Graphs. Data structures that connect a set of objects to form a kind of a network Objects are called “Nodes” or “Vertices” Connections are called “Edges”

Graphs

Graphs

• Data structures that connect a set of objects to form a kind of a

network

• Objects are called “Nodes” or “Vertices”

• Connections are called “Edges”

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Types of Graphs

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Directed vs. undirected

Undirected (Edges have no direction)

Directed (Edges have directions)

Weighted vs. unweighted

Unweighted (Edges have no cost/weight)

Weighted (Edges have associated cost/weight)

Types of Graphs (Cont’d)

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Dense vs. Sparse

Dense graphs (many edges between nodes)

Sparse graphs (few edges between nodes)

• If the graph has n vertices (nodes) Maximum # of edges is n2

• In dense graphs number of edges is close to n2

• In sparse graphs number of edges is close to n

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Some Graph Applications

transportation

Graph

street intersections

Nodes Edges

highways

communication computers fiber optic cables

World Wide Web web pages hyperlinks

social people relationships

food web species predator-prey

software systems functions function calls

scheduling tasks precedence constraints

circuits gates wires

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World Wide Web

Web graph. Node: web page. Edge: hyperlink from one page to another.

cnn.com

cnnsi.comnovell.comnetscape.com timewarner.com

hbo.com

sorpranos.com

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Social Network

Social network graph. Node: people. Edge: relationship between two people.

Reference: Valdis Krebs, http://www.firstmonday.org/issues/issue7_4/krebs

Protein Networks

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Nodes are proteins

Edges are connections (interaction between proteins)

More Formalization

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Undirected Graphs

Undirected graph. G = (V, E) V = set of nodes or vertices E = edges between pairs of nodes. Graph size parameters: n = |V|, m = |E|.

V = { 1, 2, 3, 4, 5, 6, 7, 8 }

E = { 1-2, 1-3, 2-3, 2-4, 2-5, 3-5, 3-7, 3-8, 4-5, 5-6 }

n = |V| = 8

m = |E| = 11

Graph Representation

Two main methods

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Adjacency Matrix Adjacency List

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Graph Representation: Adjacency Matrix

Adjacency matrix. V-by-V matrix (A)

A[i, j] = 1 if exists edge between node i and node j

Space proportional to V2

Checking if (u, v) is an edge takes (1) time.

Identifying all edges takes (V2) time.

For undirected graph matrix is symmetric across the diagonal

1 2 3 4 5 6 7 81 0 1 1 0 0 0 0 02 1 0 1 1 1 0 0 03 1 1 0 0 1 0 1 14 0 1 0 1 1 0 0 05 0 1 1 1 0 1 0 06 0 0 0 0 1 0 0 07 0 0 1 0 0 0 0 18 0 0 1 0 0 0 1 0

Vertices

Vertices

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Graph Representation: Adjacency List

Adjacency list. Node indexed array of lists.

Two representations of each edge.

Space proportional to O(E + V).

Checking if (u, v) is an edge takes O(deg(u)) time.

Identifying all edges takes (E + V) time.

1 2 3

2

3

4 2 5

5

6

7 3 8

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1 3 4 5

1 2 5 87

2 3 4 6

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degree = number of neighbors of u

3 7

Degree of a Node

In-degree(v): Number of edges coming to (entering) node v

Out-degree(v): Number of edges getting out (leaving) node v

For Undirected graphs In-degree = Out-degree

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In-degree(V3) = Out-degree(3) = 5In-degree(E) = 2 Out-degree(E) = 3

Graph Traversal

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Graph Traversal

Graph Traversal means visiting each node in the graph

There is a starting node (s)

Two main types of traversal Breadth-First-Search (BFS) Depth-First-Search (DFS)

Both are applicable for directed and undirected graphs

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BFS DFS

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Breadth First Search

s

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5

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3 6 9

s L1 L2 L n-1 Visit the nodes one-level at a time

Requires a queue (First-come-first-served)

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s

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5

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: s

Top of queue

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1Shortest pathfrom s

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s

2

5

4

7

8

3 6 9

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Undiscovered

Discovered

Finished

Queue: s 2

Top of queue

3

1

1

20


s

2

5

4

7

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3 6 9

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Undiscovered

Discovered

Finished

Queue: s 2 3

Top of queue

5

1

1

1

21


s

2

5

4

7

8

3 6 9

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Undiscovered

Discovered

Finished

Queue: 2 3 5

Top of queue

1

1

1

22


s

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4

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 2 3 5

Top of queue

4

1

1

1

2

23


s

2

5

4

7

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 2 3 5 4

Top of queue

1

1

1

2

5 already discovered:don't enqueue

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s

2

5

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 2 3 5 4

Top of queue

1

1

1

2

25


s

2

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 3 5 4

Top of queue

1

1

1

2

26


s

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3 6 9

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Undiscovered

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Finished

Queue: 3 5 4

Top of queue

1

1

1

2

6

2

27


s

2

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: 3 5 4 6

Top of queue

1

1

1

2

2

28


s

2

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: 5 4 6

Top of queue

1

1

1

2

2

29


s

2

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7

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: 5 4 6

Top of queue

1

1

1

2

2

30


s

2

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: 4 6

Top of queue

1

1

1

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2

31


s

2

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3 6 9

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Undiscovered

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Finished

Queue: 4 6

Top of queue

1

1

1

2

2

8

3

32


s

2

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 4 6 8

Top of queue

1

1

1

2

2

3

33


s

2

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 6 8

Top of queue

1

1

1

2

2

3

7

3

34


s

2

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7

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 6 8 7

Top of queue

1

1

1

2

2

3

9

3

3

35


s

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 6 8 7 9

Top of queue

1

1

1

2

2

3

3

3

36


s

2

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 8 7 9

Top of queue

1

1

1

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3

3

37


s

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 7 9

Top of queue

1

1

1

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3

3

38


s

2

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7

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 7 9

Top of queue

1

1

1

2

2

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3

3

39


s

2

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4

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 7 9

Top of queue

1

1

1

2

2

3

3

3

40


s

2

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 7 9

Top of queue

1

1

1

2

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3

3

41


s

2

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3 6 9

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Undiscovered

Discovered

Finished

Queue: 9

Top of queue

1

1

1

2

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3

3

42


s

2

5

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7

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: 9

Top of queue

1

1

1

2

2

3

3

3

43


s

2

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3 6 9

0

Undiscovered

Discovered

Finished

Queue: 9

Top of queue

1

1

1

2

2

3

3

3

44


s

2

5

4

7

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3 6 9

0

Undiscovered

Discovered

Finished

Queue:

Top of queue

1

1

1

2

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3

3

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s

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3 6 9

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Starting from s, we visited all (reachable) nodes

BFS forms a tree rooted at s (BFS Tree)

For each node x reachable from s we created a shortest path from s to x

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Example problems in which we use BFS

Find if node x is reachable from node y Start from node y and do BFS

Find the shortest path from node x to node y Start from node x and perform BFS

Search for a value v in the graph Start from any node and perform BFS

Always keep in mind whether we talk about undirected graph or directed graph

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Breadth First Search: Algorithm

Start at node v

Can do any processing on t

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Breadth First Search: Analysis

Theorem. The above implementation of BFS runs in O(V + E) time if

the graph is given by its adjacency list.

Proof Each node will be queued only once O(V) For each node, we visit all its out edges O(E)

– In fact each edge (u,v) is visited twice once from u’s side and once

form v’s side

Total is O(V + E)

Documents

Graphs. Data structures that connect a set of objects to form a kind of a network Objects are called “Nodes” or “Vertices” Connections are called “Edges”