Problem 3 Modify the Bellman-Ford algorithm so that it sets |$v.d|$ to |$- infty |$ for all vertices |$v|$ for which there is a negative-weight cycle on some path from the source to |$v|$. Problem 4 Show how to use the output of the FloydWarshall algorithm to detect the presence of a negative-weight cycle. Problem 5 As it appears on page 657 of the

Question

Problem 3 Modify the Bellman-Ford algorithm so that it sets

|$v.d|$

to

|$-\infty |$

for all vertices

|$v|$

for which there is a negative-weight cycle on some path from the source to

|$v|$

. Problem 4 Show how to use the output of the FloydWarshall algorithm to detect the presence of a negative-weight cycle. Problem 5 As it appears on page 657 of the text, the Floyd-Warshall algorithm requires

|$

| Theta

(n^(3))|$

| space, since it creates

|$(d)_(ij)^(^()){(k)}|$

for

|$i,j,k

|

=1,2,dots,n|$

|. Show that the procedure

$$

text(FLOYDWARSHALL')|$, which simply drops all the superscripts, is correct, and thus only

|$\Theta (n^(2))|$

space is required. |$text(FLOYD-WARSHALL')(W, n)|$

|$D

|

=(W)/(/)$

|$

| text(for )

k=1

text( to )

n|$

| |$text( ) text(for )

i=1

text( to )

n|$

| |$text( ) text( ) text(for )

j=1

text( to )

n|$

|

|$

| text( ) text( ) text( ) d_{ij} =

min{d_(ij),d_(-){ik}+d_(-){kj}}|$

|

|$

| text(return )

(D)/(/)$

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