Tutorial Sheet

1. [5-6] [8-9] Prove that if the weights on the edge of the connected undirected graph are distinct then there is a unique MST. Give an example in this regard, also discuss Kruskal MST in detail. 2. [5-6] [8-9] Explain the Floyd Warshall algorithm with example. 3. [5-6] Given a graph G={V, E} and let U and V be two distinct vertices. Explain how to modify Dijkastra’s shortest path algorithm to determine the number of distinct shortest paths from U to V. 4. [5-6] Write an algorithm for inserting a node into Fibonacci heap. 5. [5-6] What do you understand by Binomial heap? How to merge two Binomial heaps? 6. [5-6] Explain deletion procedure in B-tree. 7. [5-6][6-7] Write an algorithm of insertion in B tree. You are also required to give comments on its running time. 8. [6-7] Explain linked adjacency list representation of a graph with suitable example/diagram. 9. [6-7] What is a MST of a graph? Describe Prim’s algorithm to find MST along with its running time. 10. [6-7] Explain and write the Bellman Ford algorithm. You are also required to find the running time of the algorithm. 11. [6-7] What is a Binomial tree? Explain with suitable example. You are also required to give its properties. 12. [6-7] Explain the different conditions of getting union of two existing Binomial heaps. 13. [6-7] What are the various differences between Binomial and Fibonacci heaps. 14. Augment a binary tree, where now each node maintains the size of node (number of internal nodes including itself) with suitable example. 15. [7-8] Write an algorithm for inserting a key into a B-tree in a single pass down the tree. 16. [7-8] Show that if only the mergable heap operations are supported, the maximum degree D(m) is an n-node Fibonacci heap is at most log n. 17. [7-8] Write about linked list representation of disjoint sets. 18. [7-8] Prove that the maximum degree of any node in an n-node Binomial tree is lg n. 19. [7-8] Argue that in every n-node binary search tree there are exactly n-1 possible rotations. 20. [7-8] Show how Prim’s algorithm can be implemented using heap, what would be the time complexity of the algorithm? 21. [7-8] Prove the correctness of Kruskal algorithm. 22. [7-8] Show how to compute transitive closure of a graph using Floyd Warshall algorithm for all pair shortest path. 23. [7-8] Give an algorithm for topological sorting of a directed acyclic graph.

24. [8-9] For the graph apply Floyd warshall algorithm for constructing shortest path. Show the matrix D(k) that results each iteration. -1 1 2 2 -8 -4 3 7 5 10 25. [9-10] Suppose that graph G=(V, E) is represented as an adjacency matrix. Give a example implementation of Prim’s algorithm so that it runs in O(V2) time for this case. 26. [9-10] Give an efficient push-rebel algorithm to find a maximum matching in a bipartite graph, analyze your algorithm. 27. [9-10] Suppose we run Johnson’s algorithm on a directed graph G with weight w. show that if G contains the 0-weight cycle c, then w(U,V)=0 for every edge (U,V) in c.