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Graphs Page 1

BFH-TI: Softwareschule Schweiz

Algorithms and Data Structures

Graphs

Dr. Rolf Haenni

CAS SD01

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 2

Outline

Graphs

Implementing Graphs

Graph Algorithms

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 3Graphs

Outline

Graphs

Implementing Graphs

Graph Algorithms

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 4Graphs

What is a Graph?

I In mathematics and computer science, a graph is a pairG = (V , E ) where

Ý V is a set of n vertices (nodes)Ý E is a collection of m edges

I Vertices and edges are positions, i.e. they store elements

I Example: Flight network

HNL

SFO

LAX

ORD

DFW

LGA

PVD

MIA

1344

2555 1233

1755344

1873844

801 145

10991120

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 5Graphs

Undirected vs. Directed Graphs

I There are two different types of edges

Ý Undirected edges e = {v , w}Ý Directed edges e = (v , w), sometimes called arcs

SFO ORD1873miles SFO ORDflight

AA101

I Hence, there are three different types of graphs

Ý Undirected graphs consist of undirected edges onlyÝ Directed graphs consist of directed edges onlyÝ Mixed graphs consist of both directed and undirected edges

I An undirected edge {v , w} can be replaced by two oppositedirected edges (v , w) and (w , v)

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 6Graphs

Terminology

I General graphs

Ý v and w are adjacent, if G contains an edge {v , w} or (v , w)Ý v and w are endpoints of {v , w} or (v , w)Ý The degree of a vertex v is the number of edges {v , w},

(v , w), and (w , v) in GÝ A self-loop is an edge {v , v} or (v , v)Ý Two edges e1 and e2 are called parallel if e1 = e2

I A simple graph is free of self-loops and parallel edgesI Directed graphs

Ý v is the origin and w is the destination of (v , w)Ý (v , w) is an outgoing edge of v and an incoming edge of wÝ The out-degree of v is the number of outgoing edges of v in GÝ The in-degree of v is the number of incoming edges of v in G

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 7Graphs

Terminology: Example

U

W

V

X

Y

Z

h

a f

eb

c

g

j

di

I U and V are adjacent

Ý U and V are the endpoints of aÝ U is the origin and V the destination of a

I The degree of V is 4

Ý The indegree of V is 2 (a and e are incoming edges)Ý The outdegree of V is 2 (b and f are outgoing edges)

I j is a self-loop, c and d are parallel

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 8Graphs

Paths and Cycles

I A path is sequence of alternating vertices and edges

Ý Begins with a verticeÝ Ends with a verticeÝ Each edge is preceded and followed by its endpoints

I A cycle is circular sequence of alternating vertices and edges

Ý Each edge is preceded and followed by its endpoints

I A path or cycle is called simple, if all its vertices and edges aredistinct

I A graph is called unrooted tree, if it contains no cycles

I A graph is called connected, if there is a path between anypair of vertices

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 9Graphs

Terminology: Example

U

W

V

X

Y

Z

h

a f

ebc

g

j

di

I U, a, V , f , Y , g , X , e, V , b, W is a path that is not simple

I X , i , Z , h, Y , g , X , i , Z , . . . is a simple cycle

I The graph is not a tree

I The graph is connected

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 10Graphs

Graph ADT

I Vertices and edges are positions which store elements

I Edge operations

Ý endVertices(e)Ý isDirected(e)Ý origin(e)Ý destination(e)

I Vertex operations

Ý aVertex()Ý degree(v)Ý inDegree(v)Ý outDegree(v)Ý incidentEdges(v)Ý areAdjacent(v,w)

I General operations

Ý numVertices()Ý numEdges()Ý vertices(v)Ý edges(v)

I Update operations

Ý insertVertex(elem)Ý insertEdge(v,w,elem)Ý insertArc(v,w,elem)Ý removeVertex(v)Ý removeEdge(e)

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 11Implementing Graphs

Outline

Graphs

Implementing Graphs

Graph Algorithms

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 12Implementing Graphs

Edge List Structure

I This representation maintains two sequences (or dictionaries),one for the edges and one for the vertices of the graphs

I Each edge object consists of:

Ý ElementÝ Reference to first endpoint (origin)Ý Reference to second endpoint (destination)

I The edges are connected to the vertices, but not vice versa

a b cU

W

Va

b

c

U V W

E

V

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 13Implementing Graphs

Adjacency List Structure

I Extends the edge list structure by linking the vertices back tothe edges

I Each vertex object consists of:

Ý ElementÝ Sequence of incident edges in which it appears

I Regards the edge-vertex relations from both sides

U

W

Va

b

c

a b c

U V W

E

V

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 14Implementing Graphs

Adjacency Matrix Structure

I The vertices of a simple graph are numbered from 0, . . . , n− 1

I Vertex i is stored at rank i (vector) or with key i (dictionary)I Edges are stored in a 2-dimensional array A[i , j ] of size n × n

Ý A directed edge (i , j) is stored at A[i , j ]Ý An undirected edge {i , j} is stored at A[i , j ] and A[j , i ]Ý All other array cells refer to NULL

I Edges should contain references to its endpoints

I In graph algorithms, adjacency matrices are often reduced tosimple 0, 1-matrices

A[i , j ] =

{1, if (i , j) or {0, 1} is an edge

0, otherwise

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 15Implementing Graphs

Adjacency Matrix Structure: Example

U

W

V

X

Y

Z

h

a f

eb

c

g

j

di

Key Vertex0 U1 V2 W3 X4 Y5 Z

0 1 2 3 4 50 a1 b f2 c3 e d i4 g5 h j

0 1 2 3 4 50 0 1 0 0 0 01 0 0 1 0 1 02 0 0 0 1 0 03 0 1 1 0 0 14 0 0 0 1 0 05 0 0 0 0 1 1

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 16Implementing Graphs

Performance

Edge Adjacency AdjacencyGraph Operation List List Matrix

vertices() n n nedges() m m n2

incidentEdges(v) m deg(v) nareAdjacent(v,w) m min(deg(v), deg(w)) 1insertVertex(elem) 1 1 n2 ∗

removeVertex(v) m deg(v) n2 ∗∗

insertEdge(elem) 1 1 1removeEdge(e) 1 1 1

Space n + m n + m n2

∗ O(1) if array size N > n ∗∗ O(n) if N > n is allowed

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 17Graph Algorithms

Outline

Graphs

Implementing Graphs

Graph Algorithms

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 18Graph Algorithms

General Graph Problems

Graph Traversal: Visit all vertices and edges

Connectivity: Determine whether a graph is connected

Cycle Finding: Find a cycle if one exists

Path Finding: Find a path between two vertices (e.g. theshortest one)

Minimum Cut: Determine the minimal set of edges which, ifremoved, disconnect two vertices

Spanning Tree: Find a connected acyclic subgraph whichcontains all vertices (e.g. the smallest one)

Reachability: Determine whether there is a directed pathbetween two vertices in a digraph

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 19Graph Algorithms

Depth-First-Traversal

I Many graph problems can be solved by traversing the graph

I In a depth-first-traversal, the graph is traversed by exploringpaths as far as possible along each branch before backtracking

I Backtracking means to return to the most recent vertex thathadn’t finished exploring its branches

I Backtracking can be realized in two ways:

Ý Recursion: visit each unexplored adjacent vertex recursivelyÝ Iteration: Push unexplored adjacent vertices to a stack

I In both cases, it is necessary to mark the vertices as VISITED

I Runs in O(n+m) time and uses O(h) space, where h is thelength of the longest simple path

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 20Graph Algorithms

Depth-First-Search: Example

v

v

v v v

v v

v v

v v v

v v

v v v

v v v

v v

v v v

v v v

v v v

v v v

v v v

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 21Graph Algorithms

Breadth-First-Traversal

I In a breadth-first-traversal, all adjacent vertices are visitedimmediately after visiting a vertex

I It is usually implemented by inserting the adjacent verticesinto a queue (starting with an arbitrary node)

I Explored vertices are marked as VISITED

I Can be turned into depth-first-traversal by replacing thequeue by a stack

I Runs in O(n+m) time and uses O(n+m) space in the worstcase

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 22Graph Algorithms

Breadth-First-Search: Example

v v v

v

v v

v v

v

v v

v v v

v v

v v v

v v v

v v v

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 23Graph Algorithms

Weighted Graph Problems

I In a weighted graph, a weight w(e) is attributed to each edgeI Shortest Path: Find the path between two vertices with

minimal total weight

Ý Dijkstra-Algorithm (for positive weights): O(n2) or O(m log n)Ý A*-Algorithm (for positive weights): depends on heuristicÝ Bellman-Ford-Algorithm (for general weights): O(n·m)

I Minimum Spanning Tree: Find a spanning tree with minimaltotal weight

Ý Kruskal-Algorithm: O(m log n)Ý Prim-Algorithm: O(m log n) or O(m + n log n)Ý Baruvka-Algorithm: O(m log n)

I Weighted graphs have applications in transportation networks,planning problems, routing protocols, strategic games, etc.

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 24Graph Algorithms

Dijkstra-Algorithm

I The Dijkstra-Algorithm starts from a source vertex s andcomputes the minimal distances (lengths of the shortestpaths) to all other vertices

I The algorithm is a graph traversal implemented with a priorityqueue, in which the vertices with the shortest distance to sare explored first

I The running time depends on the implementation of thepriority queue

Ý Sequence-based (sorted or unsorted): O(n2)Ý Heap-based: O(m log n)

I Applied in routing protocols such as IS-IS and OSPF

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 25Graph Algorithms

Dijkstra-Algorithm: Example

0 4

2

4

1

7

23

1

1

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32

0 4

2 9

3

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2 9

3 5

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1

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32

0 4

2 8 6

3 5 7

4

1

7

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1

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32

0 4 9

2 7 6

3 5 7

4

1

7

23

1

12

32

0 4

1

7

23

1

1

2

32

s s s

s s s

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 26Graph Algorithms

Flow Network Problems

I A flow network is a directed graph with a capacity c(e)attributed to each edge

I Maximum Flow: Find the maximal total flow capacity fbetween two vertices

Ý Ford-Fulkerson-Algorithm: O(m·f ), for f ∈ NÝ Edmonds-Karp-Algorithm: O(n·m2)

I Minimum Cut: Find a cut with minimal total capacitybetween two vertices

I Flow networks have many applications in computer networks,transportation networks, electrical distribution systems, etc.

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures

Graphs Page 27Graph Algorithms

Maximum Flow: Example

3

5

1

32

3

12

85

s t 0/3

3/5

1/1

3/32/2

3/3

1/10/2

4/84/5

s t

Berner Fachhochschule Rolf Haenni

Technik und Informatik Algorithms and Data Structures