ECE 242 Spring 2003 Data Structures in Java

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ECE 242 Spring 2003Data Structures in Java

• http//rio.ecs.umass.edu/ece242
• Concepts of Graphs and Graph Traversal
• Prof. Lixin Gao

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Todays Topics

• Graphs
• Definition
• Directed and Undirected graph
• Simple path and cycle
• Connected and Unconnected graph
• Weighted graph
• Graph Representation
• Graph Traversal

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Abstract Data Type

• We have discussed
• List
• Tree
• Today we will talk about Graph

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Northwest Airline Flight
Anchorage
Boston
Minneapolis
Seattle
Hartford
SF
Atlanta
Austin
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Course Prerequisites
Freshman
Sophomore
Junior
Senior
ECE 122
ECE 242
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Computer Network Or Internet
MCI
ATT
Regional Network
Campus Umass
Intel
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Application

• Salemans Route
• Postmans Route
• Shortest route for saleman, we call it as Travel
  Saleman Problem

Start
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Interesting Problem

• Graph Traversal
• Shortest Path between two nodes
• Anything else?

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Concepts of Graphs
node or vertex
edges (weight)
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Graph Definition

• G (V, E)
• V is the vertex set
• Vertices are also called nodes or points
• E is the edge set
• Each edge connects two vertices
• Edges are also called arcs or lines

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Undirected vs. Directed Graph

• Undirected Graph
• no oriented edge

• Directed Graph
• every edge has oriented vertex

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Applications --- Communication Network
2
3
8
1
10
4
5
9
11
6
7

• Vertex city, edge communication link.

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Subgraph

• Subgraph
• subset of vertices and edges

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Simple Path

• A simple path traverse a node no more than once
• ABCD is a simple path

B
C
A
path
D
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Cycle

• A cycle is a path that starts and ends at the
  same point
• CBDC is a cycle

B
C
A
D
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Connected vs. Unconnected Graph
Connected Graph
Unconnected Graph
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Directed Acyclic Graph

• Directed Acyclic Graph (DAG) directed graph
  without cycle
• Examples
• Course Requirement Graph DAG

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Directed Acyclic Graph

• This is not DAG
• ABCD is a cycle

D
C
A
B
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Weighted Graph

• Weighted graph a graph with numbers assigned to
  its edges
• Weight cost, distance, travel time, hop, etc.

0
10
20
1
1
2
4
5
3
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Representation Of Graph

• Two representations
• Adjacency Matrix
• Adjacency List

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Adjacency Matrix

• N nodes in graph
• Use Matrix A0N-10N-1
• if vertex i and vertex j are adjacent in graph,
  Aij 1,
• otherwise Aij 0
• if vertex i has a loop, Aii 1
• if vertex i has no loop, Aii 0

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Example of Adjacency Matrix
0
1
2
3
0 1 1 0 1 0 1 1 1 1
0 1 0 1 1 0
So, Matrix A
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Undirected vs. Directed

• Undirected graph
• adjacency matrix is symmetric
• Aij always equals Aji
• Directed graph
• adjacency matrix may not be symmetric
• Aij may not equal Aji

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Directed Graph Matrix Representation
0
1
2
3
0 1 1 1 0 0 0 1 0 0
0 1 0 0 0 0
So, Matrix A
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Weighted Graph
0
10
20
2
1
1
0 20 10 1 20 0 0 5 10
0 0 4 1 5 4 0
4
5
3
So, Matrix A
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Adjacency List

• An array of list
• the ith element of array is a list of vertices
  that connect to vertex i

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Example of Adjacency List
0
1
2
3
vertex 0 connect to vertex 1, 2 and 3 vertex 1
connects to 3 vertex 2 connects to 3
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Weighted Graph

• Weighted graph extend each node with an addition
  field weight

0
1
10
2
20
3
1
0
10
20
1
0
10
3
4
1
1
2
2
0
20
3
5
4
3
5
0
1
1
4
2
5
3
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Compare Two Representations

• Given two vertices u, v
• find out if u and v are adjacent
• Given a vertex u
• enumerate all neighbors of u
• For all vertices
• enumerate all neighbors of each vertex

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Comparison Of Representations
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Complete Graph

• There is an edge between any two vertices

Total number of edges in graph E N(N-1)/2
O(N2)
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Sparse Graph

• There are very small no. of edges in the graph

• For example
• E N-1 O(N)

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Space Requirement of Representations

• Memory space
• adjacency matrix O(N2)
• adjacency list O(E)
• Sparse graph
• adjacency list is better
• Dense graph
• same running time

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Graph Traversal

• List out all cities that United Airline can reach
  from Hartford Airport

CHI
Hartford
SF
NYC
LA
W. DC
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Graph Traversal

• From vertex u, list out all vertices that u can
  reach via graph G
• Set of nodes to expand
• Each node has a flag to indicate visited or not

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Demos For Traversal Algorithm

• Step 1 Hartford
• find neighbors of Hartford
• Hartford, NYC, CHI

CHI
Hartford
SF
NYC
LA
W. DC
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Demos For Traversal Algorithm

• Step 2 Hartford, NYC, CHI
• find neighbors of NYC, CHI
• Hartford, NYC, CHI, LA, SF

CHI
Hartford
SF
NYC
LA
W. DC
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Demos For Traversal Algorithm

• Step 3 Hartford, NYC, CHI, LA, SF
• find neighbors of LA, SF
• no other new neighbors

CHI
Hartford
SF
NYC
LA
W. DC
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Demos For Traversal Algorithm

• Finally we get all cities that United Airline can
  reach from Hartford Airport
• Hartford, NYC, CHI, LA, SF

CHI
Hartford
SF
NYC
LA
W. DC
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Algorithm of Graph Traversal

• Mark all nodes as unvisited
• Pick a starting vertex u, add u to probing list
• While ( probing list is not empty)
• Remove a node v from probing list
• Mark node v as visited
• For each neighbor w of v, if w is unvisited,
  add w to the probing list

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Graph Traversal Algorithms

• Two algorithms
• Depth First Traversal
• Breadth First Traversal

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Depth First Traversal

• Probing List is implemented as stack (LIFO)
• Example
• As neighbor B, C, E
• Bs neighbor A, C, F
• Cs neighbor A, B, D
• Ds neighbor E, C, F
• Es neighbor A, D
• Fs neighbor B, D
• start from vertex A

A
B
E
C
D
F
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Initial State
• Visited Vertices
• Probing Vertices A
• Unvisited Vertices A, B, C, D, E, F

stack
A
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is A, mark it as
  visited
• Find As first unvisited neighbor, push it into
  stack
• Visited Vertices A
• Probing vertices A, B
• Unvisited Vertices B, C, D, E, F

B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is B, mark it as
  visited
• Find Bs first unvisited neighbor, push it in
  stack
• Visited Vertices A, B
• Probing Vertices A, B, C
• Unvisited Vertices C, D, E, F

C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is C, mark it as
  visited
• Find Cs first unvisited neighbor, push it in
  stack
• Visited Vertices A, B, C
• Probing Vertices A, B, C, D
• Unvisited Vertices D, E, F

D
C
C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is D, mark it as
  visited
• Find Ds first unvisited neighbor, push it in
  stack
• Visited Vertices A, B, C, D
• Probing Vertices A, B, C, D, E
• Unvisited Vertices E, F

E
D
D
C
C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is E, mark it as
  visited
• Find Es first unvisited neighbor, no vertex
  found, Pop E
• Visited Vertices A, B, C, D, E
• Probing Vertices A, B, C, D
• Unvisited Vertices F

E
D
D
C
C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is D, mark it as
  visited
• Find Ds first unvisited neighbor, push it in
  stack
• Visited Vertices A, B, C, D, E
• Probing Vertices A, B, C, D, F
• Unvisited Vertices F

F
D
D
C
C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is F, mark it as
  visited
• Find Fs first unvisited neighbor, no vertex
  found, Pop F
• Visited Vertices A, B, C, D, E, F
• Probing Vertices A, B, C, D
• Unvisited Vertices

F
D
D
C
C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is D, mark it as
  visited
• Find Ds first unvisited neighbor, no vertex
  found, Pop D
• Visited Vertices A, B, C, D, E, F
• Probing Vertices A, B, C
• Unvisited Vertices

D
C
C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is C, mark it as
  visited
• Find Cs first unvisited neighbor, no vertex
  found, Pop C
• Visited Vertices A, B, C, D, E, F
• Probing Vertices A, B
• Unvisited Vertices

C
B
B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is B, mark it as
  visited
• Find Bs first unvisited neighbor, no vertex
  found, Pop B
• Visited Vertices A, B, C, D, E, F
• Probing Vertices A
• Unvisited Vertices

B
A
A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Peek a vertex from stack, it is A, mark it as
  visited
• Find As first unvisited neighbor, no vertex
  found, Pop A
• Visited Vertices A, B, C, D, E, F
• Probing Vertices
• Unvisited Vertices

A
stack
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Depth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Now probing list is empty
• End of Depth First Traversal
• Visited Vertices A, B, C, D, E, F
• Probing Vertices
• Unvisited Vertices

stack
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Breadth First Traversal

• Probing List is implemented as queue (FIFO)
• Example
• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D
• start from vertex A

A
B
E
C
D
F
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Initial State
• Visited Vertices
• Probing Vertices A
• Unvisited Vertices A, B, C, D, E, F

A
queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Delete first vertex from queue, it is A, mark it
  as visited
• Find As all unvisited neighbors, mark them as
  visited, put them into queue
• Visited Vertices A, B, C, E
• Probing Vertices B, C, E
• Unvisited Vertices D, F

A
B
E
C
queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Delete first vertex from queue, it is B, mark it
  as visited
• Find Bs all unvisited neighbors, mark them as
  visited, put them into queue
• Visited Vertices A, B, C, E, F
• Probing Vertices C, E, F
• Unvisited Vertices D

B
E
C
C
F
E
queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Delete first vertex from queue, it is C, mark it
  as visited
• Find Cs all unvisited neighbors, mark them as
  visited, put them into queue
• Visited Vertices A, B, C, E, F, D
• Probing Vertices E, F, D
• Unvisited Vertices

C
F
E
E
D
F
queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Delete first vertex from queue, it is E, mark it
  as visited
• Find Es all unvisited neighbors, no vertex found
• Visited Vertices A, B, C, E, F, D
• Probing Vertices F, D
• Unvisited Vertices

E
D
F
F

D
queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Delete first vertex from queue, it is F, mark it
  as visited
• Find Fs all unvisited neighbors, no vertex found
• Visited Vertices A, B, C, E, F, D
• Probing Vertices D
• Unvisited Vertices

F

D
D


queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Delete first vertex from queue, it is D, mark it
  as visited
• Find Ds all unvisited neighbors, no vertex found
• Visited Vertices A, B, C, E, F, D
• Probing Vertices
• Unvisited Vertices

D





queue
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Breadth First Traversal (Cont)

• As neighbor B C E
• Bs neighbor A C F
• Cs neighbor A B D
• Ds neighbor E C F
• Es neighbor A D
• Fs neighbor B D

A
B
E
C
D
F

• Now the queue is empty
• End of Breadth First Traversal
• Visited Vertices A, B, C, E, F, D
• Probing Vertices
• Unvisited Vertices

queue
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Difference Between DFT BFT

• Depth First Traversal (DFT)
• order of visited A, B, C, D, E, F
• Breadth First Traversal (BFT)
• order of visited A, B, C, E, F, D

A
B
E
C
D
F
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Complexity of Graph Traversal

• Adjacency Matrix
• Cost O(NN) O(N2)
• Adjacency List
• For each node v, degree of node v
• Sum of degree(v) 2E
• Cost O(E)

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Application of Graph Traversal

• Find out all connected components of a graph
• For each node v of the graph,
• if ( v does not belong to any existing component
  )
• depth-first traversal (v)
• all nodes from v form a connected component