Give a simple example of a graph such that the set of all edges that are light edges crossing some cut in the graph does not form a minimum spanning tree.
- Show that a graph has a unique minimum spanning tree if, for every cut of the graph, there is a unique light edge crossing the cut.
- Show that the converse is not true by giving a counterexample.
Chapter 24 presents two algorithms to solve the problem of finding a minimum spanning tree of an undirected graph. Here, we shall see how mergable heaps can be used to devise a different minimum-spanning-tree algorithm.
The following pseudocode, which can be proven correct using techniques
from Section 24.1, constructs a minimum spanning tree T (you do
not have to prove the correctness here). It maintains a
partition 
of the vertices of V and, with each set Vi, a
set
of edges incident on vertices in Vi.
MST-Mergeable-Heap(G) 12 for each vertex
3 do
4
5 while there is more than one set Vi 6 do choose any set Vi 7 extract the minimum-weight edge (u,v) from Ei 8 assume without loss of generality that
and
9 if
10 then
11
, destroying Vj 12
Describe how to implement this algorithm using the mergeable-heap operations given in Figure 20.1. Give the running time of your implementation, assuming that the mergeable heaps are implemented by binomial heaps (Please be very specific about which lines in the code above correspond to which specific mergeable heap operations. )
Suppose edge weights in a graph are integers in the range from 1 to W. Show that you can make Prim's algorithm run in O(WV+E) time. For what values of W does this result improve on the bound using Fibonacci Heaps? (Hint: We do not want you to re-design Prim's algorithm. You should only re-design the priority queue implementation to achieve the new bound. To get going, think about what you could do if W=1. What if W=2? )