# lecture04 - 4 Randomness and Simulation L Olson September 8...

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# 4 Randomness and Simulation L. Olson September 8, 2015 Department of Computer Science University of Illinois at Urbana-Champaign 1

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randomness reproducibility designing an experiment 2
the scientific method 3

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accuracy How do I classify my method? Goal: determine how the error | f ( x ) - p n ( x ) | behaves relative to n (and f ). Goal: determine how the cost of computing p n ( x ) behave relative to n (and f ). for f ( x ) = 1 1 - x we have p n = n X k =0 x k = 1 + x + x 2 + . . . so e n = | f ( x ) - p n ( x ) | Is e n 1 / n r ? Is e n 1 / n ? Is e n 1 / n !? 4 approx. algorithm 两边同乘x， 两式相减
timing mymethod() takes x seconds How long does it take in general? If the data input is of size n , how long should it take? n 2 ? n !? 10 n ? 5

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big-o How to measure the impact of n on algorithmic cost? O ( · ) Let g ( n ) be a function of n . Then define O ( g ( n )) = { f ( n ) | ∃ c , n 0 > 0 : 0 f ( n ) cg ( n ) , n n 0 } That is, f ( n ) ∈ O ( g ( n )) if there is a constant c such that 0 f ( n ) cg ( n ) is satisfied. assume non-negative functions (otherwise add | · | ) to the definitions f ( n ) ∈ O ( g ( n )) represents an asymptotic upper bound on f ( n ) up to a constant example: f ( n ) = 3 n + 2 log n + 8 n + 85 n 2 ∈ O ( n 2 ) 6 in O(n^3)
big-o (omicron) O ( · ) Let g ( n ) be a function of n . Then define O ( g ( n )) = { f ( n ) | ∃ c , n 0 > 0 : 0 f ( n ) cg ( n ) , n n 0 } That is, f ( n ) ∈ O ( g ( n )) if there is a constant c such that 0 f ( n ) cg ( n ) is satisfied. 7

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big-omega Ω( · ) Let g ( n ) be a function of n . Then define Ω( g ( n )) = { f ( n ) | ∃ c , n 0 > 0 : 0 cg ( n ) f ( n ) , n n 0 } That is, f ( n ) Ω( g ( n )) if there is a constant c such that 0 cg ( n ) f ( n ) is satisfied.
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