# Exercise 1 1 do exercise 23 5 on page 37 in clrs

• Notes
• 5

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Exercise 1-1. Do Exercise 2.3-5 on page 37 in CLRS. Exercise 1-2. Do Exercise 2.3-7 on page 37 in CLRS.

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2 Handout 7: Problem Set 1 Exercise 1-3. Do Exercise 3.1-1 on page 50 in CLRS. Exercise 1-4. Do Exercise 4.1-6 on page 67 in CLRS. Exercise 1-5. Rank the following functions by order of growth; that is, find an arrangement g 1 , g 2 , . . . , g 30 of the functions satisfying g 1 = Ω( g 2 ) , g 2 = Ω( g 3 ) , . . . , g 29 = Ω( g 30 ) . Parti- tion your list into equivalence classes such that f ( n ) and g ( n ) are in the same class if and only if f ( n ) = Θ( g ( n )) . lg(lg * n ) ( 2) lg n n 2 n ! e n lg * ( n n ) 3 n n 3 lg 2 n lg( n !) n 2+sin n n 1 / lg n 1 lg * (lg n ) n · 2 n n lg lg n ln n ln ln n 3 lg n (lg n ) lg n 2 n n lg n n X k =1 1 k n Y k =2 1 - 1 k Problem 1-1. Asymptotic notation for multivariate functions The generalization of asymptotic notation from one variable to multiple variables is surprisingly tricky. One proper generalization of O -notation for two variables is the following: Definition 1 O ( g ( m, n )) = { f ( m, n ) : there exist positive constants m 0 , n 0 , and c such that 0 f ( m, n ) cg ( m, n ) for all m m 0 or n n 0 } . Consider the following alternative definition: Definition 2 O 0 ( g ( m, n )) = { f ( m, n ) : there exist positive constants m 0 , n 0 , and c such that 0 f ( m, n ) cg ( m, n ) for all m m 0 and n n 0 } . (a) Explain why Definition 2 is a “bogus” definition. That is, what anomalies does the definition of O 0 permit that are counterintuitive? You may find it helpful to illustrate your answer with a diagram of relevant regions of the m × n plane. Remarkably, famous computer scientists have used Definition 2 without being aware of its deficien- cies. Nevertheless, their theorems and analyses carry over to Definition 1, because the functions they analyzed satisfy two key properties. The first property is “monotonicity”:
Handout 7: Problem Set 1 3 Definition 3 A two-variable function f ( m, n ) is monotonically increasing if f ( m, n ) f ( m + 1 , n ) and f ( m, n ) f ( m, n + 1) for all nonnegative m and n . (b) Explain this definition in plain English.

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• Fall '01
• CharlesE.Leiserson
• Algorithms, Analysis of algorithms, CLRS

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