LAST NAME:
(according to BB)

       FIRST NAME: 
(according to BB)


Note: * is times, / is divide by, ^ is powering/exponentiating
Note: 26 points total



(1) 1 point each
For each of the following, indicate whether f(n) = O(g(n)) and/or g(n) = O(f(n)).  

1a) f(n) = n^2 + 5*n + 3 and g(n) = 5*n^(1.5) + 6*n + 7






1b) f(n) = n * (log(n))^2 and g(n) = n^(1.5)






1c) f(n) = 2^n and g(n) = 2^(2n).  







(2) 2 points
Pick 1a, 1b, or 1c and use the definition of big-O to prove what you claimed.
































(3) 2 points each
Solve the given recurrence relations.  Assume for each that T(1) = 1.

3a) T(n) = 3 * T(n/2) + n^2
















3b) T(n) = 4 * T(sqrt(n)) + (log(n))^2




















(4) 2 points
Use induction to prove that the sum 
    (1 + 2 + 4 + 8 + ... 2^n) 
is <= 2^(n+1).
























(5) 5 points
ICPC programming question.
The first line of input has two numbers.  The first indicates
how many test cases there will be.  The second indicates how many
numbers will be listed with each test case.  For each test case, 
you should read in the numbers and count how many are positive, and
then output both the case number and the count.

Sample Input:
      3 7
      3 5.6 7 -3.2 0 -1.2 123.3
      -234 -44 -22 98 88.2 -88.3 -11.78
      2344 4 2 1 88 7 33 42.788

Sample Output:
      Case 1: 2
      Case 2: 5
      Case 3: 0





















































(6) 5 points
Graph programming question.
Suppose you have some code that has a 2-dimensional array that stores
an adjacency matrix.  It was declared as 
   int G[100][100];
and then values were filled in for some graph.
Write C/C++ code to compute how many vertices are isolated (do not have 
any edges), and to compute the maximum degree (largest number of edges for 
a single vertex).  The code should output the number of isolated 
vertices and the maximum degree.

If G were the complete graph, the correct output would be:
    Isolated vertices: 0
    Maximum degree: 100

If G were a graph with no edges, the correct output would be:
    Isolated vertices: 100
    Maximum degree: 0



















































(7) 5 points
For one of the following algorithms, give a basic description of the 
algorithm.  You must fit your description in the space provided.  I will
not read or count anything written on the back of the page, etc.

Algorithms: breadth first search, dijkstra's algorithm, maximum subarray 
   divide and conquer, subset sum dynamic programming.

Description...

7a) What is the input data for the algorithm?





7b) What problem does the algorithm solve?





7c) Basic description of how the algorithm works.














7d) Running time of the algorithm.





7e) Strength or weakness of the algorithm not already mentioned above.