7/27/2019 impp

1/27 1Graphs

GRAPHS

Definitions

The Graph ADT

Data structures for graphs

LAX

PVD

LAX

DFW

FTL

STL

HNL

7/27/2019 impp

2/272Graphs

What is a Graph?

A graph G = (V,E) is composed of:

V: set ofverticesE: set ofedges connecting the vertices in V

An edge e = (u,v) is a pair ofvertices

Example:

a b

c

d e

V= {a,b,c,d,e}

E=

{(a,b),(a,c),(a,d),

(b,e),(c,d),(c,e),(d,e)}

7/27/2019 impp

3/273Graphs

Applications

electronic circuits

find the path of least resistance to CS16

networks (roads, flights, communications)

CS16

start

LAX

PVD

LAX

DFW

FTL

STL

HNL

7/27/2019 impp

4/274Graphs

mo better examplesA Spike Lee Joint Production

scheduling (project planning)

wake up

eat

work

cs16 meditation

more cs16

play

make cookiesfor cs16 HTA

sleep

dream of cs16

cs16 program

A typical student day

cxhextris

7/27/2019 impp

5/275Graphs

Graph Terminology

adjacent vertices: connected by an edge

degree (of a vertex): # of adjacent vertices

path: sequence of vertices v1,v2,. . .vk such thatconsecutive vertices vi and vi+1 are adjacent.

a b

c

d e

a b

c

d e

a b e d c b e d c

3

3 3

3

2 deg(v) = 2(# edges)vV

Since adjacent verticeseach count the

adjoining edge, it willbe counted twice

7/27/2019 impp

6/27

7/27/2019 impp

7/277Graphs

Even More Terminology

connected graph: any two vertices are connected bysome path

subgraph: subset of vertices and edges forming agraph

connected component: maximal connectedsubgraph. E.g., the graph below has 3 connectedcomponents.

connected not connected

7/27/2019 impp

8/278Graphs

Caramba! AnotherTerminology Slide!

(free) tree - connected graph without cycles forest - collection of trees

tree

forest

tree

tree

tree

7/27/2019 impp

9/279Graphs

Connectivity

Let n = #verticesm = #edges

- complete graph - all pairs of vertices are adjacent

m= (1/2)deg(v) = (1/2)(n - 1) = n(n-1)/2vV vV

Each of the n vertices is incident to n - 1 edges,however, we would have counted each edge twice!!!Therefore, intuitively, m = n(n-1)/2.

Therefore, if a graph isnot complete,m < n(n-1)/2

n = 5m = (5 4)/2 = 10

7/27/2019 impp

10/2710Graphs

More Connectivity

n = #verticesm = #edges

For a tree m = n - 1

Ifm < n - 1, G is not connected

n = 5m = 4

n = 5m = 3

7/27/2019 impp

11/2711Graphs

Spanning Tree

A spanning tree ofG is a subgraph which

- is a tree

- contains all vertices ofG

Failure on any edge disconnects system (least faulttolerant)

G spanning tree ofG

7/27/2019 impp

12/2712Graphs

AT&T vs. RT&T(Roberto Tamassia & Telephone)

Roberto wants to call the TAs to suggest an

extension for the next program...

One fault will disconnect part of graph!!

A cycle would be more fault tolerant and only

requires n edges

TA

TA

TA

TA

TA

But Plant-Opsaccidentally cutsa phone cable!!!

7/27/2019 impp

13/2713Graphs

Euler and the Bridges ofKoenigsberg

Consider if you were a UPS driver, and you didntwant to retrace your steps.

In 1736, Euler proved that this is not possible

A

B

C

DPregal River

Can one walk across each bridgeexactly once and return at thestarting point?

Gilligans Isle?

7/27/2019 impp

14/2714Graphs

Graph Model(with paralleledges)

Eulerian Tour: path that traverses every edge

exactly once and returns to the first vertex Eulers Theorem: A graph has a Eulerian Tour if and

only if all vertices have even degree

Do you find such ideas interesting?

Would you enjoy spending a whole semester doingsuch proofs?

Well, look into CS22!if you dare...

C

A

B

D

7/27/2019 impp

15/2715Graphs

The Graph ADT

The G ra p h A D T is a positional container whosepositions are the vertices and the edges ofthe graph.

- size() Return the number of vertices plus thenumber of edges ofG.

- isEmpty()

- elements()

- positions()- swap()

- replaceElement()

Notation: Graph G; Vertices v, w; Edge e; Object o

- numVertices()

Return the number of vertices ofG.- numEdges()

Return the number of edges ofG.

- vertices() Return an enumeration of the verticesofG.

- edges() Return an enumeration of the edges ofG.

7/27/2019 impp

16/2716Graphs

The Graph ADT (contd.)- directedEdges()

Return an enumeration of all directededges in G.

- undirectedEdges()Return an enumeration of allundirected edges in G.

- incidentEdges(v)Return an enumeration of all edges

incident on v.- inIncidentEdges(v)

Return an enumeration of all theincoming edges to v.

- outIncidentEdges(v)Return an enumeration of all theoutgoing edges from v.

- opposite(v, e)Return an endpoint ofe distinct from v

- degree(v)Return the degree ofv.

- inDegree(v)Return the in-degree ofv.

- outDegree(v)Return the out-degree ofv.

7/27/2019 impp

17/2717Graphs

More Methods ...- adjacentVertices(v)

Return an enumeration of the vertices

adjacent to v.- inAdjacentVertices(v)

Return an enumeration of the verticesadjacent to v along incoming edges.

- outAdjacentVertices(v)Return an enumeration of the vertices

adjacent to v along outgoing edges.- areAdjacent(v,w)

Return whether vertices v and w areadjacent.

- endVertices(e)

Return an array of size 2 storing theend vertices ofe.

- origin(e)Return the end vertex from which eleaves.

- destination(e)

Return the end vertex at which earrives.

- isDirected(e)Return true iffe is directed.

7/27/2019 impp

18/2718Graphs

Update Methods- makeUndirected(e)

Set e to be an undirected edge.

- reverseDirection(e)Switch the origin and destinationvertices ofe.

- setDirectionFrom(e, v)Sets the direction ofe away from v, oneof its end vertices.

- setDirectionTo(e, v)Sets the direction ofe toward v, one ofits end vertices.

- insertEdge(v, w, o)Insert and return an undirected edgebetween v and w, storing o at thisposition.

- insertDirectedEdge(v, w, o)Insert and return a directed edgebetween v and w, storing o at thisposition.

- insertVertex(o)Insert and return a new (isolated)vertex storing o at this position.

- removeEdge(e)Remove edge e.

7/27/2019 impp

19/2719Graphs

Data Structures for Graphs

A Graph! How can we represent it?

To start with, we store the vertices and the edges intotwo containers, and we store with each edge objectreferences to its endvertices

Additional structures can be used to performefficiently the methods of the Graph ADT

J FK

B OS

MIA

ORD

LAXD FW

S FO

TW 45

AA411

AA13

87

A

A9

03

DL

247

AA523

NW35

UA

877

DL

335

AA49

UA120

J FK

B OS

MIA

ORD

LAXD FW

S FO

7/27/2019 impp

20/2720Graphs

Edge List

The e d g e list structure simply stores the vertices andthe edges into unsorted sequences.

Easy to implement.

Finding the edges incident on a given vertex isinefficient since it requires examining the entireedge sequence

DFWB OS ORDMIA SFOJ FKLAX

D L 247 D L 335 U A 877NW 35 AA 523 AA 411 TW 45UA 120AA 49 AA 903AA 1387

E

V

7/27/2019 impp

21/2721Graphs

Performance of the Edge ListStructure

Operation Timesize, isEmpty, replaceElement, swap O(1)

numVertices, numEdges O(1)

vertices O(n)

edges, directedEdges, undirectedEdges O(m)

elements, positions O(n+m)

endVertices, opposite, origin, destination,

isDirected, degree, inDegree, outDegree

O(1)

incidentEdges, inIncidentEdges, outInci-

dentEdges, adjacentVertices, inAdja-

centVertices, outAdjacentVertices,

areAdjacent

O(m)

insertVertex, insertEdge, insertDirected-

Edge, removeEdge, makeUndirected,

reverseDirection, setDirectionFrom, setDi-

rectionTo

O(1)

removeVertex O(m)

7/27/2019 impp

22/2722Graphs

Adjacency List(traditional)

adjacency list of a vertex v:sequence of vertices adjacent to v

represent the graph by the adjacency lists of all thevertices

Space = (N + deg(v)) = (N + M)

a b

c

d e

b

b

c

c

c

d

a e

a d e

a e

d

a

b

c

d

e

7/27/2019 impp

23/2723Graphs

Adjacency List(modern)

The a d ja c e nc y list structure extends the edge liststructure by adding incidence containers to eachvertex.

The space requirement is O(n + m).

D L 247 D L 335 U A 877NW 35 AA 523 AA 411 TW 45UA 120AA 49 AA 903AA 1387

E

in out in out in out in out in out in out in out

NW 35

DL 247

AA 49

AA 411

UA 120 AA1387

AA 523

UA 877

DL335

AA 49

NW 35 AA1387

AA 903

TW 45

DL 247

AA 903

AA523

AA 411

UA 120

DL 335

UA 877 TW 45

DFWB OS ORDMIA SFOJ FKLAXV

7/27/2019 impp

24/2724Graphs

Performance of the AdjacencyList Structure

Operation Time

size, isEmpty, replaceElement, swap O(1)

numVertices, numEdges O(1)

vertices O(n)

edges, directedEdges, undirectedEdges O(m)

elements, positions O(n+m)endVertices, opposite, origin, destina-

tion, isDirected, degree, inDegree, out-

Degree

O(1)

incidentEdges(v), inIncidentEdges(v),

outIncidentEdges(v), adjacentVerti-ces(v), inAdjacentVertices(v), outAdja-

centVertices(v)

O(deg(v))

areAdjacent(u, v) O(min(deg(u),

deg(v)))

insertVertex, insertEdge, insertDirected-Edge, removeEdge, makeUndirected,

reverseDirection,

O(1)

removeVertex(v) O(deg(v))

7/27/2019 impp

25/2725Graphs

Adjacency Matrix(traditional)

matrix M with entries for all pairs of vertices

M[i,j] = true means that there is an edge (i,j) in thegraph.

M[i,j] = false means that there is no edge (i,j) in thegraph.

There is an entry for every possible edge, therefore:Space = (N2)

F T T T FT F F F TT F F T TT F T F TF T T T F

a b

c

d e

a b c d e

abcd

e

7/27/2019 impp

26/2726Graphs

Adjacency Matrix(modern)

The adjacency matrix structures augments the edgelist structure with a matrix where each row andcolumn corresponds to a vertex.

BOS DFW JFK LAX MIA ORD SFO0 1 2 3 4 5 6

The space requirement is O(n2 + m)

0 1 2 3 4 5 6

0 NW

35

DL

247

1 AA49

DL

335

2 AA1387

AA

903

TW

45

3 UA

120

4 AA523

AA

411

5 UA877

6

7/27/2019 impp

27/27

Performance of the AdjacencyMatrix Structure

Operation Timesize, isEmpty, replaceElement, swap O(1)

numVertices, numEdges O(1)

vertices O(n)

edges, directedEdges, undirectedEdges O(m)

elements, positions O(n+m)

endVertices, opposite, origin, destination,

isDirected, degree, inDegree, outDegree

O(1)

incidentEdges, inIncidentEdges, outInci-

dentEdges, adjacentVertices, inAdja-

centVertices, outAdjacentVertices,

O(n)

areAdjacent O(1)

insertEdge, insertDirectedEdge, remov-

eEdge, makeUndirected, reverseDirection,

setDirectionFrom, setDirectionTo

O(1)

insertVertex, removeVertex O(n2)