CS2420: Lecture 36

1
CS2420 Lecture 36

• Vladimir Kulyukin
• Computer Science Department
• Utah State University

2
Outline

• Graph Algorithms (Chapter 9)

3
Weighted Graphs

• Weighted graphs and weighted digraphs (networks)
  have weights assigned to their edges.
• Weighted graphs can be represented as adjacency
  matrices or adjacency lists.

4
Weighted Graphs Example
5
A
B
7
1
4
C
D
2
5
Paths

• A path from vertex u to vertex v in a graph G is
  a sequence of adjacent (connected by an edge)
  vertices that starts with u and ends in v.
• If all vertices in a path are distinct (except
  possibly for the first and last one), the path is
  simple.
• The length of the path is the number of edges in
  the path.

6
Cycles

• A cycle is a simple path of positive length that
  starts and ends in the same vertex.
• A graph with cycles is cyclic.
• A graph without cycles is acyclic.
• DAG stands for directed acyclic graph.

7
Graphs Connectivity

• An undirected graph G is connected if, for every
  pair of vertices u and v, there is a path from u
  to v.
• A directed graph with the same property is called
  strongly connected.
• If every vertex is a ball and every edge is a
  string, one can pick up any ball and lift the
  entire graph without losing any other balls.

8
Topological Sorting

• Let G (V, E) be a directed acyclic graph (dag).
• Find a list of vertices such that for every edge
  (u, v) in E, u is listed before v.

9
Topological Sorting Motivation

• There are five required courses for a student to
  take.
• The courses are C1, C2, C3, C4, and C5.
• C1 and C2 have no prerequisites.
• C3s prerequisites are C1 and C2.
• C4 has C3 as its prerequisite.
• C5 has C3 and C4 as its prerequisites.

10
Topological Sorting Motivation
C4
C1
C3
C2
C5
11
Topological Sorting

• Edges
• (C1, C3), (C2, C3), (C3, C4), (C3, C5), (C4, C5)
• This graph has two possible topological sortings
• C1, C2, C3, C4, C5
• C2, C1, C3, C4, C5

12
Topological Sorting Algorithm

• A source is a vertex with no incoming edges, i.e.
  the indegree of the source is 0.
• Identify a source.
• Delete it with all its outgoing edges from the
  graph.
• Keep going until no more sources are left.
• If, at any point, a source cannot be found, stop
  and return false.
• Make sure that all vertices are included in the
  topological sorting.

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Topological Sorting Pseudocode

• TopologicalSort(G)
• Initialiaze Q int counter 0
• for each vertex V in G
• if ( V.indegree 0 ) Q.Push(V)
• while ( !Q.IsEmpty() )
• V Q.Pop()
• V.TopNum counter
• for each W adjacent to V
• W.indegree--
• if ( W.indegree 0 )
• Q.Pop(W)
• if ( counter ! NUM_VERTICES )
• return an error, because G has a
  cycle

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Topological Sorting Example
C4
C1
C3
C2
C5
C1 is a source Delete C1 and its outgoing edges.
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Topological Sorting Example
C4
C3
C2
C5
C2 is a source Delete C2 and its outgoing edges.
16
Topological Sorting Example
C4
C3
C5
C3 is a source Delete C3 and its outgoing edges.
17
Topological Sorting Example
C4
C5
C4 is a source Delete C4 and its outgoing edges.
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Topological Sorting Example
C5
C5 is a source, and we are done.
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Topological Sort Run Time

• O(V E)