CSC212 Data Structure - Section RS Lecture 23 Introduction to Graphs Instructor: Zhigang Zhu Department of Computer Science City College of New York.

CSC212 Data Structure - Section RS

CSC212 Data Structure - Section RS

Lecture 23 Lecture 23

Introduction to GraphsIntroduction to Graphs

Instructor: Zhigang ZhuInstructor: Zhigang Zhu

Department of Computer Science Department of Computer Science

City College of New YorkCity College of New York

ReviewReview

•Linked Lists

•Binary Trees

10 15

7

null

13

V

T

LQ

E A

Examples: Texas Road NetworkExamples: Texas Road Network

•62 cities (nodes)

•120 major roads (edges)

•Minimum distance between Dallas and Corpus Christi?

Examples: Delaunay Triangulation Network From a Point Data Set Examples: Delaunay Triangulation Network From a Point Data Set

Examples: Internet Topology 01/16/2006Examples: Internet Topology 01/16/2006http://www.eee.bham.ac.uk/com_test/img%5Cdsnl%5Cinternet15jan06.png

Examples: Backbone of science With 212 clusters comprising 7000 journals

Examples: Backbone of science With 212 clusters comprising 7000 journals

Biochemistry, Cell Biology, Biophyiscs

Mathematics

Computer Science

Geosciences, Geochemistry, Mineralogy

Kevin W Boyack, Richard Klavans, Katy Börner, Mapping the backbone of science, Scientometrics, Vol. 64, No. 3. (2005), pp. 351-374.

TerminologiesTerminologies

A graph G = (V, E)A graph G = (V, E) V: verticesV: vertices E : edges, pairs of vertices from V E : edges, pairs of vertices from V V V

Undirected Graph: Undirected Graph: (u,v) is same as (v,u)(u,v) is same as (v,u)

Directed GraphDirected Graph (u,v) is different from (v,u)(u,v) is different from (v,u) Source (u)Source (u) Target (v)Target (v)

A

E F G H

DCB

A

E F G H

DCB

I

More Terminologies – P 728More Terminologies – P 728

Loop: an edge that connects a vertex to itself. Loop: an edge that connects a vertex to itself. Path: a sequence of vertices, pPath: a sequence of vertices, p00,p,p11, …p, …pmm, such that , such that

each adjacent pair of vertices peach adjacent pair of vertices pii and p and pi+1i+1 are are connected by an edge.connected by an edge.

Multiple Edges: two or more edges connecting the Multiple Edges: two or more edges connecting the same two vertices in the same direction. same two vertices in the same direction.

Simple graph: have no loops and no multiple Simple graph: have no loops and no multiple edges – required for many applications. edges – required for many applications.

Weighted graph and unweighted graphWeighted graph and unweighted graph

Representations- Adjacency Matrix Representations- Adjacency Matrix

An adjacency matrix represents the graph as a n x n matrix A:A[i, j] = 1 if edge (i, j) E (or weight of edge)

= 0 if edge (i, j) E

01

23

0 1 2 3

0 11 00 11 00

1 11 00 00 00

2 00 00 00 11

3 11 00 11 00

Space Complexity with respect to |V| and/or |E|?

Representations-Linked ListRepresentations-Linked List

01

23

A directed graph with n vertices can be represented by n different linked lists. List number i provides the connections for vertex i. To be specific: for each entry j in the list number i, there is an edge from i to j.

00

11

22

33

22

00

00

00

33

22Space Complexity with respect to |V| and/or |E|?

Graph TraversalsGraph Traversals

Traversal: Traversal: Tree traversals (ch 10): visit all of a tree’s nodes and do Tree traversals (ch 10): visit all of a tree’s nodes and do

some processing at each nodesome processing at each node Types of Graph traversalsTypes of Graph traversals

Depth First Search (DFS)Depth First Search (DFS) Breadth First Search (BFS)Breadth First Search (BFS)

Issues to considerIssues to consider There is no root – need a start vertexThere is no root – need a start vertex Be careful and do not enter a repetitive cycle – mark Be careful and do not enter a repetitive cycle – mark

each vertex as it is processedeach vertex as it is processed

Graph Traversal-Recursive DFSGraph Traversal-Recursive DFS DFS(DFS(startstart))

Initialize the boolean Initialize the boolean visitedvisited array array Rec_DFS(Rec_DFS(startstart))

Rec_DFS(Rec_DFS(startstart)) For each of unvisited neighbor For each of unvisited neighbor nextnext of of startstart

Rec_DFS(Rec_DFS(nextnext))

0

1

6

2

4

3 5

00

1

6

2

4

3 5

0,10

1

6

2

4

3 5

0,1,3

Graph Traversal-DFSGraph Traversal-DFS

0

1

6

2

4

3 5

0,1,3,50

1

6

2

4

3 5

0,1,3,5,6

0

1

6

2

4

3 5

0,1,3,5,6,4

0

1

6

4

3 5

•2 is never visited

•DFS search/spanning tree

•Non-recursive version? (using a stack)

Graph Traversal-BFSGraph Traversal-BFS

BFS(BFS(startstart)) Initialize the boolean visited arrayInitialize the boolean visited array Add Add startstart to an empty queue Q to an empty queue Q Visited[Visited[startstart]=true;]=true; While(!Q.empty())While(!Q.empty())

u=Q.dequeue () //top+pop/get_frontu=Q.dequeue () //top+pop/get_front For each unvisited neighbor v of u For each unvisited neighbor v of u

Q.enqueue(v) //pushQ.enqueue(v) //push Visited[v]=trueVisited[v]=true

Graph Traversal-BFSGraph Traversal-BFS

0

1

6

2

4

3 5

0

F R

0

1

6

2

4

3 5

0

1 4

F R

0

1

6

2

4

3 5

0,1

4 3

F R

Graph Traversal-BFSGraph Traversal-BFS

0

1

6

2

4

3 5

0,1,4

3

F R

0

1

6

2

4

3 5

0,1,4,3

5 6

F R

0

1

6

2

4

3 5

0,1,4,3,5

6

F R

Graph Traversal-BFSGraph Traversal-BFS

0

1

6

2

4

3 5

0,1,4,3,5,6

F R

0

1

6

4

3 5

BFS Tree/Spanning Tree

Further Information Further Information

Path Algorithms (Ch 15.4)Path Algorithms (Ch 15.4) Dijkstra single source pathDijkstra single source path CSc 220 AlgorithmCSc 220 Algorithm

Alternative Bellman-Ford algorithm for graphs with Alternative Bellman-Ford algorithm for graphs with negative weights negative weights http://en.wikipedia.org/wiki/Bellman-Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman-Ford_algorithm

All-pair shortest path algorithmsAll-pair shortest path algorithms Floyd–Warshall algorithm Floyd–Warshall algorithm http://en.wikipedia.org/wiki/Floyd-Warshall_algorithmhttp://en.wikipedia.org/wiki/Floyd-Warshall_algorithm

Shortest path in dynamic networks (graphs)Shortest path in dynamic networks (graphs) Graph Partition Graph Partition http://en.wikipedia.org/wiki/Graph_partitioninghttp://en.wikipedia.org/wiki/Graph_partitioning

Graph Layout/DrawingGraph Layout/Drawing