1.      [Comparing Running Times] Text problem 1-1, p. 13 measuring f(n) in nanoseconds.

2.      [Asymptotic Growth Rates]

a.   Text problem 3-2, p. 58. Do only for Big-Oh, Big-Omega, and Big-Theta.

b. Follow the instructions for problem 3-3 (a) for the following functions:

  3.     [Solution vs. Validation] (CSE 5350 do only Part 1 for 10 points / CSE 7350 do Parts 1 and 2 for 20 points)

Express the following five loosely described problems in Part 1 carefully in������������ { Instance, Question } form as utilized in "Computers and Intractability". For each problem discuss the best time and space complexity you are aware of for solving the problem (from scratch) along with a few words naming or describing the method. Then describe a P-key having the best time and space complexity you can find for validating an answer for each problem along with a few words naming or describing the method. In describing space you may assume the data is processed "on-line" and need not be stored unless necessary.

Part 1

�(i)���� Determining that a graph is not connected.

(ii)���� Finding the median� of n = 2 k + 1 integers.

(iii)��� Finding the 2 largest and 2 smallest of n integers.

�(iv)�� Determining the number of ways to score 13 on 3 dice.

�(v)��� Determining the greatest common divisor of (n,m).

Part 2

For the following just describe the P-key and the method and efficiency of checking the answer using the P-key. Problem numbers and pages refer to Gary and Johnson.

                                    (vi)��� Determining that the maximum number of paths between vertices v and w in a graph is less than k.

(vii)�� Clique of size J (see page 47 of Gary and Johnson).

(viii)� SP3 (p. 221)

(ix)        GT27 (p. 197)

(x)��� Graph Isomorphism (discuss interactive probabilistic verification using the���� zero knowledge proof technique).

  4.     [Implementation Testing] (20 points)

You are to implement and test a program for summing 1/x as x runs over all approximately eight million (23 fraction bits) single precision floating point numbers in the interval [1,2). You are to do this on a server, PC(or Mac) of your choice. You are first asked to predetermine estimates of your implementation’s computation time (architecture and compiler dependant), result accuracy (algorithm and round-off dependant), and result value (real valued sum estimate). Specifically:

a.       Discuss the computational environment for your tests, including the compiler, operating system, machine MHz and cycle times for appropriate instructions and whether pipelining affects your execution time.

b.      Predetermine an estimate of the time utilizing single precision for the variables for all computations from any documentation you can find from the hardware manufacturer and/or compiler and system provider.

c.   Predetermine a rough estimate of the exact sum (hint: how many terms are being added and how large is an “average term”).

d.       Predetermine an estimate of the accuracy. Single precision computation should be done in round to nearest mode as provided by standard C implementations.  By accuracy of the sum we mean the difference between the rounded sum of rounded values compared to the exact sum of exact values.

e.      Give the measured running time and the computed result for your implementation. Compare the results with your estimates of running time and approximate sum. Compute the sum in double precision and single precision and compare to give a reasonable value for the accuracy of the single precision sum. Compare your result with another student’s results who might have performed the sum in a different order. Can you explain the size of the approximation error?

  5.      [Algorithm Complexity] 

            (CSE5350 do only Part 1 for 20 points / CSE7350 do Parts 1 and 2 for 30 points)

                          Write an essay (4-5 word processor pages) discussing the issues involved in implementation independent measurement of the space and time complexity of algorithms. Include an analysis from your current perspective of the following fundamental issues:

Part 1 

a.       bit/basic element level space measurement

b.      polynomial/exponential growth in time or space measurement

c.       absolute/relative growth time measurement

d.      upper/lower asymptotic bounds

e.       worst/average case analysis

Part 2

f.     deterministic/probabilistic algorithm

g.    deterministic/nondeterministic algorithm

h.    solution vs. verifying

i.      verification by “zero knowledge proof” (interactive algorithm)

Note: This exercise is designed to familiarize you with basic concerns in the analysis of algorithms and deep comprehension is not expected at this time. This same question later in the course should elicit a more comprehensive response.