As it appears on page $657$ of the text, the Floyd$-$Warshall algorithm requires $\backslash$Theta $(3)$

space, since it creates $()$

for $,,=1,2,...,$

$.$ Show that the procedure FLOYD$-$WARSHALL'

$,$ which simply drops all the superscripts, is correct, and thus only $\backslash$Theta $(2)$

space is required.