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CSC2100BTutorial 10

Graph

Jianye Hao

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Outline Graph Adjacency Representation Topological Sort Minimum Spanning Tree

Kruskal’s Algorithm Prim’s Algorithm

Shortest Path Dijkstra’ Algorithm

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Graph - adjacency representation

Adjacency matrix

G = (V, E)V = { A, B, C, D, E }E = { (A, C), (A, E), (B, D), (C, B), (D, C), (E, C), (E, D)

}

ABCDE

0 0 1 0 10 0 0 1 00 1 0 0 00 0 1 0 00 0 1 1 0

A B C D E

A

B

C D

E

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Degree of a vertex v number of edges incident on that vertex For directed graph, Degree = InDegree + OutDegree

InDegree of a vertex v sum of column v

OutDegree of a vertex v sum of row v

A

BC D

E

For example, InDegree for C is 3, OutDegree for C is 1

Graph - adjacency representation

ABCDE

0 0 1 0 10 0 0 1 00 1 0 0 00 0 1 0 00 0 1 1 0

A B C D E

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Adjacency matrix

G = (V, E)V = { A, B, C, D, E }E = { (A, C), (A, E), (B, D), (C, B), (D, C), (E, C), (E, D) }

ABCDE

0 0 1 0 10 0 1 1 01 1 0 1 10 1 1 0 11 0 1 1 0

A B C D E

A

B

C D

E

Graph - adjacency representation

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Adjacency list

ABCDE

A

B

C D

E

C EDBCC D

Graph - adjacency representation

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Topological Sort A topological sort of a DAG (Directed Acyclic

Graph) G is a linear ordering of all its vertices such that if G contains an edge (u, v), then u appears before v in the ordering.

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Topological Sort

V1 V2

V7

V3

V6

V4 V5

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How to find the topologicalordering? Define the indegree of a vertex v as the # of

edges (u,v). We compute the indegrees of all vertices in the graph.

Find any vertex with no incoming edges (or the indegree is 0).

We print this vertex, and remove it, along with its edges, from the graph.

Then we apply this same strategy to the rest of the graph.

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V1 V2

V7

V3

V6

V4 V5

V2

V7

V3

V6

V4 V5

V7

V3

V6

V4 V5

V7

V3

V6

V4

V7

V3

V6

V7V6 V6

Topological Order V1 V2 V5 V4 V3 V7 V6

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Real Life Application

Course prerequisite in university studies A directed edge (u,v) indicates that course u must be

completed before course v A topological ordering of these courses is any course

sequence that does not violate the prerequisite requirement.

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Notes Is topological ordering possible with a cyclic

graph? No, since for two vertices v and w on the

cycle, v precedes w and w precedes v. Is the ordering unique?

It is not necessarily unique; any legal ordering will do (e.g., v1, v2, v5, v4, v3, v7, v6 and v1, v2, v5, v4, v7, v3, v6 are topological orderings).

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Spanning Tree

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Minimum Spanning Tree

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Real Life Application One example would be a cable

TV company laying cable to a new neighborhood.

The graph represents which houses are connected by those cables.

A spanning tree for this graph would be a subset of those paths that has no cycles but still connects to every house.

There might be several spanning trees possible. A minimum spanning tree would be one with the lowest total cost.

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MST Algorithm

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Kruskal’s Algorithm

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4Edge Weight<h, g> 1<c, i> 2<g, f> 2<a, b> 4<c, f> 4<g, i> 6<c, d> 7<h, i> 7<a, h> 8<b, c> 8<d, e> 9<e, f> 10<b, h> 11<d, f> 14

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Kruskal’s Algorithm

Edge Weight<h, g> 1<c, i> 2<g, f> 2<a, b> 4<c, f> 4<g, i> 6<c, d> 7<h, i> 7<a, h> 8<b, c> 8<d, e> 9<e, f> 10<b, h> 11<d, f> 14

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Algorithm will stop here since there are already (n-1) edges found.

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Prim’s Algorithm

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Prim’s Algorithm

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The EndAny Questions?