Cost cost of computing a solution to a problem of a

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Cost § Cost of computing a solution to a problem of a particular size: (Work) Space How should this be measured? Time How should this be measured?
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§ How to measure time cost? § Want to measure the algorithm , not the implementation § i.e., independent of: § computer speed § programming language § compiler quality, etc. Cost
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Elementary Operation C c T op
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Efficiency § Efficiency is how cost grows with the difficulty of the instance § “Difficulty” means “size” of the instance § i.e., the number of parameters needed to completely characterize the instance § Size of a list, matrix, etc. § Example: we expect the cost of sorting a list of 100 numbers to be worse than sorting 10 numbers ) ( ) ( n C c n T op
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Orders of Growth § Suppose § How much longer will the algorithm run if we double the input size? 1 ( ) ( 1) 2 C n n n = - 2 2 2 1 2 1 2 1 ) 1 ( 2 1 ) ( n n n n n n C - = - = 4 2 1 ) 2 ( 2 1 ) ( ) 2 ( ) ( ) 2 ( 2 2 = n n n C c n C c n T n T op op
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Orders of Growth § Implementation constant cop is irrelevant § Also, the coefficient ½ on C(n) cancelled § Order of growth describes the functional form of C(n) up to a constant multiple as n goes to infinity 4 2 1 ) 2 ( 2 1 ) ( ) 2 ( ) ( ) 2 ( 2 2 = n n n C c n C c n T n T op op
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  • Spring '08
  • Jones,M
  • Computer Science, Computational complexity theory, Elementary Operations

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