1. Text Book: Introduction to algorithms By C L R S
Graph Algorithms
Text Book: Introduction to algorithms
By C L R S

2. Graphs - Definitions
A directed graph (or digraph) G is a pair (V,E), where V is a finite set and E is a binary relation on V .
The set V is called the vertex set of G, and its elements are called vertices
The set E is called the edge set of G, and its elements are called edges.
Vertices are usually represented by circles and edges are represented by arrows.

3. Graphs - Definitions
In an undirected graph G = (V,E), the edge set E consists of unordered pairs of vertices, rather than ordered pairs.
If (u,v) is an edge in a directed graph G = (V,E), we say that (u,v) is incident from or leaves vertex u and is incident to or enters vertex v.
If (u,v) is an edge in an undirected graph G =(V,E), we say that (u,v) is incident on vertices u and v.

4. Graphs - Definitions
If (u,v) is an edge in a graph G =(V,E), we say that vertex v is adjacent to vertex u.
When the graph is undirected, the adjacency relation is symmetric.
The degree of a vertex in an undirected graph is the number of edges incident on it.
A vertex whose degree is 0 is isolated.

5. Graphs - Definitions
In a directed graph, the out-degree of a vertex is the number of edges leaving it, and the in-degree of a vertex is the number of edges entering it.
The degree of a vertex in a directed graph is its in-degree plus its out-degree.

6. Graphs - Definitions
A path of length k from a vertex u to a vertex u’ in a graph G=(V,E) is a sequence <v0, v1, ,vk> of vertices such that u=v0, u’=vk, and (vi-1,vi) є E for i= 1, 2, 3, , k.
An undirected graph is connected if every vertex is reachable from all other vertices. The connected components of a graph are the equivalence classes of vertices under the “is reachable from” relation.

7. Graphs - Definitions
An undirected graph is connected if it has exactly one connected component.
A directed graph is strongly connected if every two vertices are reachable from each other.

8. Representation of graphs

9. Representation of graphs

10. Representation of graphs

11. Representation of graphs

12. BFS

13. BFS

14. BFS Example

15. BFS Example

16. BFS Example

17. Properties of BFS Running time O(V + E)
BFS computes shorted path distance
Algorithm BFS builds a breadth first tree.

18. Printing shortest path

19. DFS

20. DFS

21. DFS Example

22. DFS Example

23. DFS Example

24. DFS Example

25. DFS Example

26. DFS – Example 2

27. DFS – Example 2

28. parenthesis theorem
In any depth-first search of a (directed undirected) graph G=(V,E), for any two vertices u and v, exactly one of the following three conditions holds:
the intervals [u.d, u.f] and [v.d, v.f] are entirely disjoint, and neither u nor v is a descendant of the other in the depth-first forest,
the interval [u.d, u.f] is contained entirely within the interval [v.d, v.f], and u is a descendant of v in a depth-first tree, or
the interval [v.d, v.f] is contained entirely within the interval [u.d, u.f], and v is a descendant of u in a depth-first tree.

29. Classification of edges
Tree edges are edges in the depth-first forest Gpi. Edge (u,v) is a tree edge if v was first discovered by exploring edge (u,v)
Back edges are those edges (u,v) connecting a vertex u to an ancestor v in a depth-first tree. We consider self-loops, which may occur in directed graphs, to be back edges.
Forward edges are those nontree edges (u,v) connecting a vertex u to a descendant v in a depth-first tree.
Cross edges are all other edges. They can go between vertices in the same depth-first tree, as long as one vertex is not an ancestor of the other, or they can go between vertices in different depth-first trees.

30. Classification of edges
when we first explore an edge (u,v), the color of vertex v
WHITE indicates a tree edge,
GRAY indicates a back edge, and
BLACK indicates a forward or cross edge.

31. Topological sort

32. Topological sort - algorithm

33. Topological sort

34. Strongly connected components
Strongly connected component of a directed graph G = (V,E) is a maximal set of vertices C where C is a subset of V such that for every pair of vertices u and v in C, vertices u and v are reachable from each other.

35. Strongly connected components- Algorithm
STRONGLY-CONNECTED-COMPONENTS(G) 1 call DFS(G) to compute finishing times u.f for each vertex u 2 compute GT 3 call DFS(GT), but in the main loop of DFS, consider the vertices in order of decreasing u.f (as computed in line 1) 4 output the vertices of each tree in the depth-first forest formed in line 3 as a separate strongly connected component

36. Strongly connected components- Example

37. Strongly connected components- Example

38. Component Graph

39. Minimal Spanning tree

40. Minimal Spanning tree

41. Kruskal’s Algorithm Example

42. Kruskal’s Algorithm Example

43. Kruskal’s Algorithm Example

44. Kruskal’s Algorithm Example

45. Kruskal’s Algorithm Example

46. Kruskal’s Algorithm Example

47. Kruskal’s Algorithm Example

48. Kruskal’s Algorithm Example

49. Kruskal’s Algorithm Example

50. Kruskal’s Algorithm Example

51. Kruskal’s Algorithm Example

52. Kruskal’s Algorithm Example

53. Kruskal’s Algorithm Example

54. Kruskal’s Algorithm Example

55. Kruskal’s Algorithm
Running time O(E lg V)

56. Prim’s Algorithm

57. Prim’s Algorithm - Example

58. Prim’s Algorithm - Example

59. Prim’s Algorithm - Example

60. Prim’s Algorithm - Example

61. Prim’s Algorithm - Example

62. Prim’s Algorithm - Example

63. Prim’s Algorithm - Example

64. Prim’s Algorithm - Example

65. Prim’s Algorithm - Example

66. Generic MST Algorithm

67. Light Edge
A cut (S, V – S) of an undirected graph G=(V,E) is a partition of V .
An edge (u,v) є E crosses the cut (S, V-S) if one of its endpoints is in S and the other is in V - S.
A cut respects a set A of edges if no edge in A crosses the cut.
An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

68. Safe Edge - Theorem
Let G=(V,E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V-S) be any cut of G that respects A, and let (u,v) be a light edge crossing (S, V-S). Then, edge (u,v) is safe for A.

69. Cut - Example

70. Cut – Another View

71. Shortest Path
The weight w(p) of path p = <v0, v1, , vk> is the sum of the weights of its constituent edges:

72. Shortest Path Problems
single-source shortest-paths problem
Single-destination shortest-paths problem
Single-pair shortest-path problem
All-pairs shortest-paths problem

73. Shortest Path Problems
Property
Subpaths of shortest paths are shortest paths
Graphs with negative edges
Graphs with cycles
Can a shortest path contain a cycle?

74. Are shortest paths unique?

75. Are shortest paths unique?

76. Are shortest paths unique?

77. Shortest Path Algorithms
v.d is a shortest path estimate to v from s

78. Relaxing
The process of relaxing an edge (u,v) consists of testing whether we can improve the shortest path to v found so far by going through u and, if so, updating
v.d and v.π.

79. Relaxing

80. Properties of shortest paths and relaxation

81. Properties of shortest paths and relaxation

82. Bellman Ford Algorithm

83. Bellman Ford Example

84. Bellman Ford Example

85. Bellman Ford Example

86. Bellman Ford Example

87. Bellman Ford Example

88. shortest path in DAG

89. Dijkstra’s algorithm

90. Dijkstra’s algorithm
Can be used only when all weights are non-negative
It maintains a set S of vertices whose final shortest-path weights from the source s have already been determined.
Algorithm uses a greedy strategy. It chooses the “lightest” or “closest” vertex in V - S

91. Dijkstra’s algorithm - Example

92. Dijkstra’s algorithm - Example

93. Dijkstra’s algorithm - Example

94. Dijkstra’s algorithm - Example

95. Dijkstra’s algorithm - Example

96. Dijkstra’s algorithm - Example

97. The Floyd-Warshall algorithm
The Floyd-Warshall algorithm considers the intermediate vertices of a shortest path, where an intermediate vertex of a simple path p = <v1,v2,…vl> is any vertex of p other than v1 or vl

98. The Floyd-Warshall algorithm

99. The Floyd-Warshall algorithm

100. Printing Shortest Path

101. Filling predecessor Matrix

102. Transitive closure of a graph
We define the transitive closure of G as the graph G* = (V,E*), where E* = { (i,j): there is a path from vertex i to vertex j in G} One way to compute the transitive closure of a graph is to assign a weight of 1 to each edge of E and run the Floyd-Warshall algorithm.

103. Transitive closure Algorithm

104. Transitive closure Algorithm

105. Flow Networks G = (V,E) directed.
Each edge (u,v) has a capacity c(u,v) >= 0
If (u,v) not є E, then c(u,v) = 0.
If (u,v) є E, then reverse edge (v,u) not є E.
Source vertex s, sink vertex t , assume s  v  t for all v є V , so that each vertex lies on a path from source to sink.

106. Flow Network

107. Another Flow Network

108. Flow A function f: V x V -> R satisfying Capacity constraint
For all u,v є V, 0 <= f(u,v) <= c(u,v)
Flow Constraint

109. Flows

110. Another Flow

111. Multiple sources

112. Maximum Flow Problem The value |f| of a flow f is defined as
In the maximum-flow problem, we are given a flow network G with source s and sink t , and we wish to find a flow of maximum value.

113. Network with anti parallel edges

114. Converted graph

115. The Ford-Fulkerson method
FORD-FULKERSON-METHOD (G, s, t)
1 initialize flow f to 0
2 while there exists an augmenting path p in the residual network Gf
3 augment flow f along p
4 return f

116. Residual network

117. 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

118. Residual network

119. Augmenting flow
If f is a flow in G and f’ is a flow in the corresponding residual network Gf , we define f ↑ f’, the augmentation of flow f by f’, to be a function from V x V to R, defined by

120. Augmenting paths
An augmenting path p is a simple path from s to t in the residual network Gf
Residual Capacity of path p
The maximum amount by which we can increase the flow on each edge in an augmenting path p
cf(p) = min { cf(u,v): (u,v) is on p}

121. Updated Flow Network

122. Cuts of flow networks
A cut (S,T) of flow network G = (V,E) is a partition of V into S and T = V - S such that s є S and t є T .
If f is a flow, then the net flow f(S,T) across the cut (S,T) is defined to be
Capacity of cut (S,T) is

123. Example of a cut

124. Flows through a cut
Lemma: Let f be a flow in a flow network G with source s and sink t, and let (S,T) be any cut of G. Then the net flow across (S,T) is f(S,T) = |f|
Corollary
The value of any flow f in a flow network G is bounded from above by the capacity of any cut of G.

125. Max-flow Min-cut Theorem
If f is a flow in a flow network G = (V,E) with source s and sink t , then the following conditions are equivalent:
1. f is a maximum flow in G.
2. The residual network Gf contains no augmenting paths.
3|f| = c(S,T) for some cut (S,T) of G.

126. Ford Fulkerson Algorithm

127. Ford Fulkerson - Example

128. Ford Fulkerson - Example

129. Ford Fulkerson - Example

130. Ford Fulkerson - Example

131. Ford Fulkerson - Example

132. Ford Fulkerson - Example

133. Ford Fulkerson - Example

134. Ford Fulkerson - Example

135. Ford Fulkerson - Example

136. Ford Fulkerson - Example

137. Ford Fulkerson - Example

138. Ford Fulkerson - Complexity
O( E. |f|)

139. Maximum bipartite matching
Given a graph G=(V,E)
A matching is a subset of edges M of E such that for all vertices v in V, at most one edge of M is incident on v.
A maximum matching is a matching of maximum cardinality

140. Maximum bipartite matching

141. Maximum bipartite matching

142. Maximum bipartite matching