Run update35 to get copies of some useful files added to
your cs35/labs/03 directory. You will not need to turn in any
code this week. Instead you will turn in written solutions to the
problems below as well as some graphs showing the results of various
running time experiments.
You may work with a partner this week. However, you should work
together on each problem rather than each working on only half of the
problems. Make sure that both names are on the pages that you hand in.
Write solutions to the following problems

C3.14 An array A contains n1 unique integers in the range 0 to n1. Thus exactly one number in this range is missing in the array. Design an O(n) time algorithm for finding this missing number. However, you are only allowed O(1) extra space. You cannot make extra arrays of size n, but you can store a constant number of extra variables.

Based on C3.20 The image below shows an n by n grid that is stored in memory as a two dimensional array. The element at row i column j can be indexed in constant time using grid[i][j]. Each cell in the grid contains either a 1 or a 0 (indicated in the figure with green and white blocks). In any column, all the one's appear before any of the zeros. Given such a grid, design an O(n) algorithm for finding the column with the most ones (tallest green tower). Note there are O(n^2) cells, so you cannot check every cell in the grid.
 Suppose I want to find the bigOh runtime of a certain algorithm, but I do not have access to the description of the algorithm. All I know is that the runtime of the algorithm is given by the following recursive definition: T(n)=T(n1)+O(n), meaning the time it takes to process an input of size n is equivalent to the time it takes to process an input of size n1 plus a O(n) term. Assume T(0) is a constant. We could write T(1) <= T(0)+c*1 and T(2) <= T(1)+c*2 <= T(0)+c(1+2). Determine a bigOh bound on the runtime of T(n) and prove your result using induction.
 Describe an algorithm for finding both the minimum and maximum of n numbers using fewer than 3n/2 comparisions. The following psuedocode finds the max using n1 comparisions.
input: Array A of size n>0
max=A[0]
for(i=1; i<n; i++)
if(A[i] > max)
max=A[i]
Note I am only counting comparisions of the form A[i] < x. Don't count the comparisions in the loop (i<n).

Give the best bigOh bounds on the runtime of the following functions.
Ex1(n)
for(i=0; i<n; i++)
a = i
Ex2(n)
for(i=0; i<n; i+=2)
a = i
Ex3(n)
for(i=0; i<n*n; i++)
a = i
Ex4(n)
for(i=0; i<n; i++)
for(j=0; j<=i; j++)
a = i
Ex5(n)
for(i=0; i<n*n; i++)
for(j=0; j<=i; j++)
a = i
Ex6(n)
k=1
for(i=0; i<n; i++)
for(j=0; j<k; j++)
a = j
k=k*2
Running time experiments
Perform an experimental analysis that compares the relative
running times of the functions you considered in Ex1 through Ex6 above.
In running these experiments, you may need to use very different
sizes of n, depending on the example being tested. Fast
algorithms may not start showing any distinctions until n
gets to be greater than 100,000. Slower algorithms may begin showing
differences when n is nearer to 100. One really slow algorithm might not be able to process n=100 in the lifetime of the universe.
In your cs35/labs/03 directory there are some starting
point files to help you with this task.
 There is a Timer class that works like a stop watch.
You can start one of these objects before some code for which
you want to measure the running time. Then stop it after the
code you are measuring, and print the elapsed time.

There is a test.cpp file that contains a main program that
demonstrates how to use the Timer class on a particular
problem.
 Note: the examples in the book use integer types for i,j, and n. For some values of n, n*n is larger than the maximum value of an integer. Instead of int n use long n for your function prototypes and local variables.

There is a ex4results file that shows how you can create a
text file that is readable by a script called pgraph
that views your results graphically. Be sure to put your name (or names)
in the title and to label the data with an appropriate name.
To see the results do:
pgraph ex4results
 To save the graph, click the save button at the bottom, select "Postscript (*.ps)" format,
type a filename, e.g., mygraph.ps, and click save. To print this file type:
lpr mygraph.ps
on the command line