Problem $3$

Modify the Bellman$-$Ford algorithm so that it sets $|$$$v.d|$$ to $|$$$-|$$ 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 $$ text$($FLOYDWARSHALL$\text{'})|$$$,$ which simply drops all the superscripts, is correct, and thus only $|$$$(n^2)|$$ space is required.

$|$$text$($FLOYD$-$WARSHALL'$)($W$,$ n$)|$$

$|$$$D|=\frac{W}{\frac{?}{?}}$$

$|$$$|$ text$($for $)k=1$ text$($ to $)n|$$$|$

$|$$text$()$ text$($for $)i=1$ text$($ to $)n|$$$|$

$|$$text$()$ text$()$ text$($for $)j=1$ text$($ to $)n|$$$|$

$|$$$|$ text$()$ text$()$ text$()$ d$\_\{$ij$\}=$ min$\{d_{ij},d_{-}\{ik\}+d_{-}\{kj\}\}|$$$|$

$|$$$|$ text$($return $)\frac{D}{\frac{?}{?}}$$