSgNBEOyNrxhfUY9ehgQhIoTdXPQY9OLNCOYByRpmJ7PJmNmZWZWCEv+wYMeFPHq/3jL3zh5HDSxoKGo6qa7K4g508Z1J05mbX1jcyu7ndvZ3ds/yB8eNbRMFKF1IrlUrQBrypmgdcMp61YURwFnDaD4fXUbz5RpZkU92YUz/CfcFCRrCxUuO2JB4qZ9180S27M6BV4i1IsVronL9MqNaN/d6UmSRFQYwrHWbc+NjZ9iZRjhdJzrJrGmAxn7YtFTi2k9n147RqV6KJTKljBopv6eSHGk9SgKbGeEzUAve1PxP6+dmPDST5mIE0MFmS8KE46MRNPXUY8pSgwfWYKJYvZWRAZYWJsQDkbgrf8ipVMqeW/bubBpXMEcWTqAJfDgAqpwAzWoA4FHeIY3eHek8+p8OJ/z1oyzmDmGP3C+fgB7kpFf</latexit> <latexit sha1_base64="fUWbN3kzsxJRSRE8/MVNcuo0i0=">AB7XicbVDLSgNBEOyNrxhfUY9ehgQhIoTdXPQY9OLNCOYByRpmJ7PJmNmZWZWCEv+wYMeFPHq/3jL3zh5HDSxoKGo6qa7K4g508Z1J05mbX1jcyu7ndvZ3ds/yB8eNbRMFKF1IrlUrQBrypmgdcMp61YURwFnDaD4fXUbz5RpZkU92YUz/CfcFCRrCxUuO2JB4qZ9180S27M6BV4i1IsVronL9MqNaN/d6UmSRFQYwrHWbc+NjZ9iZRjhdJzrJrGmAxn7YtFTi2k9n147RqV6KJTKljBopv6eSHGk9SgKbGeEzUAve1PxP6+dmPDST5mIE0MFmS8KE46MRNPXUY8pSgwfWYKJYvZWRAZYWJsQDkbgrf8ipVMqeW/bubBpXMEcWTqAJfDgAqpwAzWoA4FHeIY3eHek8+p8OJ/z1oyzmDmGP3C+fgB7kpFf</latexit> <latexit sha1_base64="A9q8Orv9CfeHWIoueTctuCfpw=">AB7XicbVA9T8MwEL2Ur1K+CowsFhVSWaqkC4wVLGwUiX5Ibagc12lNHTuyHaQq6n9gYQAhVv4PG/8Gp80ALU86em9O93dC2LOtHdb6ewtr6xuVXcLu3s7u0flA+P2lomitAWkVyqboA15UzQlmG026sKI4CTjvB5DrzO09UaSbFvZnG1I/wSLCQEWys1L6tiof6+aBcWvuHGiVeDmpQI7moPzVH0qSRFQYwrHWPc+NjZ9iZRjhdFbqJ5rGmEzwiPYsFTi2k/n187QmVWGKJTKljBorv6eSHGk9TQKbGeEzVgve5n4n9dLTHjp0zEiaGCLBaFCUdGoux1NGSKEsOnlmCimL0VkTFWmBgbUMmG4C2/vEra9Zrn1rw7t9K4yuMowgmcQhU8uIAG3EATWkDgEZ7hFd4c6bw4787HorXg5DPH8AfO5w9fqY5Q</latexit> <latexit sha1_base64="7k1r5ohl/h4Qdp1yWFRDybMUWwc=">AB7XicbVDLSgNBEOyNrxhfqx69DAlCRAi7XvQY9OLNCOYByRpmJ7PJmNmZWZWCEv+wYsHRcSb/+PNv3HyOGhiQUNR1U13V5hwpo3nfTu5ldW19Y38ZmFre2d3z90/aGiZKkLrRHKpWiHWlDNB64YZTluJojgOW2Gw6uJ3ykSjMp7swoUGM+4JFjGBjpcZNObwXJ1235FW8KdAy8ekVC12Tj8AoNZ1vzo9SdKYCkM41rte4kJMqwMI5yOC51U0wSTIe7TtqUCx1QH2fTaMTq2Sg9FUtkSBk3V3xMZjrUexaHtjLEZ6EVvIv7ntVMTXQZE0lqCzRVHKkZFo8jrqMUWJ4SNLMFHM3orIACtMjA2oYEPwF19eJo2ziu9V/FubxiXMkIcjKEIZfDiHKlxDepA4AGe4AVeHek8O2/O+6w158xnDuEPnM8fuwCQCQ=</latexit> <latexit sha1_base64="pLelT/3n1ernarEdpl+dJC4Vvfo=">AB7XicbVDLSgNBEOyNrxhfUY9ehgQhIoRdL3oMevFmBPOAZA2zk9lkdHZmZkVliX/4EPinj1f7zlb5w8DpY0FBUdPdFcScaeO6Yye3srq2vpHfLGxt7+zuFfcPmlomitAGkVyqdoA15UzQhmG03asKI4CTlvB49XEbz1RpZkUdyaNqR/hgWAhI9hYqXlTCe7FSa9YdqvuFGiZeHNSrpW6py/jWlrvFb+7fUmSiApDONa647mx8TOsDCOcjgrdRNMYk0c8oB1LBY6o9rPptSN0bJU+CqWyJQyaqr8nMhxpnUaB7YywGepFbyL+53USE174GRNxYqgs0VhwpGRaPI6jNFieGpJZgoZm9FZIgVJsYGVLAheIsvL5PmWdVzq96tTeMSZsjDEZSgAh6cQw2uoQ4NIPAz/AG7450Xp0P53PWmnPmM4fwB87XD8RqkY8=</latexit> <latexit sha1_base64="pLelT/3n1ernarEdpl+dJC4Vvfo=">AB7XicbVDLSgNBEOyNrxhfUY9ehgQhIoRdL3oMevFmBPOAZA2zk9lkdHZmZkVliX/4EPinj1f7zlb5w8DpY0FBUdPdFcScaeO6Yye3srq2vpHfLGxt7+zuFfcPmlomitAGkVyqdoA15UzQhmG03asKI4CTlvB49XEbz1RpZkUdyaNqR/hgWAhI9hYqXlTCe7FSa9YdqvuFGiZeHNSrpW6py/jWlrvFb+7fUmSiApDONa647mx8TOsDCOcjgrdRNMYk0c8oB1LBY6o9rPptSN0bJU+CqWyJQyaqr8nMhxpnUaB7YywGepFbyL+53USE174GRNxYqgs0VhwpGRaPI6jNFieGpJZgoZm9FZIgVJsYGVLAheIsvL5PmWdVzq96tTeMSZsjDEZSgAh6cQw2uoQ4NIPAz/AG7450Xp0P53PWmnPmM4fwB87XD8RqkY8=</latexit> <latexit sha1_base64="lh+QN5vYVCNTaViZri/ieStTftY=">AB7XicbVBNSwMxEJ31s9avqkcvwSLUS8l60WPRizcr2A9o15JNs21sNlmSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvTAQ3FuNvb2V1bX1js7BV3N7Z3dsvHRw2jUo1ZQ2qhNLtkBgmuGQNy61g7UQzEoeCtcLR9dRvPTFtuJL3dpywICYDySNOiXVS87YSPsizXqmMq3gGtEz8nJQhR71X+ur2FU1jJi0VxJiOjxMbZERbTgWbFLupYQmhIzJgHUcliZkJstm1E3TqlD6KlHYlLZqpvycyEhszjkPXGRM7NIveVPzP6Q2ugwyLpPUMkni6JUIKvQ9HXU5pRK8aOEKq5uxXRIdGEWhdQ0YXgL768TJrnVR9X/Ttcrl3lcRTgGE6gAj5cQA1uoA4NoPAIz/AKb57yXrx372PeuLlM0fwB97nD6iBjoA=</latexit> <latexit sha1_base64="8g+c8Fm4V71cuVu27a2oG6AMhA=">AB8nicbVDLSsNAFL3xWeur6tJNaBEqQknc6LoxmWFviCJZTKdtEMnkzBzI5TQz3Djwgdu/Rp3/o3Tx0JbD1w4nHMv94TpoJrdJxva219Y3Nru7BT3N3bPzgsHR23dZIpylo0EYnqhkQzwSVrIUfBuqliJA4F64Sj26nfeWRK80Q2cZyICYDySNOCRrJ85tDhqQaPsjzXqni1JwZ7FXiLkilXvYv3gCg0St9+f2EZjGTSAXR2nOdFIOcKORUsEnRzRLCR2RAfMlSRmOshnJ0/sM6P07ShRpiTaM/X3RE5ircdxaDpjgkO97E3F/zwvw+g6yLlM2SzhdFmbAxsaf/232uGEUxNoRQxc2tNh0SRSialIomBHf5VXSvqy5Ts29N2ncwBwFOIUyVMGFK6jDHTSgBRQSeIXeLXQerberY9565q1mDmBP7A+fwCQI5I+</latexit> <latexit sha1_base64="1j5ev6ilyFSc2YovtpREqyQAMPs=">AB8nicbVDLSsNAFJ3UV62vqks3Q4tQEUriRpdFNy4r9AVJLJPpB06mYSZGyGE/oVuXCji1q9x179x+lho64ELh3Pu5d57gkRwDbY9tQobm1vbO8Xd0t7+weFR+fiko+NUdamsYhVLyCaCS5ZGzgI1ksUI1EgWDcY38387hNTmseyBVnC/IgMJQ85JWAk12uNGJBa8Cgv+uWqXbfnwOvEWZJqo+JdPk8bWbNf/vYGMU0jJoEKorXr2An4OVHAqWCTkpdqlhA6JkPmGipJxLSfz0+e4HOjDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlEomBGf15XSuao7dt15MGncogWK6AxVUA056Bo10D1qojaiKEYv6A29W2C9Wh/W56K1YC1nTtEfWF8/mY2TxA=</latexit> <latexit sha1_base64="1j5ev6ilyFSc2YovtpREqyQAMPs=">AB8nicbVDLSsNAFJ3UV62vqks3Q4tQEUriRpdFNy4r9AVJLJPpB06mYSZGyGE/oVuXCji1q9x179x+lho64ELh3Pu5d57gkRwDbY9tQobm1vbO8Xd0t7+weFR+fiko+NUdamsYhVLyCaCS5ZGzgI1ksUI1EgWDcY38387hNTmseyBVnC/IgMJQ85JWAk12uNGJBa8Cgv+uWqXbfnwOvEWZJqo+JdPk8bWbNf/vYGMU0jJoEKorXr2An4OVHAqWCTkpdqlhA6JkPmGipJxLSfz0+e4HOjDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlEomBGf15XSuao7dt15MGncogWK6AxVUA056Bo10D1qojaiKEYv6A29W2C9Wh/W56K1YC1nTtEfWF8/mY2TxA=</latexit> <latexit sha1_base64="a8UYW0Su6kS0T6+bYsH29pm5wpg=">AB8nicbVA9SwNBEJ2LXzF+RS1tFoMQm3Bno2XQxjJCviA5w95mL1myt3fszgnhyM+wsVDE1l9j579xk1yhiQ8GHu/NMDMvSKQw6LrfTmFjc2t7p7hb2ts/ODwqH5+0TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wSTu7nfeLaiFg1cZpwP6IjJULBKFqp12+OdJq8KguB+WKW3MXIOvEy0kFcjQG5a/+MGZpxBUySY3peW6CfkY1Cib5rNRPDU8om9AR71mqaMSNny1OnpELqwxJGtbCslC/T2R0ciYaRTYzoji2Kx6c/E/r5dieONnQiUpcsWi8JUEozJ/H8yFJozlFNLKNPC3krYmGrK0KZUsiF4qy+vk/ZVzXNr3oNbqd/mcRThDM6hCh5cQx3uoQEtYBDM7zCm4POi/PufCxbC04+cwp/4Hz+AH2kLU=</latexit> <latexit sha1_base64="2Nu7S0mxhr8c+CikqDKY8Bf0+5M=">AB8nicbVDJSgNBEK2JW4xb1KOXIUGICGEmFz0GvXiMkA0mY+jp9CRNenqG7hohDPkMLx5c8OrXePNv7CwHTXxQ8Hiviqp6QSK4Rsf5tnIbm1vbO/ndwt7+weFR8fikreNUdaisYhVNyCaCS5ZCzkK1k0UI1EgWCcY3878ziNTmseyiZOE+REZSh5yStBIXq85Ykgq8qF20S+Wnaozh71O3CUp10u9yzcAaPSLX71BTNOISaSCaO25ToJ+RhRyKti0Es1SwgdkyHzDJUkYtrP5idP7XOjDOwVqYk2nP190RGIq0nUWA6I4IjverNxP8L8Xw2s+4TFJki4WhamwMbZn/9sDrhFMTGEUMXNrTYdEUompQKJgR39eV10q5VXafq3ps0bmCBPJxBCSrgwhXU4Q4a0AIKMTzBC7xaD1b79bHojVnLWdO4Q+szx9HS5IO</latexit> <latexit sha1_base64="JtNKk2OifBCJsu/nv8Q9CeO7Jno=">AB8nicbVDLSsNAFJ34rPVdelmaBEqQkm60WXRjcsKfUESy2Q6aYdOJmHmRgihf6EbF4q49Wvc9W+cPhbaeuDC4Zx7ufeIBFcg21PrY3Nre2d3cJecf/g8Oi4dHLa0XGqKGvTWMSqFxDNBJesDRwE6yWKkSgQrBuM72Z+94kpzWPZgixhfkSGkoecEjCS67VGDEhVPtYv+6WKXbPnwOvEWZJKo+xdPU8bWbNf+vYGMU0jJoEKorXr2An4OVHAqWCTopdqlhA6JkPmGipJxLSfz0+e4AujDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlIomBGf15XSqdcu+Y8mDRu0QIFdI7KqIocdI0a6B41URtRFKMX9IbeLbBerQ/rc9G6YS1nztAfWF8/ULWTlA=</latexit> <latexit sha1_base64="JtNKk2OifBCJsu/nv8Q9CeO7Jno=">AB8nicbVDLSsNAFJ34rPVdelmaBEqQkm60WXRjcsKfUESy2Q6aYdOJmHmRgihf6EbF4q49Wvc9W+cPhbaeuDC4Zx7ufeIBFcg21PrY3Nre2d3cJecf/g8Oi4dHLa0XGqKGvTWMSqFxDNBJesDRwE6yWKkSgQrBuM72Z+94kpzWPZgixhfkSGkoecEjCS67VGDEhVPtYv+6WKXbPnwOvEWZJKo+xdPU8bWbNf+vYGMU0jJoEKorXr2An4OVHAqWCTopdqlhA6JkPmGipJxLSfz0+e4AujDHAYK1MS8Fz9PZGTSOsCkxnRGCkV72Z+J/nphDe+DmXSQpM0sWiMBUYjz7Hw+4YhREZgihiptbMR0RSiYlIomBGf15XSqdcu+Y8mDRu0QIFdI7KqIocdI0a6B41URtRFKMX9IbeLbBerQ/rc9G6YS1nztAfWF8/ULWTlA=</latexit> <latexit sha1_base64="WALeiKi4PWgAumKikdV7B6ZkfQ=">AB8nicbVA9SwNBEJ3zM8avqKXNYRBiE+7SaBm0sYyQL7icYW+zSZbs7R67c0I48jNsLBSx9dfY+W/cJFdo4oOBx3szMyLEsENet63s7G5tb2zW9gr7h8cHh2XTk7bRqWashZVQuluRAwTXLIWchSsm2hG4kiwTjS5m/udJ6YNV7KJ04SFMRlJPuSUoJWCXnPMkFTkY+2qXyp7VW8Bd534OSlDjka/9NUbKJrGTCIVxJjA9xIM6KRU8FmxV5qWELohIxYKkMTNhtjh5l5aZeAOlbYl0V2ovycyEhszjSPbGRMcm1VvLv7nBSkOb8KMyRFJuly0TAVLip3/r874JpRFNLCNXc3urSMdGEok2paEPwV19eJ+1a1feq/oNXrt/mcRTgHC6gAj5cQx3uoQEtoKDgGV7hzUHnxXl3PpatG04+cwZ/4Hz+ADTMkIU=</latexit> <latexit sha1_base64="jMSL2OINSwlIU1DFc5CHfF+CiMs=">AB8HicbVDLSgNBEOz1GeMr6tHLkiBEhLDrRY9BLx4j5CXZJcxOsmQmdlZlYIS7CiwdFvfo53vwbJ4+DJhY0FXdHdFCWfaeN63s7a+sbm1ndvJ7+7tHxwWjo6bOk4VxQaNeazaEdHImcSGYZjO1FIRMSxFY1up37rEZVmsaybcYKhIAPJ+owSY6WHoD5EQ8ryvFsoeRVvBneV+AtSqhaDi3cAqHULX0EvpqlAaSgnWnd8LzFhRpRhlOMkH6QaE0JHZIAdSyURqMNsdvDEPbNKz+3HypY07kz9PZERofVYRLZTEDPUy95U/M/rpKZ/HWZMJqlBSeL+il3TexOv3d7TCE1fGwJoYrZW106JIpQYzPK2xD85ZdXSfOy4nsV/96mcQNz5OAUilAGH6gCndQgwZQEPAEL/DqKOfZeXM+5q1rzmLmBP7A+fwBHFSRag=</latexit> <latexit sha1_base64="gdLfQI6euH6g8XZvgkCxRKZJspI=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF20psMmZldZmaFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7woQzbVx36uQ2Nre2d/K7hb39g8Oj4vFJS8epotCkMY9VJyQaOJPQNMxw6CQKiAg5tMPR3cxvP4HSLJYNM04gEGQgWcQoMVZ69BtDMKQiL3rFslt158DrxFuScq3kX75Ma+N6r/jt92OaCpCGcqJ13MTE2REGUY5TAp+qiEhdEQG0LVUEgE6yOYHT/C5Vfo4ipUtafBc/T2REaH1WIS2UxAz1KveTPzP6YmugkyJpPUgKSLRVHKsYnx7HvcZwqo4WNLCFXM3orpkChCjc2oYEPwVl9eJ62rqudWvQebxi1aI/OUAlVkIeuUQ3dozpqIoEekZv6N1Rzqvz4XwuWnPOcuYU/YHz9QMlvpLw</latexit> <latexit sha1_base64="gdLfQI6euH6g8XZvgkCxRKZJspI=">AB8HicbVDLSgNBEJyNrxhfUY9ehgQhIoRdL3oMevEYIS/JLmF20psMmZldZmaFsOQr9OBEa9+jrf8jZPHQRMLGoqbrq7woQzbVx36uQ2Nre2d/K7hb39g8Oj4vFJS8epotCkMY9VJyQaOJPQNMxw6CQKiAg5tMPR3cxvP4HSLJYNM04gEGQgWcQoMVZ69BtDMKQiL3rFslt158DrxFuScq3kX75Ma+N6r/jt92OaCpCGcqJ13MTE2REGUY5TAp+qiEhdEQG0LVUEgE6yOYHT/C5Vfo4ipUtafBc/T2REaH1WIS2UxAz1KveTPzP6YmugkyJpPUgKSLRVHKsYnx7HvcZwqo4WNLCFXM3orpkChCjc2oYEPwVl9eJ62rqudWvQebxi1aI/OUAlVkIeuUQ3dozpqIoEekZv6N1Rzqvz4XwuWnPOcuYU/YHz9QMlvpLw</latexit> <latexit sha1_base64="oB6jlXGIkO9IKqkRvd0Cu68+Q6c=">AB8HicbVA9SwNBEJ2LXzF+RS1tDoMQm3Bno2XQxjJCviQ5wt5mL1myu3fszgnhyK+wsVDE1p9j579xk1yhiQ8GHu/NMDMvTAQ36HnfTmFjc2t7p7hb2ts/ODwqH5+0TZxqylo0FrHuhsQwRVrIUfBuolmRIaCdcLJ3dzvPDFteKyaOE1YIMlI8YhTglZ67DfHDElVXQ7KFa/mLeCuEz8nFcjRGJS/+sOYpIpIY0/O9BIOMaORUsFmpnxqWEDohI9azVBHJTJAtDp65F1YZulGsbSl0F+rviYxIY6YytJ2S4NisenPxP6+XYnQTZFwlKTJFl4uiVLgYu/Pv3SHXjKYWkKo5vZWl46JhRtRiUbgr/68jpX9V8r+Y/eJX6bR5HEc7gHKrgwzXU4R4a0AIKEp7hFd4c7bw4787HsrXg5DOn8AfO5w8J1Y/h</latexit> <latexit sha1_base64="DK/wbEiUiPA2hr2keU+G1aLr5M=">AB9XicbVDLSgNBEOyNrxhfUY9ehgQhIoRdL3oMevEYIS/IrmF2MpsMmZ1ZmaVsOQ/vAgq4tV/8ebfOHkcNLGgoajqprsrTDjTxnW/ndza+sbmVn67sLO7t39QPDxqaZkqQptEcqk6IdaUM0GbhlO4miOA45bYejm6nfqBKMykaZpzQIMYDwSJGsLHSvd8YUoMrPpcDJM56xbJbdWdAq8RbkHKt5J+/AEC9V/zy+5KkMRWGcKx13MTE2RYGUY4nRT8VNMEkxEe0K6lAsdUB9ns6gk6tUofRVLZEgbN1N8TGY61Hseh7YyxGeplbyr+53VTE10FGRNJaqg80VRypGRaBoB6jNFieFjSzBRzN6KyBArTIwNqmBD8JZfXiWti6rnVr07m8Y1zJGHEyhBTy4hBrcQh2aQEDBE7zCm/PoPDvzse8NecsZo7hD5zPH40ok1o=</latexit> <latexit sha1_base64="RHa1m03WbZ7W9E8ltOzuGdhNgY=">AB9XicbVDLSgNBEOz1GeMr6tHLkCBEhLDrRY9BLx4j5AXZNcxOZpMhszPLzKyhPyFBy8eFPHqv3jL3zh5HDSxoKGo6qa7K0w408Z1J87a+sbm1nZuJ7+7t39wWDg6bmqZKkIbRHKp2iHWlDNBG4YZTtuJojgOW2Fw9up3qkSjMp6iZLaBDjvmARI9hY6cGvD6jBZ/LPhLn3ULJrbgzoFXiLUipWvQvnifVrNYtfPs9SdKYCkM41rjuYkJRlgZRjgd5/1U0wSTIe7TjqUCx1QHo9nVY3RmlR6KpLIlDJqpvydGONY6i0PbGWMz0MveVPzP6Qmug5GTCSpoYLMF0UpR0aiaQSoxQlhmeWYKYvRWRAVaYGBtU3obgLb+8SpqXFc+tePc2jRuYIwenUIQyeHAFVbiDGjSAgIXeIN358l5dT6cz3nrmrOYOYE/cL5+AJaSlOA=</latexit> <latexit sha1_base64="RHa1m03WbZ7W9E8ltOzuGdhNgY=">AB9XicbVDLSgNBEOz1GeMr6tHLkCBEhLDrRY9BLx4j5AXZNcxOZpMhszPLzKyhPyFBy8eFPHqv3jL3zh5HDSxoKGo6qa7K0w408Z1J87a+sbm1nZuJ7+7t39wWDg6bmqZKkIbRHKp2iHWlDNBG4YZTtuJojgOW2Fw9up3qkSjMp6iZLaBDjvmARI9hY6cGvD6jBZ/LPhLn3ULJrbgzoFXiLUipWvQvnifVrNYtfPs9SdKYCkM41rjuYkJRlgZRjgd5/1U0wSTIe7TjqUCx1QHo9nVY3RmlR6KpLIlDJqpvydGONY6i0PbGWMz0MveVPzP6Qmug5GTCSpoYLMF0UpR0aiaQSoxQlhmeWYKYvRWRAVaYGBtU3obgLb+8SpqXFc+tePc2jRuYIwenUIQyeHAFVbiDGjSAgIXeIN358l5dT6cz3nrmrOYOYE/cL5+AJaSlOA=</latexit> <latexit 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

Recommend

Efficiency Announcements Measuring Efficiency Recursive Computation of the Fibonacci Sequence

Efficiency Announcements Measuring Efficiency Recursive Computation of the Fibonacci Sequence

Efficiency Announcements Measuring Efficiency Recursive Computation of the Fibonacci Sequence def fib (n): Our first example of tree recursion: if n == 0 : return 0 elif n == 1 : fib(5) return 1 else : return fib(n- 2 ) + fib(n- 1 ) fib(3)

338 views • 20 slides
Announcements Efficiency Recursive Computation of the Fibonacci Sequence Our first example of

Announcements Efficiency Recursive Computation of the Fibonacci Sequence Our first example of

Announcements Efficiency Recursive Computation of the Fibonacci Sequence Our first example of tree recursion: def fib (n): if n == 0 : return 0 elif n == 1 : fib(5) return 1 else : return fib(n- 2 ) + fib(n- 1 ) Measuring Efficiency fib(3)

302 views • 3 slides
The Challenges of Measuring Transportation Efficiency Jim Sullivan, PE, MS Research Analyst

The Challenges of Measuring Transportation Efficiency Jim Sullivan, PE, MS Research Analyst

The Challenges of Measuring Transportation Efficiency Jim Sullivan, PE, MS Research Analyst March 12, 2010 The Challenges of Measuring Transportation Efficiency Motivation Literature Review Definition of Transportation Efficiency

537 views • 12 slides
Efficiency and Growth T. Gutowski 2.83 and 2.813 1 Efficiency and Growth Can efficiency

Efficiency and Growth T. Gutowski 2.83 and 2.813 1 Efficiency and Growth Can efficiency

Efficiency and Growth T. Gutowski 2.83 and 2.813 1 Efficiency and Growth Can efficiency improvements out run growth? Historical review of 10 activities including materials and electricity production and consumer services (Dahmus &amp;

647 views • 62 slides
Computational Logic Efficiency Issues in Prolog 1 Efficiency In general, efficiency

Computational Logic Efficiency Issues in Prolog 1 Efficiency In general, efficiency

Computational Logic Efficiency Issues in Prolog 1 Efficiency In general, efficiency savings: Not only time (number of unifications, reduction steps, LIPS, etc.) Also memory General advice: Use the best algorithms Use

196 views • 19 slides
Improving Algorithmic Efficiency 15-112 Big Ideas Efficiency in Algorithms Now that we know

Improving Algorithmic Efficiency 15-112 Big Ideas Efficiency in Algorithms Now that we know

Improving Algorithmic Efficiency 15-112 Big Ideas Efficiency in Algorithms Now that we know how to calculate the efficiency of a program, we need to consider efficiency during algorithm design . Small changes in a program can lead to big

470 views • 30 slides
Energy Efficiency Measuring for Success: Coordination, Collaboration, and Transparency

Energy Efficiency Measuring for Success: Coordination, Collaboration, and Transparency

Energy Efficiency Measuring for Success: Coordination, Collaboration, and Transparency Comments by Lucas Davis Associate Professor UC Berkeley Haas School of Business ldavis@haas.berkeley.edu March 6, 2013 Lucas Davis (UC Berkeley) Measuring

360 views • 8 slides
ECON 4100: Industrial Organization Lecture 2- Efficiency 1 Overview Efficiency and markets

ECON 4100: Industrial Organization Lecture 2- Efficiency 1 Overview Efficiency and markets

ECON 4100: Industrial Organization Lecture 2- Efficiency 1 Overview Efficiency and markets Pareto Efficiency Consumer Surplus and Producer Surplus revisited A non-surplus approach to efficiency 2 Efficiency and Surplus Can we

132 views • 12 slides
Measuring Heterogeneity and Efficiency of Firms within the same Industry: a C++ plugin for Stata

Measuring Heterogeneity and Efficiency of Firms within the same Industry: a C++ plugin for Stata

Measuring Heterogeneity and Efficiency of Firms within the same Industry: a C++ plugin for Stata for computing the Zonotope Marco Co Cococ occion oni * Pisa University, Marco Grazzi Catholic University of Sacro Cuore, Le Li Chuo University

377 views • 22 slides
Water Efficiency as a Partner to Energy Efficiency Mary Ann Dickinson President and CEO

Water Efficiency as a Partner to Energy Efficiency Mary Ann Dickinson President and CEO

Water Efficiency as a Partner to Energy Efficiency Mary Ann Dickinson President and CEO November 19, 2019 AWE: A Voice for Water Efficiency Our mission is to promote an efficient and sustainable water future A unique network and forum

331 views • 22 slides
T2 Track Reconstruction and Classification 09/07/10 Matti Leinonen 1 Overview of presentation

T2 Track Reconstruction and Classification 09/07/10 Matti Leinonen 1 Overview of presentation

T2 Track Reconstruction and Classification 09/07/10 Matti Leinonen 1 Overview of presentation Track reconstruction Measuring efficiency Changes made Efficiency plots Track classification Measuring efficiency Cuts used

250 views • 14 slides
Deep Efficiency Acquisition: All Fuels Steps in the Process 4. Dynamics of Deep Efficiency

Deep Efficiency Acquisition: All Fuels Steps in the Process 4. Dynamics of Deep Efficiency

Topics: 1. What is Happening in Rhode Island VEIC Role 2. Update on Efficiency Vermont Electric 3. Update on Efficiency Vermont Deep Efficiency Acquisition: All Fuels Steps in the Process 4. Dynamics of Deep Efficiency Presentation

428 views • 5 slides
Evaluating the effectiveness of ODEX in measuring true energy efficiency achievements: Case study

Evaluating the effectiveness of ODEX in measuring true energy efficiency achievements: Case study

University College Cork Department of Civil &amp; Environmental Engineering Sustainable Energy Research Group Evaluating the effectiveness of ODEX in measuring true energy efficiency achievements: Case study Irish Industry Caiman Cahill, Brian

741 views • 20 slides
Generic types Measuring Efficiency Minesweeper start-up Grader comments for JUnit and

Generic types Measuring Efficiency Minesweeper start-up Grader comments for JUnit and

Generic types Measuring Efficiency Minesweeper start-up Grader comments for JUnit and BigRational assignments should be in your repository. There seems to be a problem there with JUnit. Should be resolved this afternoon Details about

230 views • 10 slides
Opportunities &amp; Challenges for Energy Efficiency in 111(d) Energy Efficiency in EPA's Clean

Opportunities &amp; Challenges for Energy Efficiency in 111(d) Energy Efficiency in EPA's Clean

Opportunities &amp; Challenges for Energy Efficiency in 111(d) Energy Efficiency in EPA's Clean Power Plan: Using Building Block #4 to Set and Meet Emissions Goals November 4, 2014 Dian Grueneich (California PUC Commissioner Emeritus) Senior

321 views • 10 slides
Recursion Announcements Recursive Functions Recursive Functions 4 Recursive Functions

Recursion Announcements Recursive Functions Recursive Functions 4 Recursive Functions

Recursion Announcements Recursive Functions Recursive Functions 4 Recursive Functions Definition : A function is called recursive if the body of that function calls itself, either directly or indirectly 4 Recursive Functions Definition

1.11k views • 89 slides
61A Lecture 6 Announcements Recursive Functions Recursive Functions 4 Recursive Functions

61A Lecture 6 Announcements Recursive Functions Recursive Functions 4 Recursive Functions

61A Lecture 6 Announcements Recursive Functions Recursive Functions 4 Recursive Functions Definition: A function is called recursive if the body of that function calls itself, either directly or indirectly 4 Recursive Functions Definition:

1.08k views • 88 slides
Reconstruction Efficiency for Reconstruction Efficiency for NC radiative sample NC

Reconstruction Efficiency for Reconstruction Efficiency for NC radiative sample NC

Reconstruction Efficiency for Reconstruction Efficiency for NC radiative sample NC radiative sample Events Events per Relative Total 6.6e20 POT Efficiency Efficiency Total 7972 270 - - True nu vtx TPC 3546 120 44 % 44 %

316 views • 3 slides
Overview Bureau of Energy Efficiency 1 Bureau of Energy Efficiency, Ministry of Power,

Overview Bureau of Energy Efficiency 1 Bureau of Energy Efficiency, Ministry of Power,

Energy Efficiency Labels for Residential Buildings Overview Bureau of Energy Efficiency 1 Bureau of Energy Efficiency, Ministry of Power, Government of India Indias Electricity Scenario Electricity Consumption in 2016-17= 1066 BU TOTAL

768 views • 30 slides
Analysis of Informa.on - III Efficiency of Graphic The efficiency of a graphic is determined

Analysis of Informa.on - III Efficiency of Graphic The efficiency of a graphic is determined

Analysis of Informa.on - III Efficiency of Graphic The efficiency of a graphic is determined as: To obtain a correct and complete answer to a given ques.on, all other things being equal, one graphic requires a shorter observa.on .me than

716 views • 49 slides
1 How Does Efficiency Targeting in School Aid Affect Efficiency and Equity in School Spending and

1 How Does Efficiency Targeting in School Aid Affect Efficiency and Equity in School Spending and

1 How Does Efficiency Targeting in School Aid Affect Efficiency and Equity in School Spending and Performance Scores? Jay E. Ryu Professor of Public Policy and Administration Department of Political Science Bentley Annex 237, Ohio University

1.42k views • 86 slides
Predictability and Efficiency in Predictability and Efficiency in Wireless Sensor Networks

Predictability and Efficiency in Predictability and Efficiency in Wireless Sensor Networks

Predictability and Efficiency in Predictability and Efficiency in Wireless Sensor Networks Lothar Thiele Jan Beutel Federico Ferrari, Matthias Keller, Roman Lim, Andreas Meier, Clemens Moser, MustafaYuecel, Matthias Woehrle P

758 views • 63 slides
Maximizing the Efficiency Potential Maximizing the Efficiency Potential in New Hampshire N

Maximizing the Efficiency Potential Maximizing the Efficiency Potential in New Hampshire N

Maximizing the Efficiency Potential Maximizing the Efficiency Potential in New Hampshire N Natalie Hildt li Hild EES E Board Meeting November 20 2009 November 20, 2009 0 NORTHEAS NORTHEAS T ENERGY EFFICIENCY P T ENERGY EFFICIENCY P

149 views • 12 slides
Productivity, quality &amp; efficiency &amp; efficiency increase in high temperature processes

Productivity, quality &amp; efficiency &amp; efficiency increase in high temperature processes

Productivity, quality &amp; efficiency &amp; efficiency increase in high temperature processes Archivierungsangaben: K:\3pp\0PP\OE\Marketing\Akquifolien\Praesentation-englisch.ppt J. Demmel, 7th Nov. 2011 Contents 1. Introduction:

340 views • 16 slides

More recommend