1-4 CHARACTERS CODE(if you want your final grade posted)

 
CIS 680: Final Exam
 
1:50 minutes
Open Notes, Open Books
The exam consists of five problems
Answers not appearing in designated spaces WILL NOT be graded

• Problem #1 (10 points) In Kruskal‘s algorithm, the union-find algorithm merges the longer linked lists at the end of the shorter linked lists. The algorithm requires O(|V | log |V |) time. What contribution the union-find algorithm provides to the time complexity of Kruskel’s algorithm, if the union operation merges the lists in arbitrary manner, without taking into account their lengths or other ranks.
• Problem #2 (10 points) Consider the closest pair algorithm. How many distances the algorithm computes, in each of the following cases.
  1. The points are at locations (0,0), (1,0), (2,0), (3,0), (4,0), (5,0), (6,0), (7,0).
  2. The points are at locations (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1).
• Problem #3 (10 points) Consider the Graham’s Scan algorithm for solving the Convex Hull problem. For each of the following cases, draw a set of 8 points satisfying the specified condition.
  1. The algorithm constructs the convex hull without ever backtracking.

  2. The algorithm constructs a convex hull containing a minimal number of boundary points

  3. The algorithm constructs a convex hull containing 5 boundary nodes, and doesn’t visit the omitted nodes consecutively.
• Problem #4 (10 points) For each of the following problems, provide a tree generator of a solution space.
  1. The assignment problem assumes n jobs to be assigned to n workers, and a function c(i, j) which specifies the cost of assigning job i to worker j. The problem is interested in finding an assignment having a minimal cost.

  2. The partition problem assumes a set of items, and a function v(i) assigning to each item a value. The problem asks whether the set can be partitioned into two subsets having the same total value.
• Problem #5 (10 points) The number of ways of choosing k items from a set of n items is represented by the following recurrence relation.

  C(n, k) = C(n - 1, k - 1) + C(n - 1, k)

  1. Provide a dynamic programming algorithm for calculating C(n, k).

  2. What calculations the dynamic programming algorithm saves in computing C(5, 2)