Discussion #30 1/23 Discussion #30 Graph Definitions and Applications.

Discussion #30 1/23

Discussion #30

Graph Definitions and Applications

Discussion #30 2/23

Topics

• Motivation — graph applications

• Basic definitions

Discussion #30 3/23

Graphs: Very Useful Abstractions

• Graphs apply to many diverse areas: social sciences, linguistics, physical sciences, communication engineering, and many more.

• Why? Because they are a useful abstraction.– Abstractions capture the essence of a real-world problem.– Abstractions discard irrelevant details and let us focus on the

essence of the problem to be solved.– Abstractions lend themselves to precise (mathematical)

specification.– Mathematically defined abstractions lead to correct computer

implementations.

Discussion #30 4/23

Graphs in Programming

• Testing: paths in programs:

• Call graphs in programs• Software products consist of modules that invoke

each other.• Deadlock in operating systems

…

T F T

FT

F

Sequence If-then-else While Repeat-until Case

Discussion #30 5/23

Graphs in Transportation& Product Delivery

• Airline connections: getting from here to there

• Highways with mileage between cities

• Traveling salesman problem: visit all cities with minimal distance

• Traffic networks studied by transportation and city planners

• Pipelines for delivering water or gas

Discussion #30 6/23

Graphs in Networks• Social networks

– Friend of a friend– Co-collaborators

• Computer networks– Connect all, minimal number of connections– (Minimal) spanning trees, LANs, WANs

• Scheduling networks– Workflow from source to sink– Critical path, slack time– Earliest starting time, latest completion time

Discussion #30 7/23

Graphs in Database Design• Database design

– hypergraphs– equivalence classes (of 1-1 relationships) plus functional relationships

yield a DB with no redundancy.

• Generated, redundancy free relations– snap(StudentID, Name, Address, Phone); csg(Course, StudentID, Grade)– cr(Course, Room); cdh(Course, Day, Hour); cp(Course, Prerequisite)

Prerequisite

Course

StudentID

Grade

Room

Phone

NameAddress

Day

Hour

Discussion #30 8/23

Problems that Don’t Initially Look Like Graph Problems

Have you ever had to work your way through a maze of interpreters at an international conference?

Swedish and Indonesian delegates wish to talk to each other, but among the sixteen interpreters, not one speaks both Swedish and Indonesian. A way to solve this problem is to form a chain of interpreters.

There are, however, only four booths; interpreters must be in booths, and each booth can contain only one interpreter.

Problem: Does a solution exist? If so, which four interpreters will you place in the booths?

Discussion #30 9/23

Interpreters at the Great World Conference

Swedish—(

Booth #1 Booth #2 Booth #3 Booth #4

)—Indonesian

(1)Portuguese

Indonesian

(2)Polish

English

German

(3)Italian

Norwegian

English

(4)Korean

Turkish

(5)English

Polish

(6)Swedish

Turkish

(7)Spanish

Chinese

Japanese

(8)French

Portuguese

(9)Dutch

German

(10)Swedish

Norwegian

(11)Japanese

Indonesian

Russian

(12)Dutch

French

Chinese

Russian

Portuguese

(13)Polish

Italian

Swedish

(14)German

Chinese

Korean

(15)Dutch

Spanish

Indonesian

(16)French

English

Discussion #30 10/23

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B

Chinese 1 1 1

Dutch 1 1 1 1

English 1 1 1 1

French 1 1 1

German 1 1 1

Indonesian 1 1 1 1

Italian 1 1

Japanese 1 1

Korean 1 1

Norwegian 1 1

Polish 1 1 1

Portuguese 1 1 1

Russian 1 1

Spanish 1 1

Swedish 1 1 1 1

Turkish 1 1

Table of Spoken Languages


A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B

A 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0

1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1

2 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0

3 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0

4 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0

5 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0

6 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0

7 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0

8 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0

9 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0

10 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0

11 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1

12 0 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0

13 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0

14 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0

15 0 1 0 0 0 0 0 1 0 1 0 1 1 0 0 0 0 1

16 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 0

B 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

“Can Speak to” Matrix


Graph of “Can Speak to” Matrix

10 3Norwegian English

2

Eng

lish

English

13Polish

Swed

ish

6

Swedish

Sw

edis

h

5

Polish

Polis

h

English

Eng

lish

14

Ger

man

9German

Germ

an

4

Turkish

Korean

7

Dutch

Dutch

DutchChinese

French

Chinese Chinese

Span

ish

8

1

11

Port

ugue

se

Portuguese

Portuguese

French

Japa

nese

Indonesian

Indonesian

Rus

sian

Indo

nesi

an

Italian

Swedish

Indonesian15

12

A B

Indonesian

Indo

nesi

an

Swedish

Swedish

16

French


Project 5: Graph-Based Optimizationh :- P1, P2,…h,…Pn

• Acyclic: partial ordering, single pass

• Cyclic: multiple passes, least fixed point

• In both cases, we can ignore all rules not in the graph

h

P1 P2 … Pnh1 :- …h2,…h2 :- …h3,…h3 :- …h1,…

h1

h2

h3

h :- P1, P2, …, Pn

P1 :- … h

P1 P2 … Pn


Graph Definition• A graph G = (V, E, f) is

– a set of vertices V (sometimes also called nodes)– a set of edges E (sometimes also called arcs)– a function f from the set of edges E to the set of ordered

and unordered pairs of elements of V.

• ExampleV = {1, 2, 3}, E = {e1, e2, e3, e4, e5},

f = { (e1,(1,2)), (e2,{1,2}), (e3,(1,2)), (e4,(1,1)),(e5,{2,2}) }

1 2

3

e1

e2

e3

e4 e5

• Hypergraph: (e6,{1,2,3})

e6

Multiset: set with duplicates


“Weird” Variations of Graphs

• All kinds of “weird” variations and vocabulary:– Hypergraph: has edges with more than 2 nodes– Multigraph: has parallel edges– Mixed graph: some nodes directed, some nodes

undirected

• We are not going to study “weird” graphs.

• How do we make graphs simple?


Simple Graph = Graph• Eliminate f• Make edges either all ordered or all unordered.

G = (V, E), whereV is a set of nodes, andE is a set of all ordered or all

unordered pairs over V.• Example

Directed Graph or Digraph

Unordered graph, with multiset (e.g. {2, 2}) if needed1

2 3

V = {1, 2, 3}

E = {(1, 2), (2, 3), (3, 3)}


Vocabulary for Simple Graphs

– d,u incident edge: edge “touching” a node

– d initiating node,originating node,tail node,predecessor

– d terminating node,ending node,head node,successor

(“d” for directed and “u” for undirected)


Vocabulary (continued…)

– d out degreen…

m …– d in degree

– d,u total degree = in + out degree = degree (Note: a node with only a loop has

degree 2.)– d,u subgraph:H is a subgraph of G, H G,

if H is a graph and V(H) V(G)

and E(H) E(G).


Vocabulary (continued…)– u adjacent nodes: if there is an edge connecting them– d adjacent nodes: node b is adjacent to node a if there

is a directed edge from a to b– u complete: every node adjacent to every other node

(or to every node)

K1 K3K2

– u bipartite:- Two groups of nodes (often

different kinds)- Edges only connect between

groups (the two kinds)

K1K2 K4

… …


Vocabulary (continued…)– d,u planar: can be drawn with no crossing lines

A graph is planar if and only if it does not contain either K5 or B3 as a subgraph.

K5

B3


Vocabulary (continued…)

– d path:

– sequence of edges such that the head of the previous is the tail of the next

– usually written as a sequence of nodes

<1, 3, 2, 4> is a path1

2

4

3

5

<1, 3> is a path

<5, 4, 3, 2, 4, 3> is a path

– d length of path = number of edges = number of nodes 1. (<1> is a degenerate path of

length 0.)

– u path & length: same idea (except can go both ways)


Vocabulary (continued…)– d,u simple path (elementary path): all nodes distinct– d cycle: path that starts and stops at the same node

e.g. <4, 3, 2, 4>, <2, 4, 3, 2, 4, 3, 2>– d simple cycle: no edge appears more than once

e.g. <4, 3, 2, 4> is simple, <2, 4, 3, 2, 4, 3, 2> is not– d elementary cycle: no node (except the first/last) appears

twicee.g. <1, 2, 4, 2, 3, 1> is simple but not elementary;

<1, 2, 3, 1> and <2, 4, 2> are both elementary– u cycle: path that starts and stops at the same node; no edge

traversed in both directions.– u simple cycle: path of length 3 where only the first and

last nodes are repeated.

not a simple cycle not a simple cycle simple cycle– d,u acyclic: no cycle in graph


Vocabulary (continued…)– d path relation: {(x,y)| a path exists from x to

y}– d,u reachability of y from x: a path exists from x to y– d,u minimum path from x to y: a path of minimum

length from x to y– d,u distance from x to y: minimum path length– u connected: any two nodes reachable from

each other– d connected: ignore direction and use the

definition for undirected graphs

disconnected connected

Discussion #30 1/23 Discussion #30 Graph Definitions and Applications.

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