LECTURE 32:GRAPH TRAVERSALS

CSC 213 – Large Scale Programming

Today’s Goals

Make Britney sad through my color choices

Revisit issue of graph terminology and usage Subgraphs, spanning, & connected graphs,

oh my! Problems solved or why you should care

about these Just rats in a maze: using graphs to find

the exit Confident in your decisions: how to travel

depth-first Using bread-first searching to cover all the

bases 2 algorithms both alike in dignity: how do

they differ?

Today’s Goals

Make Britney sad through my color choices

Revisit issue of graph terminology and usage Subgraphs, spanning, & connected graphs,

oh my! Problems solved or why you should care

about these Just rats in a maze: using graphs to find

the exit Confident in your decisions: how to travel

depth-first Using bread-first searching to cover all the

bases 2 algorithms both alike in dignity: how do

they differ?

Graphs Solve Many Problems

Subgraphs

Subgraph of G contains edges & vertices from G

Graph

Subgraphs

Subgraph of G contains edges & vertices from G

Subgraph

Subgraphs

Subgraph of G contains edges & vertices from G

Spanning subgraph is subgraph containing all vertices in G

Subgraph

Graph

Subgraphs

Subgraph of G contains edges & vertices from G

Spanning subgraph is subgraph containing all vertices in G

Subgraph

Spanning subgraph

Connected Graphs & Subgraphs Connected if path

exists between all vertex pairs Requires path, not

edge, between vertices

Connected Graph

Connected Graphs & Subgraphs Connected if path

exists between all vertex pairs Requires path, not

edge, between vertices

Disconnected Graph

Connected Graphs & Subgraphs Connected if path

exists between all vertex pairs Requires path, not

edge, between vertices

Connected component is subgraph containing all connected vertices Often called

“maximally connected”

Disconnected Graph

Graph

Connected Graphs & Subgraphs Connected if path

exists between all vertex pairs Requires path, not

edge, between vertices

Connected component is subgraph containing all connected vertices Often called

“maximally connected”

Disconnected Graph

Graph with 2 connected components

Connected Graphs & Subgraphs Connected if path

exists between all vertex pairs Requires path, not

edge, between vertices

Connected component is subgraph containing all connected vertices Often called

“maximally connected”

Disconnected Graph

Graph with 2 connected components

Tree

Connected acyclic graph Resembles Tree

structure

Graph

Tree

Connected acyclic graph Resembles Tree

structure

Tree

Tree

Connected acyclic graph Resembles Tree

structure

Don’t lose forest for trees Forest is graph with

multiple trees

Tree

Graph

Tree

Connected acyclic graph Resembles Tree

structure

Don’t lose forest for trees Forest is graph with

multiple trees

Tree

Tree

Tree

Connected acyclic graph Resembles Tree

structure

Don’t lose forest for trees Forest is graph with

multiple trees

Tree

Forest

Spanning Tree

Subgraph that is both spanning subgraph & tree Contains all vertices in graph spanning

subgraph Tree connected without any cycles

Graph

Spanning Tree

Subgraph that is both spanning subgraph & tree Contains all vertices in graph spanning

subgraph Tree connected without any cycles

Tree

Spanning Tree

Subgraph that is both spanning subgraph & tree Contains all vertices in graph spanning

subgraph Tree connected without any cycles

Spanning subgraph

Spanning Tree

Subgraph that is both spanning subgraph & tree Contains all vertices in graph spanning

subgraph Tree connected without any cycles

Spanning tree

Spanning Tree

Subgraph that is both spanning subgraph & tree Contains all vertices in graph spanning

subgraph Tree connected without any cycles

Tree Than Spans Royals?

Depth- & Breadth-First Search Common graph traversal algorithms DFS & BFS traversals have common

traits Visits all vertices and edges in G Determines whether of not G is connected Finds connected components if G not

connected Heavily used to solve graph problems

DFS and Maze Traversal

Classic maze strategy Intersection, corner,

& dead end are vertices

Corridors become edges

Traveled corridors marked

Marks trace back to start

DFS Example

DB

A

C

E

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

DFS Example

DB

A

C

E

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

DFS Example

DB

A

C

E???

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

DFS Example

DB

A

C

E

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

DFS Example

DB

A

C

E

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

DFS Example

DB

A

C

E

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

DFS Example

DB

A

C

E

discovery edgeback edge

A visited vertex

A unexplored vertex

unexplored edge

Properties of DFS

Connected component’s vertices & edges visited

Edges visited form spanning tree for component

DB

A

C

E

Properties of DFS

Connected component’s vertices & edges visited

Edges visited form spanning tree for component

DB

A

C

E

L0

BFS Example

DB

A

C

E

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

F

cross edge

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L2

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L2

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L2

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L2

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

L2

L1

L0

BFS Example

DB

A

C

E

F

discovery edge

A visited vertex

A unexplored vertex

unexplored edge

cross edge

Properties of BFS

Some properties similar to DFS: Visits all vertices & edges in connected

component Component’s spanning tree from discovery

edges For each vertex v in Li

Spanning tree has exactly i edges on path from s to v

All paths in G from s to v have at least i edges

L2

L1

L0

B

A

C

D

C

E

F

DFS vs. BFS

DFS BFS

Back edge (v,w) w is ancestor of v in tree

of discovery edges

Cross edge (v,w) v in same or previous

level as w in tree of discovery edges

L2

L1

L0

B

A

C

D

C

E

F

DB

A

C

E

For Next Lecture

Weekly assignment available & due Tuesday Gives you good chance to check your

understanding

Next Fri. 1st check-in for program assignment #3 Planning now saves time later!

Read 13.4.1 – 13.4.3 for class on Monday When using directed edges, what changes

needed? Other computations possible for directed

graphs?