A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 3.13 Exercises 913.3 Arrange the following expressions by growth rate from slowest to fastest.4n 2 log 3 n n! 3 n 20n 2 log 2 n n 2/3See Stirling’s approximation in Section 2.2 for help in classifying n!.3.4 (a) Suppose that a particular algorithm has time complexity T(n) = 3 ×2 n , and that executing an implementation of it on a particular machinetakes t seconds for n inputs. Now suppose that we are presented with amachine that is 64 times as fast. How many inputs could we process onthe new machine in t seconds?(b) Suppose that another algorithm has time complexity T(n) = n 2 , andthat executing an implementation of it on a particular machine takest seconds for n inputs. Now suppose that we are presented with a machinethat is 64 times as fast. How many inputs could we process onthe new machine in t seconds?(c) A third algorithm has time complexity T(n) = 8n. Executing an implementationof it on a particular machine takes t seconds for n inputs.Given a new machine that is 64 times as fast, how many inputs couldwe process in t seconds?3.5 Hardware vendor XYZ Corp. claims that their latest computer will run 100times faster than that of their competitor, Prunes, Inc. If the Prunes, Inc.computer can execute a program on input of size n in one hour, what sizeinput can XYZ’s computer execute in one hour for each algorithm with thefollowing growth rate equations?n n 2 n 3 2 n3.6 (a) Find a growth rate that squares the run time when we double the inputsize. That is, if T(n) = X, then T(2n) = x 2(b) Find a growth rate that cubes the run time when we double the inputsize. That is, if T(n) = X, then T(2n) = x 33.7 Using the definition of big-Oh, show that 1 is in O(1) and that 1 is in O(n).3.8 Using the definitions of big-Oh and Ω, find the upper and lower bounds forthe following expressions. Be sure to state appropriate values for c and n 0 .(a) c 1 n(b) c 2 n 3 + c 3(c) c 4 n log n + c 5 n(d) c 6 2 n + c 7 n 63.9 (a) What is the smallest integer k such that √ n = O(n k )?