CS330 Introduction to Algorithms
November 12, 2012
Spanning trees, single source shortest paths II.
1. Run the DAG single source shortest paths algorithm in the following graph starting
from S . The graph is given by its adjacency list along with edge weights.
S:A:3→B:3→C:1
A:D:4→E:1
B:D:1→E:2
C:B:1
E:
2. How can you tell after at most n iterations of the Bellman-Ford algorithm, whether
there is a negative cycle in the graph?
3. Give an algorithm that can count the total number of paths (not necessary shortest
paths) from the source to all other nodes in the above graph, while traversing the
graph in the same order as in exercise 1.
4. (CLRS 23.1-1) Let (u, v) be a minimum-weight edge in a connected graph G. Show
that (u, v) belongs to some minimum spanning tree of G.
5. Show that a subpath of a shortest path is always a shortest path.
6. Let G be a graph with weighted edges, so that negative edge weights are allowed
but G does not contain any non-positive weight cycles. Show that in this case all
shortest paths are simple. (A path is simple if it doesn't contain cycles.)
7. Let G be a directed graph where all edges have positive edge weights. Show how
you can apply any of the single source shortest paths algorithms to nd the longest
paths from a given source to all other nodes.
8. (CLRS 24.3-5) Show an example of a directed graph where Dijkstra's algorithm
could relax the edges of a shortest path out of order.
9. (CLRS 23.1-8) Let T be a minimum spanning tree of graph G, and let L be the
sorted list of the edge weights of T . Show that for any minimum spanning tree T 0
of G, the list L is also a sorted list of edge weights for T 0 .