Solutions_to_Introduction_to_Algorithm_3nd_Exercis

2019-04-13 11:06发布

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3.1-1 Let f(n) and g(n) be asymptotically nonnegative functions. Using the basic definition of Θ-notation, prove that max(f(n),g(n))=Θ(f(n)+g(n)).
　　Let f(n),g(n) be asymptotically nonnegative. Show that max(f(n),g(n)) = Θ(f(n)+g(n)). This means that there exists positive constants c1,c2 and n0 such that, 0<= c1(f(n)+g(n))<=max(f(n),g(n))<=c2(f(n)+g(n)), for all n>=n0.
　　Selecting c2=1 clearly shows the third inequality since the maximum must be smaller than the sum. c1 should be selected as 1/2 since the maximum is always greater than the weighted average of f(n) and g(n). Note tha significance of the "asymptotically nonnegative" assumption. The first inequality could be satisfied otherwise

 3.1-2 Show that for any real constants a and b, where b>0, 　　To show that , we want to find constants c1,c2,n0>0 such that  　　Note that  and Thus,Since b>0, the inequality still holds when all parts are raised to the power b:

Thus, satisfy the definition.
3.1-3 Explain why the statement, "The running time of algorithm A is at least," is meaningless. 　　Let the running time be T(n).   means that  for some function f(n) in the set , This statement holds for any running time T(n), since the function g(n)=0 for all n is in,and running times are always nonnegative. Thus, the statement tells us nothing about the running time.
3.1-4 Is ? Is ? 　　,but . 　　To show that, we must find constants c, such that . 　　Since  for all n, we can satisfy the definition with c=2 and  　　To show that , assume there exist constants c, such that . 　　Then . But no constant is greater than all , and so the assumption leads to contradiction.



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