CMSC 451 University of Maryland Biology Discussion

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CMSC 451 Homework 6
1. Using Warshall’s algorithm, compute the reflexive-transitive closure of the relation below.
Show the matrix after the reflexive closure and then after each pass of the outermost for loop
that computes the transitive closure.
0
1
0
0
[1
0
0
0
0
0
0
0
1
1
1
0
0
1
0
0
1
0
0
0
1]
2. Using the matrix in the previous problem show the final result of executing Floyd’s
algorithm on that matrix to produce a matrix containing path lengths.
3. Show the graph that corresponds to the matrix in the first problem assuming the rows and
columns correspond to the vertices a, b, c, d and e. Show its condensation graph, renaming its
vertices. Determine any topological order of that graph and create an adjacency matrix with
the vertices ordered in that topological order. Finally compute the reflexive-transitive closure
of that matrix. What characteristic of that matrix indicates that it defines a total order?
4. Using Floyd’s algorithm, compute the distance matrix for the weight directed graph defined
by the following matrix:
0
2
4
0
2

5
3 3
[
]

0 
−2  −4 0
Show the intermediate matrices after each iteration of the outermost loop.
Grading Rubric
Problem
Problem 1
Meets
10 points
Does Not Meet
0 points
Showed the correct matrix after the
reflexive closure (2)
Did not show the correct matrix after
the reflexive closure (0)
Showed the correct matrices after each
pass of the outermost for loop that
computes the transitive closure (8)
Did not show the correct matrices after
each pass of the outermost for loop
that computes the transitive closure (0)
Problem 2
Problem 3
Problem 4
10 points
0 points
Showed the correct final result of
executing Floyd’s algorithm on that
matrix to produce a matrix containing
path lengths (10)
Did not show the correct final result of
executing Floyd’s algorithm on that
matrix to produce a matrix containing
path lengths (0)
10 points
0 points
Showed the correct graph that
corresponds to the matrix in the first
problem assuming vertices a, b, c, d
and e (1)
Did not show the correct graph that
corresponds to the matrix in the first
problem assuming vertices a, b, c, d
and e (0)
Showed its correct condensation
graph, renaming its vertices (2)
Did not show its correct condensation
graph, renaming its vertices (0)
Determined a correct topological order
of that graph (2)
Did not determine a correct topological
order of that graph (0)
Created a correct adjacency matrix
with the vertices ordered in that
topological order (1)
Did not create a correct adjacency
matrix with the vertices ordered in that
topological order (0)
Correctly computed the reflexivetransitive closure of that matrix (2)
Correctly explained what characteristic
of that matrix indicates that it defines a
total order (2)
Did not correctly compute the
reflexive-transitive closure of that
matrix (0)
Did not correctly explain what
characteristic of that matrix indicates
that it defines a total order (0)
10 points
0 points
Showed the correct intermediate
matrices after each iteration of the
outermost loop using Floyd’s algorithm
(7)
Did not show the correct intermediate
matrices after each iteration of the
outermost loop using Floyd’s algorithm
(0)
Showed the correct final matrix after
executing Floyd’s algorithm (3)
Did not show the correct final matrix
after executing Floyd’s algorithm (0)
Give an example of an application of a graph, in which determining all pairs shortest paths
would be of importance. Describe what the vertices, edges and edge weights of the graph
represent. Explain the significance of the shortest path for such a graph and why it would be
important.

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