Graphs Outline

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01/24/03Algorithm TheoryHiroaki Kobayashi 1

Graphs

ORD

DFW

SFO

LAX

802

1743

1843

1233337


OutlineGraphs

DefinitionApplicationsTerminologyPropertiesADT

Data structures for graphsEdge list structureAdjacency list structureAdjacency matrix structure

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GraphA graph is a pair (V, E), where

V is a set of nodes, called verticesE is a collection of pairs of vertices, called edgesVertices and edges are positions and store elements

Example:A vertex represents an airport and stores the three-letter airport codeAn edge represents a flight route between two airports and stores the mileage of the route

ORD PVD

MIADFW

SFO

LAX

LGA

HNL

849

802

13871743

1843

10991120

1233

337

2555

142


Edge TypesDirected edge

ordered pair of vertices (u,v)first vertex u is the originsecond vertex v is the destinatione.g., a flight

Undirected edgeunordered pair of vertices (u,v)e.g., a flight route

Directed graphall the edges are directede.g., route network

Undirected graphall the edges are undirectede.g., flight network

ORD PVDflight

AA 1206

ORD PVD849miles

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John

DavidPaul

brown.edu

cox.net

cs.brown.edu

att.netqwest.net

math.brown.edu

cslab1bcslab1a

ApplicationsElectronic circuits

Printed circuit boardIntegrated circuit

Transportation networksHighway networkFlight network

Computer networksLocal area networkInternetWeb

DatabasesEntity-relationship diagram


TerminologyEnd vertices (or endpoints) of an edge

U and V are the endpoints of aEdges incident on a vertex

a, d, and b are incident on VAdjacent vertices

U and V are adjacentDegree of a vertex

X has degree 5 Parallel edges

h and i are parallel edgesSelf-loop

j is a self-loop

XU

V

W

Z

Y

a

c

b

e

d

f

g

h

i

j

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P1

Terminology (cont.)Path

sequence of alternating vertices and edges begins with a vertexends with a vertexeach edge is preceded and followed by its endpoints

Simple pathpath such that all its vertices and edges are distinct

ExamplesP1=(V,b,X,h,Z) is a simple pathP2=(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple

XU

V

W

Z

Y

a

c

b

e

d

f

g

hP2


Terminology (cont.)Cycle

circular sequence of alternating vertices and edges each edge is preceded and followed by its endpoints

Simple cyclecycle such that all its vertices and edges are distinct

ExamplesC1=(V,b,X,g,Y,f,W,c,U,a,↵) is a simple cycleC2=(U,c,W,e,X,g,Y,f,W,d,V,a,↵)is a cycle that is not simple

C1

XU

V

W

Z

Y

a

c

b

e

d

f

g

hC2

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PropertiesNotation

n number of verticesm number of edges

deg(v) degree of vertex v

Property 1

Σv deg(v) = 2mProof: each edge is

counted twice

Property 2In an undirected graph

with no self-loops and no multiple edgesm ≤ n (n − 1)/2

Proof: each vertex has degree at most (n − 1)

Examplen = 4m = 6deg(v) = 3

What is the bound for a directed graph?

A simple graph with n vertices has O(n2) edges


Main Methods of the Graph ADTVertices and edges

are positionsstore elements

Accessor methodsaVertex()incidentEdges(v)endVertices(e)isDirected(e)origin(e)destination(e)opposite(v, e)areAdjacent(v, w)

Update methodsinsertVertex(o)insertEdge(v, w, o)insertDirectedEdge(v, w, o)removeVertex(v)removeEdge(e)

Generic methodsnumVertices()numEdges()vertices()edges()

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Edge List StructureVertex object

elementreference to position in vertex sequence

Edge objectelementorigin vertex objectdestination vertex objectreference to position in edge sequence

Vertex sequencesequence of vertex objects

Edge sequencesequence of edge objects

v

u

w

a c

b

a

zd

u v w z

b c d


Adjacency List StructureEdge list structureIncidence sequence for each vertex

sequence of references to edge objects of incident edges

Augmented edge objects

references to associated positions in incidence sequences of end vertices

u

v

wa b

a

u v w

b

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Adjacency Matrix StructureEdge list structureAugmented vertex objects

Integer key (index) associated with vertex

2D-array adjacency array

Reference to edge object for adjacent verticesNull for non nonadjacent vertices

u

v

wa b

2

1

0

210

∅∅

∅

∅∅

a

u v w0 1 2

b


Asymptotic Performance

n2n + mn + mSpace

n2deg(v)mremoveVertex(v)111insertEdge(v, w, o)n211insertVertex(o)

111removeEdge(e)

1min(deg(v), deg(w))mareAdjacent (v, w)ndeg(v)mincidentEdges(v)

Adjacency Matrix

AdjacencyList

EdgeList

n vertices, m edgesno parallel edgesno self-loopsBounds are “big-Oh”

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Depth-First Search

DB

A

C

E


OutlineDefinitions

SubgraphConnectivitySpanning trees and forests

Depth-first searchAlgorithmExamplePropertiesAnalysis

Applications of DFSPath findingCycle finding

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SubgraphsA subgraph S of a graph G is a graph such that

The vertices of S are a subset of the vertices of GThe edges of S are a subset of the edges of G

A spanning subgraph of G is a subgraph that contains all the vertices of G

Subgraph

Spanning subgraph


Connectivity

A graph is connected if there is a path between every pair of verticesA connected component of a graph G is a maximal connected subgraph of G

Connected graph

Non connected graph with two connected components

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Trees and ForestsA (free) tree is an undirected graph T such that

T is connectedT has no cycles

This definition of tree is different from the one of a rooted tree

A forest is an undirected graph without cyclesThe connected components of a forest are trees

Tree

Forest


Spanning Trees and ForestsA spanning tree of a connected graph is a spanning subgraph that is a treeA spanning tree is not unique unless the graph is a treeSpanning trees have applications to the design of communication networksA spanning forest of a graph is a spanning subgraph that is a forest

Graph

Spanning tree

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Depth-First SearchDepth-first search (DFS) is a general technique for traversing a graphA DFS traversal of a graph G

Visits all the vertices and edges of GDetermines whether G is connectedComputes the connected components of GComputes a spanning forest of G

DFS on a graph with nvertices and m edges takes O(n + m ) timeDFS can be further extended to solve other graph problems

Find and report a path between two given verticesFind a cycle in the graph


DFS AlgorithmThe algorithm uses a mechanism for setting and getting “labels” of vertices and edges

Algorithm DFS(G, v)Input graph G and a start vertex v of GOutput labeling of the edges of G

in the connected component of vas discovery edges and back edges

setLabel(v, VISITED)for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLOREDw ← opposite(v,e)if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)DFS(G, w)

elsesetLabel(e, BACK)

Algorithm DFS(G)Input graph GOutput labeling of the edges of G

as discovery edges andback edges

for all u ∈ G.vertices()setLabel(u, UNEXPLORED)

for all e ∈ G.edges()setLabel(e, UNEXPLORED)

for all v ∈ G.vertices()if getLabel(v) = UNEXPLORED

DFS(G, v)

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Example

DB

A

C

E

DB

A

C

E

DB

A

C

E

discovery edgeback edge

A visited vertexA unexplored vertex

unexplored edge


Example (cont.)

DB

A

C

E

DB

A

C

E

DB

A

C

E

DB

A

C

E

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DFS and Maze Traversal The DFS algorithm is similar to a classic strategy for exploring a maze

We mark each intersection, corner and dead end (vertex) visitedWe mark each corridor (edge ) traversedWe keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack)


Properties of DFSProperty 1

DFS(G, v) visits all the vertices and edges in the connected component of v

Property 2The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v

DB

A

C

E

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Analysis of DFSSetting/getting a vertex/edge label takes O(1) timeEach vertex is labeled twice

once as UNEXPLOREDonce as VISITED

Each edge is labeled twiceonce as UNEXPLOREDonce as DISCOVERY or BACK

Method incidentEdges is called once for each vertexDFS runs in O(n + m) time provided the graph is represented by the adjacency list structure

Recall that Σv deg(v) = 2m


Path FindingWe can specialize the DFS algorithm to find a path between two given vertices u and z using the template method patternWe call DFS(G, u) with uas the start vertexWe use a stack S to keep track of the path between the start vertex and the current vertexAs soon as destination vertex z is encountered, we return the path as the contents of the stack

Algorithm pathDFS(G, v, z)setLabel(v, VISITED)S.push(v)if v = z

return S.elements()for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLOREDw ← opposite(v,e)if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)S.push(e)pathDFS(G, w, z)S.pop(e)

elsesetLabel(e, BACK)

S.pop(v)

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Cycle FindingWe can specialize the DFS algorithm to find a simple cycle using the template method patternWe use a stack S to keep track of the path between the start vertex and the current vertexAs soon as a back edge (v, w) is encountered, we return the cycle as the portion of the stack from the top to vertex w

Algorithm cycleDFS(G, v)setLabel(v, VISITED)S.push(v)for all e ∈ G.incidentEdges(v)

if getLabel(e) = UNEXPLOREDw ← opposite(v,e)S.push(e)if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)pathDFS(G, w)S.pop(e)

elseT ← new empty stackrepeat

o ← S.pop()T.push(o)

until o = wreturn T.elements()

S.pop(v)


Breadth-First Search

CB

A

E

D

L0

L1

FL2

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OutlineBreadth-first search

AlgorithmExamplePropertiesAnalysisApplications


Breadth-First SearchBreadth-first search (BFS) is a general technique for traversing a graphA BFS traversal of a graph G

Visits all the vertices and edges of GDetermines whether G is connectedComputes the connected components of GComputes a spanning forest of G

BFS on a graph with nvertices and m edges takes O(n + m ) timeBFS can be further extended to solve other graph problems

Find and report a path with the minimum number of edges between two given vertices Find a simple cycle, if there is one

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BFS AlgorithmThe algorithm uses a mechanism for setting and getting “labels” of vertices and edges

Algorithm BFS(G, s)L0 ← new empty sequenceL0.insertLast(s)setLabel(s, VISITED)i ← 0while ¬Li.isEmpty()

Li +1 ← new empty sequencefor all v ∈ Li.elements()

for all e ∈ G.incidentEdges(v)if getLabel(e) = UNEXPLORED

w ← opposite(v,e)if getLabel(w) = UNEXPLORED

setLabel(e, DISCOVERY)setLabel(w, VISITED)Li +1.insertLast(w)

elsesetLabel(e, CROSS)

i ← i +1

Algorithm BFS(G)Input graph GOutput labeling of the edges

and partition of the vertices of G

for all u ∈ G.vertices()setLabel(u, UNEXPLORED)

for all e ∈ G.edges()setLabel(e, UNEXPLORED)

for all v ∈ G.vertices()if getLabel(v) = UNEXPLORED

BFS(G, v)


Example

CB

A

E

D

discovery edgecross edge

A visited vertexA unexplored vertex

unexplored edge

L0

L1

F

CB

A

E

D

L0

L1

F

CB

A

E

D

L0

L1

F

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Example (cont.)

CB

A

E

D

L0

L1

F

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2


Example (cont.)

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2

CB

A

E

D

L0

L1

FL2

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PropertiesNotation

Gs: connected component of sProperty 1

BFS(G, s) visits all the vertices and edges of Gs

Property 2The discovery edges labeled by BFS(G, s) form a spanning tree Tsof Gs

Property 3For each vertex v in Li

The path of Ts from s to v has iedges Every path from s to v in Gs has at least i edges

CB

A

E

D

L0

L1

FL2

CB

A

E

D

F


AnalysisSetting/getting a vertex/edge label takes O(1) timeEach vertex is labeled twice

once as UNEXPLOREDonce as VISITED

Each edge is labeled twiceonce as UNEXPLOREDonce as DISCOVERY or CROSS

Each vertex is inserted once into a sequence LiMethod incidentEdges is called once for each vertexBFS runs in O(n + m) time provided the graph is represented by the adjacency list structure

Recall that Σv deg(v) = 2m

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ApplicationsUsing the template method pattern, we can specialize the BFS traversal of a graph G to solve the following problems in O(n + m) time

Compute the connected components of GCompute a spanning forest of GFind a simple cycle in G, or report that G is a forestGiven two vertices of G, find a path in G between them with the minimum number of edges, or report that no such path exists

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