Graph Theory Seminar

7/22/2019 Graph Theory Seminar

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GRAPH THEORYBy: Puneet Patil


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A model

Many structures in real world can be represented on

paper by means of a diagram consisting of a set of points

together with lines joining some or all pairs of these points.

For example,

Avertex: an objectAn e ge : a relation between two objects


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What is a graph?

A graph G consists of a set of vertices and a set of edges.

G = {V,E}

In this example, V = {A,B,C,D,E,F}, E = {AB, BC, BD, CD,

DF, DE,EF}.Any graph can be drawn on paper in many waysthe

important thing is which vertices are connected (adjacent)

to each other.


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Why study graph theory?

I Useful set of techniques for solving real-world problems

particularly for different kinds of optimization.

I Graph theory is useful for analyzing things that are

connected to other things, which applies almost

everywhere.

I Some difficult problems become easy when represented

using a graph.

I There are lots of unsolved questions in graph theory:

solve one and become rich and famous1.


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Graphs Networks

Graph

(Network)

Vertexes

(Nodes)

Edges

(Arcs)Flow

CommunicationsTelephonesexchanges,

computers, satellites

Cables, fiber optics,microwave relays Voice, video, packets

CircuitsGates, registers,processors

Wires Current

Mechanical JointsRods, beams,springs

Heat, energy

HydraulicReservoirs, pumping

stations, lakes Pipelines Fluid, oil

Financial Stocks, currency Transactions Money

TransportationAirports, rail yards,street intersections

Highways, rail beds,airway routes

Freight, vehicles,passengers


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Adjacent, Loop, Multiple edges Two vertices are adjacent and are neighbors if they are

the endpoints of an edge

Example:

Aand Bare adjacent

Aand Dare not adjacent

Loop: An edge whose endpoints are equal.

Multiple edges : Edges have the same pair of endpoints.

Simple graph : A graph has no loops

or multiple edges.


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Directed Graph

An edge e Eof a directed graph is

represented as an ordered pair (u,v),where u,

v V. Here uis the initial vertex and vis the

terminal vertex. Also assume here that u v 2

4

31

V= { 1, 2, 3, 4}, | V | = 4

E= {(1,2), (2,3), (2,4), (4,1), (4,2)}, | E|=5


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Undirected Graph

2

4

31

V= { 1, 2, 3, 4}, | V | = 4

E= {(1,2), (2,3), (2,4), (4,1)}, | E|=4

An edge e Eof an undirected graph is

represented as an unordered pair

(u,v)=(v,u),where u, v V. Also assume

that u v


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Degree of a Vertex

Degreeof a vertex in an undirected graph is the numberof edges incident on it. In a directed graph, theout

degreeof a vertex is the number of edges leaving it andthe in degreeis the number of edges entering it

2

4

31

2

4

31

The degreeof vertex 2

is 3

The indegreeof vertex 2

is 2 and the in degreeof

vertex 4 is 1


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Weighted Graph

A weighted graph is a graph for which each edge has an

associated weight, usually given by a weight function w: E

R

2

4

31

2

4

31

1.2

2.1

0.2

0.5 4

8

6

2

9


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Walks and Paths3

2

3

4

1

1

V5V4

V3V2

V1V6

4

1

A walkis an sequence of nodes (v1, v2,..., vL) such that

{(v1, v2), (v1, v2),..., (v1, v2)} E, e.g. (V2, V3,V6, V5,V3)

A simple pathis a walk with no repeated nodes, e.g. (V1,

V4,V5, V2,V3) A cycleis an walk (v1, v2,..., vL) where v1=vLwith no other

nodes repeated and L>3, e.g. (V1, V2,V5, V4,V1)

A graph is called cyclicif it contains a cycle; otherwise it iscalled acyclic


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Complete Graphs

A

D

C

B

4 nodes and (4*3)/2 edges

Vnodes and V*(V-1)/2

edges

C

A

B

3 nodes and 3*2 edges

Vnodes and V*(V-1)

edges

A complete graph is an undirected/directed graph in which

every pair of vertices is adjacent. If (u, v ) is an edge in agraph G, we say that vertex v is adjacent to vertex u.


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Connected GraphsA

D E F

B C

A B

C D

An undirected graph is connectedif

you can get from any node to any

other by following a sequence ofedges OR any two nodes are

connected by a path


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Bipartite Graph

A bipartite graph

is an undirected graph

G = (V,E) in which V can be

partitioned into 2 sets V1 and V2

such that ( u,v) E implies

either

u V1 and v V2

OR

v V1 and u V2.

u1

u2

u3

u4

v1

v2

v3

V1 V2


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Trees

A

B

D

F

C

E

Let G = (V, E) be an undirected graph.

The following statements are equivalent,

1. Gis a tree

2. Any two vertices in Gare connected

by unique simple path3. Gis connected, but if any edge is

removed from E, the resulting graph is

disconnected

4. G is connected, and | E| = | V| -15. G is acyclic, and | E| = | V| -1

6. G is acyclic, but if any edge is added

to E, the resulting graph contains a

cycle


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Planar graph

If graph can be drawn in the plane without any of its

edges crossing each other, then it is called Planar graph.

Otherwise its called as Non planar graph.

Non planar Planar


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Adjacency matrix

LetG= (V, E), |V| = nand|E|=m

The dj cency m trixofGwrittenA(G), is the n-by-nmatrix in which entry ai,jis the number of edges inG

with endpoints {vi, vj}.

a

b

c

d

e

w

x

y z

w x y z

0 1 1 0

1 0 2 0

1 2 0 1

0 0 1 0

w

x

y

z


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Incidence Matrix

LetG= (V, E), |V| = nand|E|=m

The incidence m trix M(G) is the n-by-mmatrix inwhich entrymi,jis1 ifviis an endpoint ofeiand

otherwise is 0.

a

b

c

d

e

w

x

y z

a b c d e

1 1 0 0 0

1 0 1 1 0

0 1 1 1 1

0 0 0 0 1

w

x

y

z


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Isomorphism

An isomorphism from a simple graph Gto a simplegraphHis a bijection f:V(G)V(H) such thatuvE(G) if

and only if f(u)f(v) E(H)

We say G s isomorphic to H, writtenG H

HG

w

x z

y c d

ba

f1: w x y z

c b d a

f2: w x y z

a d b c

20


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20

Example:H1, H2, andH3are subgraphs of G

c

d

a b

de

a b

c

deH1

G

H3H2

a b

cde

Subgraph


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Clique- is a complete sub graph.

Clique number (G) - is cardinality of its largest clique.

Clique cover number (G) - is the minimum number

of cliques in G needed to cover the vertex set of G. Stable set (G) - is a subset of vertices with the property

that no two vertices in the stable set are adjacent.

Chromatic number (G) - is the smallest number of

colors needed to color the vertices of so that no twoadjacent vertices share the same color.

(G) (G)

(G) (G)


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Branch & Bound algorithm

To apply BnB, one must be able to partition the solution

space into subsets. Often the subsets are arranged in a

tree structure. Leaves in the tree are solutions. Internal

nodes represent the set of solutions below them in the

tree (the subtree).


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How Branch-and-Bound Works Some nodes in the tree are active.These are the nodes

waiting to be explored. Branching refers to exploring the sub tree of an active

node.

Bounding refers to estimating a bound on the solutions in

the sub tree rooted at the active node.


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The Branch-and-Bound Algorithmbegin

activeset :=f0g;

bestval:=NULL;

currentbest:=NULL;

while activeset is not empty do

choose a branching node, node k 2 activeset;

remove node k from activeset;

generate the children of node k, child i, i=1,. . . ,nk,

and corresponding optimistic bounds obi;

for i=1 to nk do

if obi worse than bestval then kill child i;

else if child is a complete solution thenbestval:=obi, currentbest:=child i;

else add child i to activeset

end for

end while

end


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Example


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Dynamic Programming

Dynamic Programming is a general algorithm designtechnique for solving problems defined by or formulated asrecurrences with overlapping sub instances

Programming here means planning

Main idea:- set up a recurrence relating a solution to a larger instance

to solutions of some smaller instances- solve smaller instances once

- record solutions in a table- extract solution to the initial instance from that table


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Example


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Ingredients:

Instances:The possible inputs to the problem.

Solutions for Instance:Each instance has an exponentially large

set of valid solutions.

Cost of Solution:Each solution has an easy-to-compute cost or

value.

SpecificationPreconditions:The input is one instance.

Postconditions:A valid solution with optimal cost. (minimum

or maximum)

Optimization Problems


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Greedy Solutions to Optimization

Problems

Surprisingly, many important and practical

optimization problems can be solved this way.

Every two-year-old knows the greedy algorithm.

In order to get what you want,

just start grabbing what looks best.


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Example 1: Making Change

Problem:Find the minimum # of quarters, dimes,nickels, and pennies that total to a given amount.


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The Greedy Choice

Commit to the object that looks the ``best''

Must prove that this locally greedy choice

does not have negative global consequences.


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Instance:A drawer full of coins and an amount of change to return

25 25 25 25 25 25 25 25 25 25

10 10 10 10 10 10 10 10 10 10

5 5 5 5 5

5

5

5

5

5

1 1 1 1 1 1 1 1 1 1

Amount = 92

Solutions for Instance:

A subset of the coins in the drawer that total the amount

Making Change Example


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Solutions for Instance:A subset of the coins that total the amount.

25 25 25 25 25 25 25 25 25 25

10 10 10 10 10 10 10 10 10 10

5 5 5 5 5 5 5 5 5 5

1 1

1

1

1

1

1

1

1

1

Amount = 92

Cost of Solution:The number of coins in the solution = 14


Goal:Find an optimal valid solution.


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25 25 25 25 25 25 25 25 25 25

10 10 10 10 10 10 10 10 10 10

5 5 5 5 5 5 5 5 5 5

1 1 1 1 1 1 1 1 1 1

Amount = 92


Greedy Choice:

Does this lead to an optimal # of coins?

Start by grabbing quarters until exceeds amount,

then dimes, then nickels, then pennies.

Cost of Solution:7



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HardMaking Change Example

Greedy Choice:Start by grabbing a 4-cent coin.

Problem:Find the minimum # of

4, 3, and 1 cent coins to make up 6 cents.

Consequences:

4+1+1 = 6 mistake

3+3=6 better

Greedy Algorithm does not work!


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Shortest Path

Generalize distance to weighted setting

Digraph G= (V,E) with weight function W: ER(assigning real values to edges)

Weight of pathp= v1v2 vkis

Shortest path = a path of the minimum weight

Applications

static/dynamic network routing robot motion planning

map/route generation in traffic

1

1

1

( ) ( , )k

i i

i

w p w v v


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Shortest-Path Problems Shortest-Path problems

Single-source (single-destination). Find a shortest path from a

given source (vertex s) to each of the vertices.

Single-pair. Given two vertices, find a shortest path between them.

Solution to single-source problem solves this problem efficiently,

too.

All-pairs. Find shortest-paths for every pair of vertices. Dynamic

programming algorithm.


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Negative Weights and Cycles?

Negative edges are OK, as long as there are no negative

weight cycles (otherwise paths with arbitrary small

lengths would be possible)

Shortest-paths can have no cycles (otherwise we could

improve them by removing cycles) Any shortest-path in graph Gcan be no longer than n 1 edges,

where nis the number of vertices


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Relaxation For each vertex v in the graph, we maintain v.d(), the

estimate of the shortest path from s, initialized to at the

start

Relaxing an edge (u,v) means testing whether we can

improve the shortest path to vfound so far by going

through u

5

u v

vu

2

2

9

5 7

Relax(u,v)

5

u v

vu

2

2

6

5 6

Relax(u,v)

Relax (u,v,G)

if v.d() > u.d()+G.w(u,v) then

v.setd(u.d()+G.w(u,v))

v.setparent(u)


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Dijkstra's Algorithm

Non-negative edge weights

Greedy, similar to Prim's algorithm for MST

Like breadth-first search (if all weights = 1, one can simplyuse BFS)

Use Q, a priority queue ADT keyed by v.d() (BFS usedFIFO queue, here we use a PQ, which is re-organizedwhenever some ddecreases)

Basic idea

maintain a set Sof solved vertices at each step select "closest" vertex u, add it to S, and relax all

edges from u


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Dijkstras Pseudo Code

Input: Graph G, start vertex s

relaxing

edges

Dijkstra(G,s)

01 foreachvertex u G.V()

02 u.setd()

03 u.setparent(NIL)

04 s.setd(0)

05 S // Set S is used to explain the algorithm

06 Q.init(G.V()) // Q is apriority queueADT

07whilenotQ.isEmpty()08 u Q.extractMin()

09 S S {u}

10 foreachv u.adjacent() do

11 Relax(u, v, G)

12 Q.modifyKey(v)


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Dijkstras Example

0s

u v

yx

10

5

1

2 39

4 67

2

10

5

0s

u v

yx

10

5

1

2 39

4 67

2

Dijkstra(G,s)


02 u.setd()

03 u.setparent(NIL)

04 s.setd(0)05 S

06 Q.init(G.V())


09 S S {u}

10 foreachv u.adjacent() do11 Relax(u, v, G)

12 Q.modifyKey(v)


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Dijkstras Example (2)

u v

8 14

5 7

0s

yx

10

5

1

2 39

4 67

2

8 13

5 7

0s

u v

yx

10

5

1

2 39

4 67

2

Dijkstra(G,s)


02 u.setd()

03 u.setparent(NIL)

04 s.setd(0)05 S

06 Q.init(G.V())


09 S S {u}


12 Q.modifyKey(v)


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Dijkstras Example (3)

8 9

5 7

0

u v

yx

10

5

1

2 39

4 67

2

8 9

5 7

0

u v

yx

10

5

1

2 39

4 67

2

Dijkstra(G,s)


02 u.setd()

03 u.setparent(NIL)

04 s.setd(0)05 S

06 Q.init(G.V())


09 S S {u}


12 Q.modifyKey(v)


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Dijkstras Running Time

Extract-Min executed |V| time Decrease-Key executed |E| time

Time = |V| TExtract-Min+ |E| TDecrease-Key Tdepends on different Q implementations

Q T(Extract-

Min)

T(Decrease-Key) Total

array O(V) O(1) O(V 2)

binary heap O(lgV) O(lg V) O(Elg V)

Fibonacci heap O(lgV) O(1) (amort.) O(VlgV+ E)

46


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Bellman-Ford Algorithm

Dijkstras doesnt work when there are negative edges:

Intuitionwe can not be greedy any more on the assumption that

the lengths of paths will only increase in the future

Bellman-Ford algorithm detects negative cycles (returns

false) or returns the shortest path-tree


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Bellman-Ford Algorithm

Bellman-Ford(G,s)

01 foreachvertex u G.V()02 u.setd()03 u.setparent(NIL)

04 s.setd(0)

05fori 1 to |G.V()|-1 do06 foreach edge (u,v) G.E() do

07 Relax(u,v,G)

08 foreach edge (u,v) G.E() do

09 if v.d() > u.d() + G.w(u,v) then10 return false

11 return true


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Bellman-Ford Example

5

0s

zy

6

7

8-3

72

9

-2xt

-4

6

7

0s

zy

6

7

8-3

72

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-2xt

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5

6 4

7 2

0s

zy

6

7

8 -37

2

9

-2xt

-4

5

2 4

7 2

0s

zy

6

7

8 -37

2

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-2xt

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5


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Bellman-Ford Example

2 4

7 2

0s

zy

6

7

8-3

72

9

-2xt

-4

Bellman-Ford running time:

(|V|-1)|E| + |E| = Q(VE)

5


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Thank you !

Graph Theory Seminar

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