General Comments: Show all your work, otherwise no partial credit. No credit without proper justifications. State all your assumptions. No Algorithm is complete without its time and space complexity. When presenting an algorithm, first indicate if it is similar to a well-known algorithm, second describe intuitively how the algorithm works (which may be supported by examples), third give its pseudo code and finally analyze its time and space complexity. Always describe general idea of the algorithm before giving its pseudo-code. Do not reinvent the wheel, i.e., if a well-known algorithm can be modified to solve a problem efficiently, use that solution and clearly indicate the changes required. Do not unnecessarily complicate a solution, i.e., if a simple but efficient solution exists then we should use it. Finally, if you just write pseudo-code of a well-known algorithm without indicating how it applies or modified to the problem at hand, no credit will be given.
Problem R-1.6 on page 47 of the GT book.
Graph the functions listed in the first row of R-1.6 using a logarithmic scale for the x- and y-axes; that is, if the function value f(n) is y, plot this as a point with x-coordinate at log n and y-coordinate at log y. Of course assume log is base 2. What do you observe from the plots (i.e., compare the five functions with respect to their growth rates)?
Problem R-1.3 on page 47 of the GT book. Justify your answer.
Problem R-1.7 on page 47 of the GT book. Justify your answer.
Solve the recurrence relation T(n)=T(n/2)+4n when n>=2 and T(n)=3 when n<=1 using expansion method as discussed in class. After finding the closed form, express your answer using big-oh notation. ( Problem C-1.7 on page 50 of the GT book. Justify your answer. Programming Problem: Implement Algorithms 1.14 and 1.15 (prefixaverages) on pages 32-33 of the GT textbook using your favorite high-level language (e.g. C, C++, C#, Java, or even LISP, FP, etc.). For input, fill array X with random numbers between -1000 and 1000. Measure the running time of these two algorithms for 0 <=n <= 10^6, plot them, and compare them with their asymptotic time complexities. Choose appropriate increments for n so that you don’t overwhelm your machine or unnecessarily spend time in creating input yet your measurement sample is representative (i.e., show your creativity). Finally, write a brief report (at most 1-page, formatted using one of the common word-processors with a min of 11pt font). No need to say, but you should be using good conventions and programming practices in developing your programs [just in case you forgot, refresh them from some of the coding conventions etc links provided on the TopicsCovered page or dig-up your undergraduate notebooks!] More problems may be added. For easy access to problems needed in this homework, click on this link. General Instructions on submitting your homeworks. For programming assignments, submit a SINGLE zipped file of your source codes, scripts (to run your program if any) and a brief report along with a copy of a couple of sample executions of your solution. No need to say, but you should be using good conventions and programming practices in developing your programs [just in case you forgot, refresh them from some of the coding conventions etc links provided on the TopicsCovered page.] Use
show and discuss his or her solution in class, so be ready with your presentation.
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