1. ADVANCED ALGORITHMS
GRAPH
ALGORITHMS
(UNIT-2)

2. The Bellman-Ford Algorithm (BFA)
The BFA solves the single-source shortest-paths
problem in general case in which edge weights
may be negative.
Given a weighted directed graph G = (V,E).
with source s,
and weight function w,
the BFA returns a boolean value
indicating whether or not
there is a negative-weight cycle
that is reachable from the source.

3. Bellman-Ford(G,w,s)
Initialize-Single-Source(G,s)
for i = 1 to |V| -1
for each edge (u,v) ∊ E
RELAX(u,v,w)
if v.d > u.d + w(u,v)
return FALSE
8 return TRUE
RELAX (u,v,w)
if v.d > u.d + w(u,v)
2 v.d = u.d + w(u,v)
3 v.∏ = u

4. Single –Source Shortest Paths in DAG:
DAG stands for ‘Directed Acyclic Graph’.
There is no cycle in a directed acyclic graph.
Topological Sort : Topological Sort of a DAG,
G = (V,E) is a linear ordering of all its
nodes such that if G has an edge (u,v),
then u appears before v in the ordering.
The algorithm starts by topologically sorting the DAG to impose a linear ordering on the vertices.
If a DAG contains a path from u to v, then u preceds v in the topological sort.

5. DAG-SHORTEST-PATHS(G,w,s)
Topologically sort the vertices of G
INITIALIZE-SINGLE-SOURCE(G,s)
for each vertex u, taken in topologically sorted order
for each vertex v ∊ G.Adj[u]
RELAX (u, v, w)
Ex-1 : Topologically sort the following DAG : t
r x
y s z

6. Ex-2 : Draw the Graph and Solve it for the following Weight Table :
r  s  t  x  y  z
Ex-2 : Draw the Graph and Solve it for the following Weight Table :
r  s s  t t  x x  y y  z
s  x x  z r  t t  y t  z

7. 3. Johnson’s Algorithm for Sparse Graphs:
This algorithm finds shortest paths between all pairs.
This algorithm returns a matrix of shortest-path
weights for all pairs of vertices or reports that the
that the input graph contains a negative-weight cycle.
This algorithm uses the technique of ’reweighting’.
If all edge weights w in a graph G = (V,E) are
non-negative, we can find shortest paths between all pairs of vertices by running Dijkstra’s algorithm once from each matrix.
b) If G has negative –weight edges, then compute a new set of non-negative edge weights (reweighting) that allows us to use the same method.

8. JOHNSON(G,w)
Compute G’, where G’.V = G.V U {s}
G’.E = G.E U { (s,v) : v ∊ G.V
w(s,v) = 0 for all v ∊ G.V
if BELLMAN-FORD(G’, w, s) = = FALSE
print “the input graph contains a negative- weight cycle”.
else for each vertex v ∊ G’.V
set h(v) to the value of δ(s,v)
computed by the BF algorithm.
for each edge (u,v) ∊ G’. E
w’(u,v) = w(u,v) + h(u) – h(v)

9. Let D = (duv) be a new matrix. for each vertex u ∊ G.V
10. run DIJKSTRA(G,w’,u) to compute
δ’(s,v) for all v ∊ G.V
for each vertex v ∊ G.V
duv = δ’(u,v) + h(v) – h(u)
return D
Ex-3: Consider the following Graph.
G(V) = [ a, b, c, d, e]
G(E) = [ (a  b : 3), (a  c : 8), (a  e : -4),
(b  d : 1), (b  e : 7), (c  b : 4),
(d  c : -5), (d  a : 2), (e  d : 6) ]

10. Consider new vertex s.
Draw edges from s to a, b, c, d, e with weights zero.
Draw the Graph G’ (V’,E’), where
V’ = V U (s).
and E’ = E U [(s,a), (s,b), (s,c), (s,d),(s,e)]
for G, we have |V| = 5 |E| = 9
for G’, we have |V| = 6 |E| = 14
And h(a) = δ(s,a) = h(b) = δ(s,b) = -1
h(c) = δ(s,c) = h(d) = δ(s,d) = 0
h(e) = δ(s,e) = -4

11. Bellman-Ford Algorithm : TRUE
REWEIGHTING :
w’(a,b) = w(a,b) + h(a) – h(b)
= – (-1) = 4
w’(a,c) = w(a,c) + h(a) – h(c) = 13
w’(a,e) = w(a,e) + h(a) – h(e) = 0
w’(b,d) = w(b,d) + h(b) – h(d) = 0
w’(b,e) = w(b,e) + h(b) – h(e) = 10
Similarily w’(c,b) = 0 w’(d,c) = 0
w’(d,a) = 2 w’(e,d) = 2
w’(s,a) = 0 w’(s,b) = 1
w’(s,c) = 5 w’(s,d) = 0
w’(s,e) = 4

12. Draw the D-Matrix (5 x 5 ) : δ(u,v) = duv
The Output :
Draw the D-Matrix (5 x 5 ) : δ(u,v) = duv
δ(u,v) a b c d e a b c d e

13. A flow network G = (V,E) is a directed graph in
4. FLOW NETWORKS :
A flow network G = (V,E) is a directed graph in
which each edge (u,v) ∊ E has a non-negative
capacity c(u,v)  0.
There are two distinguished vertices :
source s sink t
A flow in G has two following properties :
a) Capacity Constraint : 0 ≤ f(u,v) ≤ c(u,v)
b) Flow Conservation : For all u ∊ V – {s,t}
 f(u,v) =  f(v,u)
v ∊ V v ∊ V

14. Ex-4 : Consider the following Flow Network
v v3
s t
v v4
The following is FNW for |f| = 19
v /12 v3
11/ /20
s / /9 7/7 t
8/ /4
v /14 v4

15. 4. The Ford-Fulkerson Method :
The following is FNW for |f| = 20
v /12 v3
8/ /20
S / / /7 T
12/ /4
v /14 v4
4. The Ford-Fulkerson Method :
This FF Method is used for solving the maximum-
flow problem.
This method iteratively increases the value of the
flow.

16. Residual Capacity : Residual Network : Let f be a flow in G.
Consider a pair of vertices u,v ∊ V
The Residual Capacity Cf(u,v) is :
= c(u,v) – f(u,v) if (u,v) ∊ E
Cf(u,v) = f(v,u) if (v,u) ∊ E
= otherwise
Residual Network :
Given a flow network G = (V,E) and a flow f, the residual network of G induced by f is Gf = (V,Ef), where
Ef = { (u,v) ∊ V X V : Cf(u,v) > 0 }

17. The Ford-Fulkerson Algorithm :
FORD-FULKERSON(G,s,t)
1 for each edge (u,v) ∊ E
(u,v).f = 0
3 while there exists a path p from s to t
in the residual network Gf
Cf(p) = min {Cf(u,v) : (u,v) is in p}
for each edge (u,v) in p
if (u,v) ∊ E
(u,v) .f = (u.v).f + Cf(p)
else (v,u).f = (v,u).f - Cf(p)

18. Example-5 : Consider the following FNW :

23. The above Residual network has no augmenting paths, and the flow f shown above is therefore a maximum flow.
The value of the maximum flow found is  23

24. Maximum Bipartite Matching :
Let there is an Undirected Graph : G = (V,E)
A matching is a subset of edges M  E
such that for all vertices v ∊ V,
at most one edge of M is incident on v.
A vertex v ∊ V is matched by the matching M,
if some edge in M is incident on v,
otherwise, v is unmatched.
A maximum matching is a matching of maximum cardinality, i.e., a matching M such that for any
matching M’, it is |M|  |M’|.

25. Let the vertex set V is partitioned in to L and R :
V = L U R
where L and R are disjoint and all edges in E are
in either L or R.
Ex-6 : Let there are L machines.
Let there are R tasks.
Here the edge (u,v) in E is to mean that
a particular machine u ∊ L is capable
of performing a particular task v ∊ R.
A maximum matching provides work for as many
machines as possible.

26. Finding a Maximum Bipartite Matching :
Let there is a Bipartite Graph G = (V, E) .
The Ford-Fulkerson method can be used here.
Construct a FNW, G’ = (V’,E’) , where
V’ = V U {s,t}
E’ = E U all edges from s to L
U all edges from R to t.
Here, E’ = { (s,u) : u ∊ L }
U {(u,v) : (u,v) ∊ E }
U {(v,t) ∊ R }

29. 6. Max Flow - Min Cut Theorem :
Ex-7 : Consider the following Graph :

30. A cut is a node partition (S,T) such that s is in S,
and t is in T.
capacity(S,T) = sum of weights of edges leaving S.

33. The Max-Flow of the above Graph is : 28
Consider the above Original Graph.
Find its Max-Flow :
The Max-Flow of the above Graph is :
Max-Flow Min-Cut Theorem :
The value of the max flow is equal to the capacity of the min cut.
i.e., Max-Flow = Min-Cut
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