Graph Theory Kayman Lui 20-09-2007. Overview Graph –Notation and Implementation –Tree Depth...

What is a graph?

• A set of vertices and edges– Directed/Undirected– Weighted/Unweighted– Cyclic/Acyclic

vertex

edge

Representation of Graph

• Adjacency Matrix– A V x V array, with matrix[i][j]

storing whether there is an edge between the ith vertex and the jth vertex

• Adjacency Linked List– One linked list per vertex, each

storing directly reachable vertices

• Edge List

Representation of Graphs

Adjacency Matrix

Adjacency Linked List

Edge List

Memory Storage

O(V2) O(V+E) O(V+E)

Check whether (u,v) is an edge

O(1) O(deg(u)) O(deg(u))

Find all adjacent vertices of a vertex u

O(V) O(deg(u)) O(deg(u))

deg(u): the number of edges connecting vertex u

Trees and related terms

root

siblings

descendents children

ancestors

parent

What is a tree?

• A tree is an undirected simple graph G that satisfies any of the following equivalent conditions:

– G is connected and has no simple cycles. – G has no simple cycles and, if any edge is

added to G, then a simple cycle is formed. – G is connected and, if any edge is removed

from G, then it is not connected anymore. – Any two vertices in G can be connected by a

unique simple path. – G is connected and has n − 1 edges. – G has no simple cycles and has n − 1 edges.

Graph Searching

• Given: a graph• Goal: visit all (or some) vertices

and edges of the graph using some strategy (the order of visit is systematic)

• DFS, BFS are examples of graph searching algorithms

• Some shortest path algorithms and spanning tree algorithms have specific visit order

Depth-First Search (DFS)

• Strategy: Go as far as you can (if you have not visit there), otherwise, go back and try another way– Example: a person want to visit a

place, but do not know the path

F

A

BC

D

E

DFS (Demonstration)

unvisited

visited

DFS (pseudo code)

DFS (vertex u) {mark u as visitedfor each vertex v directly reachable from u

if v is unvisitedDFS (v)

}

• Initially all vertices are marked as unvisited

“Advanced” DFS

• Apart from just visiting the vertices, DFS can also provide us with valuable information

• DFS can be enhanced by introducing:– birth time and death time of a vertex

• birth time: when the vertex is first visited• death time: when we retreat from the

vertex

– DFS tree– parent of a vertex

DFS spanning tree / forest

• A rooted tree• The root is the start vertex• If v is first visited from u, then u is the

parent of v in the DFS tree• Edges are those in forward direction of

DFS, ie. when visiting vertices that are not visited before

• If some vertices are not reachable from the start vertex, those vertices will form other spanning trees (1 or more)

• The collection of the trees are called forest

A

F

B

C

D

E

GH

DFS forest (Demonstration)

unvisited

visited

visited (dead)

A B C D E F G H

birth

death

parent

A

B

C

F

E

D

G

1 2 3 13 10 4 14

12 9 8 16 11 5 15

H

6

7

- A B - A C D C

DFS (pseudo code)

DFS (vertex u) {mark u as visited

time time+1; birth[u]=time;

for each vertex v directly reachable from u

if v is unvisitedparent[v]=u

DFS (v) time time+1; death[u]=time;

}

Classification of edges

• Tree edge• Forward edge• Back edge• Cross edge

• Question: which type of edges is always absent in an undirected graph?

A

B

C

F

E

D

G

H

Determination of edge types

• How to determine the type of an arbitrary edge (u, v) after DFS?

• Tree edge– parent [v] = u

• Forward edge– not a tree edge; and– birth [v] > birth [u]; and– death [v] < death [u]

• How about back edge and cross edge?

Determination of edge types

Tree edge Forward Edge Back Edge Cross Edge

parent [v] = u not a tree edgebirth[v] > birth[u]death[v] < death[u]

birth[v] < birth[u]death[v] > death[u]

birth[v] < birth[u]death[v] < death[u]

Applications of DFS Forests

• Topological sorting (Tsort)• Strongly-connected components

(SCC)• Some more “advanced” algorithms

Topological Sort

• Topological order: A numbering of the vertices of a directed acyclic graph such that every edge from a vertex numbered i to a vertex numbered j satisfies i<j

• Topological Sort: Finding the topological order of a directed acyclic graph

Example

• Assembly Line– In a factory, there is several process. Some

need to be done before others. Can you order those processes so that they can be done smoothly?

• Studying Order– Louis is now studying ACM materials. There

are many topics. He needs to master some basic topics before understanding those advanced one. Can you help him to plan a smooth study plan?

T-sort Algorithm

• If the graph has more then one vertex that has indegree 0, add a vertice to connect to all indegree-0 vertices

• Let the indegree 0 vertice be s• Use s as start vertice, and

compute the DFS forest• The death time of the vertices

represent the reverse of topological order

Tsort (Demonstration)

S

D

B

E F

C

G

A

S A B C D E F G

birth

death

G C F B A E D

1 2 3 4 5

67

8

91011

12 13

141516

D E A B F C G

Strongly-connected components (SCC)

• A graph is strongly-connected if– for any pair of vertices u and v, one

can go from u to v and from v to u.

• Informally speaking, an SCC of a graph is a subset of vertices that– forms a strongly-connected

subgraph– does not form a strongly-connected

subgraph with the addition of any new vertex

SCC (Illustration)

SCC (Algorithm)

• Compute the DFS forest of the graph G to get the death time of the vertices

• Reverse all edges in G to form G’• Compute a DFS forest of G’, but

always choose the vertex with the latest death time when choosing the root for a new tree

• The SCCs of G are the DFS trees in the DFS forest of G’

A

F

B

C

D

GH

SCC (Demonstration)

A

F

B

C

D

E

GH

A B C D E F G H

birth

death

parent

1 2 3 13 10 4 14

12 9 8 16 11 5 15

6

7

- A B - A C D C

D

G

A E B

F

C

H

SCC (Demonstration)

D

G

A E B

F

C

H

A

F

B

C

D

GH

E

DFS Summary

• DFS spanning tree / forest

• We can use birth time and death time in DFS spanning tree to do varies things, such as Tsort, SCC

• Notice that in the previous slides, we related birth time and death time. But in the discussed applications, birth time and death time can be independent, ie. birth time and death time can use different time counter

Breadth-First Search (BFS)

• Instead of going as far as possible, BFS goes through all the adjacent vertices before going further (ie. spread among next vertices)– Example: set a house on fire, the fire

will spread through the house

• BFS makes use of a queue to store visited (but not dead) vertices, expanding the path from the earliest visited vertices.

A

B

C

D

E

F

G

H

I

J

BFS (Demonstration)

unvisited

visited

visited (dequeued)

Queue: A B C F D E H G J I

BFS (Pseudo code)

while queue not emptydequeue the first vertex u from queuefor each vertex v directly reachable from u

if v is unvisitedenqueue v to queuemark v as visited

• Initially all vertices except the start vertex are marked as unvisited and the queue contains the start vertex only

Applications of BFS

• Shortest paths finding• Flood-fill

Flood Fill

• An algorithm that determines the area connected to a given node in a multi-dimensional array

• Start BFS from the given node, counting the total number of nodes visited

• It can also be handled by DFS

Comparisons of DFS and BFS

DFS BFS

Depth-first Breadth-first

Stack Queue

Does not guarantee shortest paths

Guarantees shortest paths

What is graph modeling?

• Conversion of a problem into a graph problem

• Sometimes a problem can be easily solved once its underlying graph model is recognized

• Graph modeling appears in many ACM problems

Basics of graph modeling

• A few steps:– identify the vertices and the edges– identify the objective of the problem– state the objective in graph terms– implementation:

• construct the graph from the input instance

• run the suitable graph algorithms on the graph

• convert the output to the required format

Examples(1)

• Given a grid maze with obstacles, find a shortest path between two given points

start

goal

Examples (2)

• A student has the phone numbers of some other students

• Suppose you know all pairs (A, B) such that A has B’s number

• Now you want to know Alpha number, what is the minimum number of calls you need to make?

Examples (2)

• Vertex: student• Edge: whether A has B’s number• Add an edge from A to B if A has

B’s number• Problem: find a shortest path

from your vertex to Alpha’s vertex

Teacher’s Problem

• Question: A teacher wants to distribute sweets to students in an order such that, if student u tease student v, u should not get the sweet before v

• Vertex: student• Edge: directed, (v,u) is a directed

edge if student v tease u• Algorithm: T-sort

Variations of BFS and DFS

• Bidirectional Search (BDS)• Iterative Deepening Search(IDS)

Bidirectional search (BDS)

• Searches simultaneously from both the start vertex and goal vertex

• Commonly implemented as bidirectional BFS

start goal

BDS Example: Bomber Man (1 Bomb)

• find the shortest path from the upper-left corner to the lower-right corner in a maze using a bomb. The bomb can destroy a wall.

S

E

Bomber Man (1 Bomb)

S

E

1 2 3

4

1234

5

4

Shortest Path length = 8

5

6 67 78 8

9

9

10

10

11

11

12

12 12

13

13

Iterative deepening search (IDS)

• Iteratively performs DFS with increasing depth bound

• Shortest paths are guaranteed

IDS (pseudo code)

DFS (vertex u, depth d) {mark u as visitedif (d>0)

for each vertex v directly reachable from uif v is unvisited

DFS (v,d-1)}

i=0Do {

DFS(start vertex,i)Increment i

}While (target is not found)

IDS Complexity (the details can be skipped)

• ( )=bm

• ( ) =bm

1

0

1

1

nnk

k

rr

r

1

(1 )

1

nnk

k

r rr

r

1

0

1

1

ddk

dk

bt b

b

0

1

0

1

2

1

1

1 (1 )

1 (1 )

m

ii

im

i

m

t

b

b

m b b

b b

0

m

ii

t

- b is branching factor- td is the number of vertices visited for depth d

11

1

m

m

bt

b

mt

Conclusion

• The complexity of IDS is the same as DFS

Other Topics in Graph Theory

• Cut Vertices & Cut Edges• Euler Path/Circuit & Hamilton

Path/Circuit• Planarity

Cut Vertices & Cut Edges

• What is a cut vertex?– The removal of a set of vertices

causes a connected graph disconnected

• What is a cut edge?– The removal of a set of edges causes

a connected graph disconnected

Euler Path & Hamilton Path

• An Euler path is a path in a graph which visits each edge exactly once

• A Hamilton path is a path in an undirected graph which visits each vertex exactly once.

Planarity

• A planar graph is a graph that can be drawn so that no edges intersect

• K5 and K3,3 are non-planar graphs

Last Question:Equation

• Question: Find the number of solution of xi, given ki,pi. 1<=n<=6, 1<=xi<=150

• Vertex: possible values of – k1x1

p1 , k1x1p1 + k2x2

p2 , k1x1p1 + k2x2

p2 + k3x3p3 ,

k4x4p4 , k4x4

p4 + k5x5p5 , k4x4

p4 + k5x5p5 + k6x6

p6

Graph problems in Uva

• 280• 336• 532• 572• 10592

Graph Theory Kayman Lui 20-09-2007. Overview Graph –Notation and Implementation –Tree Depth...

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