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**Unformatted text preview: **Outline 1 Compare the growth rate of functions 2 Limit Test 3 L’Hospital Rule 4 Stirling Formula 5 Summations 6 Integration Method 7 Solving Linear Recursive Equations c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 1 / 36 Compare the growth rate of functions c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 36 Compare the growth rate of functions We have two algorithms A 1 and A 2 for solving the same problem, with runtime functions T 1 ( n ) and T 2 ( n ) , respectively. Which algorithm is more efficient? c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 36 Compare the growth rate of functions We have two algorithms A 1 and A 2 for solving the same problem, with runtime functions T 1 ( n ) and T 2 ( n ) , respectively. Which algorithm is more efficient? We compare the growth rate of T 1 ( n ) and T 2 ( n ) . c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 36 Compare the growth rate of functions We have two algorithms A 1 and A 2 for solving the same problem, with runtime functions T 1 ( n ) and T 2 ( n ) , respectively. Which algorithm is more efficient? We compare the growth rate of T 1 ( n ) and T 2 ( n ) . If T 1 ( n ) = Θ( T 2 ( n )) , then the efficiency of the two algorithms are about the same ( when n is large ). c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 36 Compare the growth rate of functions We have two algorithms A 1 and A 2 for solving the same problem, with runtime functions T 1 ( n ) and T 2 ( n ) , respectively. Which algorithm is more efficient? We compare the growth rate of T 1 ( n ) and T 2 ( n ) . If T 1 ( n ) = Θ( T 2 ( n )) , then the efficiency of the two algorithms are about the same ( when n is large ). If T 1 ( n ) = o ( T 2 ( n )) , then the efficiency of the algorithm A 1 will be better than that of algorithm A 2 ( when n is large ). c Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 36 Compare the growth rate of functions We have two algorithms A 1 and A 2 for solving the same problem, with runtime functions T 1 ( n ) and T 2 ( n ) , respectively. Which algorithm is more efficient? We compare the growth rate of T 1 ( n ) and T 2 ( n ) . If T 1 ( n ) = Θ( T 2 ( n )) , then the efficiency of the two algorithms are about the same ( when n is large ). If T 1 ( n ) = o ( T 2 ( n )) , then the efficiency of the algorithm A 1 will be better than that of algorithm A 2 ( when n is large ). By using the definitions, we can directly show whether T 1 ( n ) = O ( T 2 ( n )) , or T 1 ( n ) = Ω( T 2 ( n )) . However, it is not easy to prove the relationship of two functions in this way....

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