Algorithm Design COMP3027/3927 Tutorial questions The University of Sydney 2019 Semester 1 Tutorial 2 School of Computer Science
Pre-tutorial questions
Do you know the basic concepts of this week’s lecture content? These questions are only to test yourself. They will not be explicitly discussed in the tutorial, and no solutions will be given to them.
1. Graph terminology
(a) What is a graph G(V, E)?
(b) Graphs are usually represented either as an adjacency matrix or as an adjacency list. Can you
explain the two representations?
(c) What are the advantages/disadvantages between the two different representations?
(d) What is a simple path in a graph?
(e) What is a cycle in a graph?
(f) What is a tree? If a tree has n vertices, how many edges does it have?
(g) What is the difference between a rooted tree and an unrooted tree?
(h) What is a bipartite graph?
2. Graph traversals
(a) What is the difference between BFS and DFS? (b) Explain the two search algorithms BFS and DFS.
Note that some of the questions are on proofs that have been covered in class. The point of these questions is to have you write the proof in your own way and, in the process, to property digest the proof.
Tutorial
Problem 1
Run a Breadth First Traversal for graph G starting at A. Write the order the nodes are explored. Assume ties are broken lexicographically (i.e., if there is more than one child, pick the one that comes first in lexicographic order).
AEF BG CH DI
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Problem 2
Run a Depth First Traversal for graph G starting at A. Write the order the nodes are explored. As before, assume ties are broken lexicographically.
AEF BG CH DI
Problem 3
Run a Breadth First Search for graph G starting at A and searching for I. Write out the path that is explored. Assuming breaking ties lexicographically.
AEF BG CH DI
Problem 4
Run a Depth First Search for graph G starting at A and searching for I. Write out the path that is explored. Assume ties are broken lexicographically.
AEF BG CH DI
Problem 5
Consider the following directed graph G:
1. Give the adjacency matrix and the adjacency list representation of this graph.
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AEF
BDG
CH
2. [Advanced] Is this graph strongly connected?
3. [Advanced] This graph has a few directed cycles. Find the minimum number of edges that you need
to delete from this graph in order to make it a directed acyclic graph (DAG).
4. [Advanced] Write down the transitive closure of the graph.
5. [Advanced] Write down a topological ordering of the resulting DAG.
Problem 6
Give an O(n) time algorithm to detect whether a given undirected graph contains a cycle. If the answer is yes, the algorithm should produce a cycle. (Assume adjacency list representation.)
Problem 7
An undirected graph G = (V,E) is said to be bipartite if its vertex set V can be partition into two sets A and B such that E ⊆ A × B. Design an O(n + m) algorithm to test if a given input graph is bipartite using the following guide:
1. Suppose we run BFS from some vertex s ∈ V and obtain layers L1,…,Lk. Let (u,v) be some edge in E. Showthatifu∈Li andv∈Lj then|i−j|≤1.
2. Suppose we run BFS on G. Show that if there is an edge (u, v) such that u and v belong to the same layer then the graph is not bipartite
3. Suppose G is connected and we run BFS. Show that if there are no intra-layer edges then the graph is bipartite
4. Put together all the above to design an O(n + m) time algorithm for testing bipartiteness.
Problem 8
[Advanced] In class we sketched a proof that a Directed Acyclic Graph always has a topological ordering. Write out a formal proof using induction.
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