CS570 Analysis of Algorithms Fall 2006 Exam 1 Name Student ID Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Maximum 20 10 10 10 10 20 20 Note The exam is closed book closed notes Received 1 20 pts Mark the following statements as TRUE or FALSE No need to provide any justification TRUE FALSE If T n is both O f n and f n then T n is f n TRUE FALSE For a graph G and a node v in that graph the DFS and BFS trees of G rooted at v always contain the same number of edges TRUE FALSE Complexity of the Decrease Key operation is always O lgn for a priority queue TRUE FALSE For a graph with distinct edge weights there is a unique MST TRUE FALSE Dynamic programming considers all the possible solutions TRUE FALSE Consider an undirected graph G V E and a shortest path P from s to t in G Suppose we add one 1 to the cost of each edge in G P will still remain as a shortest path from s to t TRUE FALSE Consider an undirected graph G V E and its minimum spanning tree T Suppose we add one 1 to the cost of each edge in G T will still remain as an MST TRUE FALSE Problems solved using dynamic programming cannot be solved thru greedy algorithms TRUE FALSE The union Find data structure can be used for an efficient implementation of the reverse delete algorithm to find an MST TRUE FALSE While there are different algorithms to find a minimum spanning tree of undirected connected weighted graph G all of these algorithms produce the same result for a given G 2 10 pts Indicate for each pair of expressions A B in the table below whether A is O or of B Assume that k and c are positive constants You can mark each box with Y yes and N no A n n n c 2n n2 3 2 B 3 n 2 n k n 2log n O 3 10 pts a What is the minimum and maximum numbers of elements in a heap of height h b What is the number of leaves in a heap of size n c Is the sequence 23 7 14 6 13 10 1 5 17 12 a max heap If not show how to heapify the sequence d Where in a max heap might the smallest element reside assuming that all elements are distinct 4 10 pts Prove or disprove the following The shortest path between any two nodes in the minimum spanning tree T V E of connected weighted undirected graph G V E is a shortest path between the same two nodes in G Assume the weights of all edges in G are unique and larger than zero 5 10 pts Suppose that you divided a graph G V E into two sub graphs G1 V1 E1 and G2 V2 E2 And we can find M1 which is a MST of G1 and M2 which is MST of G2 Then M1 U M2 U minimum weight edge among those connecting two graph G1 and G2 always gives MST of G Prove it or disprove it 6 20 pts There are n workers in the factory with heights of h1 h2 hn and n workingclothes with height sizes of s1 s2 sn The problem is to find best matching strategy such that we minimize the following average differences 1 hi si n Present an efficient algorithm to solve this problem along with its proof of correctness 7 20 pts Given an unlimited supply of coins of denominations x1 x2 xn we wish to make change for a value v that is we wish to find a set of coins whose total value is v This might not be possible for example if the denominations are 5 and 10 then we can make change for 15 but not for 12 Give an O nv algorithm to determine if it is possible to make change for v using coins of denominations x1 x2 xn Additional Space Additional Space