Tutorial questionsAlgorithms
2019 Semester 1 Tutorial 4
The University of Sydney
School of Computer Science
Problem 1
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.)
Solution:
We run DFS with a minor modification. Every time we scan the neighborhood of
a vertex uwe check if the neighbor vhas been discovered before and whether it
is different than u’s parent. If we can find such a vertex the we have our cycle:
v, u, parent[u],parent[parent[u]], . . . , v.
We only need to argue that this algorithm runs in O(n) time. Consider the execution of the
algorithm up the point when we discovered the cycle. After the O(n) time spent initializing
the arrays needed to run DFS, each call to dfs-visit takes time that is proportional to the
edges discovered. However, up until the time we find the cycle we have only discovered tree
edges. So the total number of edges is upper bounded by n−1. Thus, the overall running
time in O(n).
Problem 2
Find an amount for which the greedy coin change algorithm fails for the set of coins {1c,10c, 25c}.
Solution: For 30cthe greedy algorithm would choose 25c, 1c, 1c, 1c, 1c, 1c(6 coins), but
10c, 10c, 10c(3 coins) is more efficient.
Problem 3
Trace Prim’s algorithm on the graph in Fig. 1 starting at node a. What’s the output?
1
