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mailto:[email protected]:[email protected]

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Graphs

Introduction

definitions

representation

basic algorithms

Graph search algorithms

depth-first search

breadth-first search

Exercises

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Graphs

In mathematics, a graph is an abstractrepresentation of a set of objects where somepairs of the objects are connected by links.

the objects are vertices (or nodes or points)

the links are edges

The edges may be directed (asymmetric) orundirected (symmetric).

directed edges are usually called arcs

undirected edges can be called lines

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Graphs

Graphs are basic subject studied by graph

theory.

Examples: road networks

the web

social networks

precedence constraints

...

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Definition of graph

A graph is an ordered pair G = (V,E)comprising a set Vof nodes and aset E of edges.

Vand E are usually taken to befinite.

the order of a graph is |V|, the number

of vertices.

the size of a graph is |E|, the number

of edges.

the degree of a vertex is the number of

edges that connect to it.

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HN

HK

HU

TP

BK

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Definition of graph

A graph is a weighted graph if a

number (weight) is assigned to

each edge.

Such weight might represent costs,lengths or capacities... depending

on the problem at hand.

Adjacency relation: if (u,v) E is

an edge of G = (V,E), the vertices u

and v are said to be adjacent to

one another.

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HN

HK

HU

TP

BK

1.0

1.51.1

2.1 1.0

2.01.7

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Question

Question 1: Consider an undirected graph that

has n vertices, no parallel edges and is

connected (that is, in one piece). What is the

minimum and maximum number of edges thatthe graph could have, respectively?

1. n-1 and n(n-1)/2

2. n-1 and n2

3. n and 2n

4. n and nn

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Sparse versus dense graphs

Let n be number of vertices, m be

number of edges

In most applications, n < m

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Representation

Different data structures for the representationof graphs are used in practice.

Most common data structures are: adjacency lists: vertices are stored as records or

objects, every vertex stores a list of adjacent vertices

adjacency matrix: a two-dimensional matrix in

which the rows represent source vertices and thecolumns represent destination vertices.

unordered edge sequences: a sequence of edges,each edge contains a pair of vertices.

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Adjacency lists

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HN

HK

HU

TP

BK

HK HN TP NULL

HN HK BK HU TP NULL

BK HN TP NULL

HU HN TP NULL

TP HK HN BK HU NULL

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Adjacency lists

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0

1

5

3

2

4

6

0 1

1 2 3 5

2 3

3 0 6

4 2

5 0 3 6

6 4

Vertices are commonly

indexed by integers.

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Adjacency matrix

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HN

HK

HU

TP

BK

HK HN BK HU TP

HK 0 1 0 0 1

HN 1 0 1 1 1

BK 0 1 0 0 1

HU 0 1 0 0 1

TP 1 1 1 1 0

Symmetricmatrix

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Adjacency matrix

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0

1

5

3

2

4

6

0 1 2 3 4 5 6

0 0 1 0 0 0 0 0

1 0 0 1 1 0 1 02 0 0 0 1 0 0 0

3 0 0 0 0 0 0 1

4 0 0 1 0 0 0 0

5 1 0 0 1 0 0 1

6 0 0 0 0 1 0 0

Asymmetric

matrix

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Adjacency matrix

Represent G by a n*n matrix a where aij = 1 if

and only if G has an edge (i,j)

Variants: aij = number of (i,j) edges (if parallel edges)

aij = weight of (i,j) edge (if any)

aij = 1 if G has edge (i,j), aij=-1 if G has edge(j,i)

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Adjacency matrix

Question 2: How much space does an

adjacency matrix require, as a function of the

number n of vertices and the number m of

edges?

1. (n)

2. (m)

3. (m+n)

4. (n2)

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Unordered edge sequences

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HN

HK

HU

TP

BK

0

1

5

3

2

4

6

(HK, HN)

(HN, BK)

(HN, HU)

(BK, TP)

(HU, TP)

(HK, TP) (0,1)(1,2)

(1,3)

(1,5)

(2,3)

(3,6)(4,2)

(5,0)

(5,3)

(5,6)

(6,4)

(u,v) (v,u)

(u,v)(v,u)

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Choice of representations

Which representation is better?

The answer depends on graph density and

operations needed.

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Graph algorithms

Graph algorithms are a significant field ofinterest within computer science.

Typical high-level operations associated withgraphs aregraph search (orgraph traversal):

finding a path between two nodes

finding the shortest path from one node toanother

finding special paths (Hamiltonian path, Eulerpath)

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Graphs

Introduction

definitions

representation basic algorithms


depth-first search


Exercises

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Graph search

Graph search refers to the problem ofvisiting all nodes in a graph in a particularmanner.

a tree is a special kind of graph

tree traversal is a special case of graph search

In general graph search, each node mayhave to be visited more than once.

in tree traversal, each node is visited exactly once

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Graph search

Some motivations:

check if a network is connected -- can get to anywherefrom anywhere else

driving direction

formulate a plan (example, how to fill in a Sudokupuzzle)

nodes = a partially completed puzzle

arcs = filling in one new square

compute the pieces or components of a graph

clustering, structures of the web graph, etc.

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Graph search

Goals:

1. find everything findable from a given vertex

2. dont explore anything twice Generic algorithm (given graph G, vertex s)

initially, s explored, all other vertices unexplored

while possible choose an edge (u,v) with u explored and v

unexplored, if none halt

mark v explored

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Graph search

Claim: at end of the algorithm, v exploredif and only if G has a path from s to v.

Proof of this claim is given in the course Datastructure and algorithms

How to choose among the possibly many

frontier edges? depth-first search (using a stack -- recursion)

breadth-first search (using a queue)

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Graph search

Depth-first search:

explore aggressively like a maze, backtrack only whennecessary

compute topological ordering of directed acyclic graphs

compute connected components in directed graphs

Breadth-first search:

explore nodes in layers can compute shortest path

can compute connected components of an undirectedgraph

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Graphs

Introduction

definitions



depth-first search


Exercises

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Depth-first search

Depth-first search (DFS) is an algorithm for

searching/traversing a tree or a graph.

start at root node of a tree or at some node of a graph

explore as far as possible along each branch beforebacktracking

Formally, DFS progresses by expanding the first child

node of the search tree

go deeper and deeper until a goal node is found or until anode that has no children is found

then the search backtracks, returning to the most recentnode that has not been visited

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Depth-first search

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HN

HK

HU

TP

BK

HK HN BK HU TP

HK 0 1 0 0 1

HN 1 0 1 1 1

BK 0 1 0 0 1

HU 0 1 0 0 1

TP 1 1 1 1 0

dfs(HK): HK, HN, BK, TP, HU

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Depth-first search

Suppose that a graph G = (V,E) is representedby an adjacency matrix of size n*n: (a[i,j])n*n.

Let visit(u) is a function which visits a node u. Let visited[1..n] be a boolean array which

marks whether a node u has been alreadyvisited.

visited[u] = 1 if u has been visited

visited[u] = 0 otherwise

Initially, visited[u] = 0, for all u = 1,...,n

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Depth-first search

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void dfs(int u) { int v; visit(u); visited[u] = 1; for (v = 1; v < n; v++) if (visited[v] == 0 && a[u][v] == 1) { dfs(v); }}

v has not been visited

there exists edge (u,v)

void visit(int u) {

printf("%3d", u);}

/* Depth-first search from node u */

recursively visit v

int main(int argc, char **argv) {

initialize(); dfs(0); return 0;}

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Depth-first search

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0

1 2

5 4

3

Question 3: What is the result of dfs(0)?

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Depth-first search

Claim: at end of the algorithm, v

marked as visited if and only if there

exists a path from s to v in G. Reason: particular instantiation of

genetic search procedure.

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Graphs

Introduction

definitions



depth-first search


Exercises

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Breadth-first search

In breadth-first search (BFS), the nodes

are visited in layers:

start at root node of a tree or at some node of agraph and visit its neighboring nodes

then for each of those neighbor nodes in turn,visit their neighbor nodes which were

unvisited, and so on

All neighbor nodes are added to a queue

(FIFO data structure)

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BFS (graph G, start vertex s):

1. marks as visited

2. let Q be a queue, initialized with s

3. while Q is not empty:

remove the first node of Q, call it u

for each edge (u,v), if v is unvisited:

mark v as visited

add v to Q (at the end)

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void bfs(int s) { int u, v; visit(s); visited[s] = 1; enqueue(Q,s);while (!isEmpty(Q)) { u = dequeue(Q); for (v = 1; v < n; v++) if (visited[v] == 0 && a[u][v] == 1) { visit(v); visited[v] = 1;

enqueue(Q,v); } }}

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0

1 3

2

5

Question 4: What is the result of bfs(0)?

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Application of BFS

BFS can be applied to compute shortest paths on agraph.

Goal: compute dist(u), the fewest number of edges

on a path from s to u.

Extra code:

init: dist(u) = 0 if u = s, dist(u) = +if u s when considering edge (u,v), if v is unvisited then setdist(v) = dist(u) + 1

At termination, dist(u) = i if and only if u is in ithlayer, that is the shortest path from s to u has i edges.

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Graphs

Introduction

definitions



depth-first search


Exercises

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Exercises

Exercise 1: Implement and test BFS and

DFS algorithm on different graphs

use different graph representations

use different start vertices

Exercise 2: Application of BFS to

compute the shortest paths:

compute and print dist(u) for all vertices u

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Exercises

Exercise 3: A connectedcomponent of a an undirectedgraph is a subgraph in which

any two vertices areconnected to each other bypaths, and which is connectedto no additional vertices.

write a function to compute theconnected components of agraph using either the BFS orDFS algorithms.

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There are 3 connected

components in this graph

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