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Graphs and Digraphs - Examples

An (undirected) graph G = (V,E)

A B C D E F A F B E C D

A B C D E F

A B C D E F

0 1 0 1 0 1

1 0 0 0 1 0

0 0 0 1 0 0

1 0 1 0 1 1

0 1 0 1 0 0

1 0 0 1 0 0

B D F A E D A C B D E A D F adjacency matrix for G adjacency list for G 2 /2).

Adjacency matrix: Θ(n

2 ) space. An algorithm that examines the entire graph structure will require

Ω(n2

) time. Adjacency list: Θ(n+e) space. An algorithm that examines the entire graph structure will require Ω(n+e) time. Often , e << n 2 . In this case, the adjacency list may be preferable.

A digraph

G = ( V, E)

A B C D E F

A B C D E F

A B C D E F

0 1 0 0 0 1

0 0 0 0 1 0

0 0 0 0 0 0

1 0 1 0 0 0

0 1 0 1 0 0

0 0 0 1 0 0

B F E A C B D D adjacency matrix for G adjacency list for G In a digraph, e may be as high as n(n-1) ≈ n 2 , but otherwise the remarks on the previous page hold. A F B C D E

A weighted digraph

G = (V, E,W)

A B CDEF

A B C D E F ∞ 3 ∞ ∞ ∞ 12 ∞ ∞ ∞ ∞ 11 ∞

9 ∞ 8 ∞ ∞ ∞

∞ 14 ∞ 17 ∞ ∞ ∞ ∞ ∞21∞∞ A B C D E F B F E A C B D D adjacency matrix for G adjacency list for G In the adjacency matrix, a non-existent edge might be denoted by 0 or ∞. For example, a non-existent edge could represent i) a capacity of 0, or ii) a cost of . 12 17 9 3 11 21 14
8 A F B C D E 123
11 9 14 21
8 17

Directed Acyclic Graphs (DAGs)

In any digraph, we define a vertex v to be

a source , if there are no edges leading into v, and a sink if there are no edges leading out of v.

A directed acyclic graph

(or DAG) is a digraph that has no cycles.

Example of a DAG:

Theorem

Every finite DAG has at least one source, and at least one sink. In fact, given any vertex v, there is a path from some source to v, and a path from v to some sink. A F B C D E G a sink a source Note: This theorem need not hold in an infinite DAG. For example, this DAG has neither a source nor a sink. Note: In any digraph, the vertices could represent tasks, and the edges could represent constraints on the order in which the tasks be performed. For example, A must be performed before B, F, or G.

B must be performed before C or E.

C must be performed before G.

D must be performed before C.

E must be performed before D.

F must be performed before D.

We will see that the constraints are consistent if any only if the digraph has no cycles, i.e., is a DAG.

A topological sort

of a digraph G = (V,E) is labeling of the vertices by 1, 2, ..., |V| (or by elements of some other ordered set) such that (u,v) is a edge label(u) < label(v). We will see that a digraph has a topological sort if and only if it is a DAG. For a tasks / constraints graph, a topological sort provides an order in which the tasks can be performed serially, and conversely any valid order for performing the tasks serially gives a topological sort. ... ... -3 -2 -12 103

Strongly Connected Components of a Digraph

If G is a digraph, define a relation ~ on the vertices by: a ~ b is there is both a path from a to b, and a path from b to a. This is an equivalence relation. The equivalence classes are called the strong components of G.

G is strongly connected

if it has just one strong component.

This digraph has five strong components.

A B C D E H I G J K L Q P ON F M Given a strongly connected digraph G, we may form the component digraph G SCC as follows: i)

The vertices of G

SCC are the strongly connect components of G. ii)

There is an edge from

v to w in G SCC if there is an edge from some vertex of component v to some vertex of component w in G. Theorem: The component graph of a digraph is a DAG. Here is the component digraph for the digraph on the preceding page. 3 4 5 1 2quotesdbs_dbs17.pdfusesText_23