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Cs445 — Homework #5
All pairs shortest path, Network Flow, and
Matching in Bipartite Graphs
Due: 4/19/2005 during class meeting.
1. Let
G
(
V,E
) be a graph, with a weight assigned to its of its edges. The weight
is either positive or negative. Explain how to modify Johnson’s algorithm so
the output of the algorithm is an
n
×
n
matrix that speciﬁes for every pairs of
vertices
u,v
∈
V
, the ﬁrst edge in the shortest path from
u
to
v
. The running
time of the algorithm is
O
(

E

V

log

E

)
2. Assume
G
(
V,E
) is a graph where the weights of all edges are positive. You have
ran Johnson algorithm on this graph. What are the values
h
(
v
) and ˆ
w
(
u,v
)
given by the algorithm for each pairs of vertices
u,v
∈
V
? Prove.
3. Let
G
(
V,E
) be a graph with positive and negative weights given to its edges.
Assume that for each vertex
v
∈
V
you are also given a value
h
(
v
) with the
property that for every edge (
u,v
)
∈
E
,
w
(
u,v
) +
h
(
u
)

h
(
v
)
≥
0. (Note  the
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 Spring '06
 Williams

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