Graph algorithms

Friday, March 27, 20092
Basics
 A graph G is a pair (V, E), where
 V is a finite set called vertex set of G, and its elements are called vertices
V={v1, v2, v3}
 E is a binary relation on V, i.e. It is called the edge set
of G, and its elements are called edges
E= {(v1, v1), (v1, v2), (v1, v3),
(v2, v1), (v2, v2), (v2, v3),
(v3, v1), (v3, v2), (v3, v3)}

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Lecture #11
Adapted from slides by Dr A. Sattar
Directed Graphs
 A graph G=(V,E) with vertex set
V={v1, v2 , v3,……vn } is called
directed or digraph if
(vi, vj) ≠ (vj, vi) for i ≠ j
 In other words edges (vi, vj) and
(vj, vi), associated with any pair of
vertices vi, vj are considered
distinct
 Self loops (edges from a vertex to
itself) are possible
 For an edge (u, v), we say it is
incident from or leaves vertex u,
and is incident to or enters vertex v

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Lecture #11
Undirected Graphs
 In an undirected graph G=(V,E),
the edge set E consists of
unordered pairs of vertices rather
than ordered pairs i.e. (vi, vj) = (vj,
vi) for i ≠ j
 In an undirected graph, self loops
are forbidden and so every edge
consists of two distinct vertices
 For an edge (u, v), we say it is
incident on vertices u and v

Lecture #11
Adjacency
 If (u, v) is an edge in a graph G=(V, E), we say vertex v is adjacent to
vertex u
 When graph is undirected, the adjacency relationship is symmetric
 When graph is directed, the adjacency relationship is not necessarily
symmetric
 If v is adjacent to u in a directed graph, we say u → v

Lecture #11
Complete Graphs
 A graph in which every pair of vertices is adjacent is called a complete
graph. In other words, a graph which has all the edges is referred to as
complete
 An undirected complete graph , with n vertices, has n(n-1)/2 edges. Its
space complexity is O(n2
) . Such a graph is called dense.
 A complete directed graph with n vertices has n2
edges, including edges
associated with all individual vertices. If individual edges are excluded,
the total number of edges is n(n-1)

Lecture #11
Degree
 In an undirected graph, the number of edges incident on a vertex is
called the degree of a vertex
 A vertex with degree 0 is said to be isolated
 In a directed graph, the out-degree of a vertex is the number of edges
leaving it
 In a directed graph, the in-degree of a vertex is the number of edges
entering it
 In a directed graph, the degree of a vertex is the in-degree + the out-
degree

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Lecture #11
Path
 A path is a sequence of vertices which are connected by consecutive edges
 The number of edges in a path is called the length of the path
 In the figure, path <v1, v2, v4, v6> has length 3
 The path contains the vertices v1, v2, v3, and v4, and edges (v1, v2), (v2, v3),
(v3, v4)
 If there is a path p from u to v, we say v is reachable from u via p
 A path is simple if all vertices in the path are distinct

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Cycles
 In a directed graph, a path < v0,
v1, …. vk> forms a cycle if v0=vk
and the path contains at least one
edge
 In an undirected graph, a path <
v0, v1, …. vk> forms a cycle if
v0=vk and k>=3
 A cycle is simple if <v1, …. vk>
are distinct
 A self loop is a cycle of length 1
 A graph with no cycles is acyclic

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Lecture #11
Connected Graphs
 A undirected graph is said to be connected if there exists a path between all
pairs of vertices. In other words, each vertex is reachable from any other vertex
in the graph.
 Connectedness in an undirected graph is simple because the edges can be
traversed in either direction to reach a vertex.
 A directed graph is strongly connected if every two vertices are reachable from
each other
 A graph can have more than one connected components
 An undirected graph is connected if it has exactly one connected component

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Lecture #11
Weighted Graphs
 A graph in which values w1, w2, w3,……are associated with edges is
called weighted graph. Weights are typically costs or distances in
different applications of graphs.

Lecture #11
Graph Representation
 Two standard ways to represent directed and undirected graphs
 Adjacency list
 Adjacency matrix
 The list representation consists of an array of linked lists, each corresponding to a
vertex.
 The list stores all of the vertices that are linked to a given vertex.
 Let G=(V,E) be a graph. For each u V, the adjacency list Adj(u) contains all the
vertices v such that there is an edge (u ,v ) E
 Vertices in an adjacency list are typically stored in an arbitrary order

Lecture #11
Adjacency List Representation
 If G is a directed graph, the sum of the lengths of all adjacency lists is |E|
 If G is an undirected graph, the sum of the lengths of all adjacency lists is
2|E|

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Lecture #11

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Lecture #11

Lecture #11
Advantages and Disadvantages
 Usually preferred because it provides a compact way to represent
sparse graphs – those for which |E| is much less that |V|2
 For both directed and undirected graphs, the amount of
representation required is θ(V+E)
 Can be readily adapted to represent weighted graphs. The weight
is simply stored with vertex v in u’s adjacency list
 There is no quicker way to determine if a given edge (u, v) is
present in the graph than to search for v in the adjacency list
Adj[u]

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Lecture #11
Adjacency Matrix Representation
 One of the standard ways was of representing graph is to use a matrix to denote
links between pairs of vertices in a graph. The matrix is known as adjacency
matrix
 Let G=(V,E) be a graph, with V={v1,v2,……vn}, and E={(v1,v2),(v1,v3)….}
where n=|V|.
 The adjacency matrix A=(aij) can be represented as
 The size | V|x|V| of adjacency matrix is independent of the number of vertices.
The memory requirement is θ(|V|2
)

Lecture #11
Adjacency Matrix Representation

Lecture #11
 Preferred when the graph is dense – |E| is close to |V|2
 We can tell quickly if there is an edge connecting two given vertices
 Can be readily adapted to represent weighted graphs. The weight is
simply stored as the entry in row u and column v
 The simplicity of an adjacency matrix makes it preferable when graphs
are reasonably small. Also only one bit may be used per entry

Lecture #11
 The in-degree and out-degree of a vertex can be computed by simply
summing the corresponding column and row of the adjacency matrix.
In-degree is number of edges terminating at, and out-degree the number
of edges originating from, a vertex.
 A major disadvantage of matrix representation is inefficient utilization
of storage space if the graph is not dense, that is, there are fewer edges.
For example, the adjacency matrix of a graph with 10 vertices and four
edges would contain 100 elements, only four 1’s. The 96% storage
space would be wasted

Lecture #11
Trees
 Free Trees
 A free tree is a connected, acyclic, undirected graph
 We often omit ‘free’ when we say that a graph is a tree
 If an undirected graph is acyclic but possibly disconnected, it is a
forest

Lecture #11
Properties of Free Trees
 Let G=(V,E) be an undirected graph. The following statements
are equivalent:
 G is a free tree
 Any two vertices in G are connected by a unique simple path
 G is connected, but if any edge is removed from E, the resulting
graph is disconnected
 G is connected and |E|=|V|-1
 G is acyclic and |E|=|V|-1
 G is acyclic, but if any edge is added to E, the resulting graph
contains a cycle

Lecture #11
Rooted Trees
 A rooted tree is a free tree in which one of the vertices is
distinguished from the others
 This distinguished vertex is called the root of the tree
 Consider a node x in rooted tree T with root r. Any node y on the
unique path from r to x is called an ancestor of x
 x is a descendant of y

Graph algorithms

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