Graphs

1 Definition

2 Terminology

3 Properties

4 Internal representation

• Adjacency list

• Adjacency matrix

5 Exploration algorithms

6 Other algorithms

Definition

A graph G is a finite set V of vertices and a finite set E of edges connecting pairs of vertices:

G = (V,E)

Directed vs. undirected

• G is undirected if its edges are undirected (top fig.),

• G is directed if its edges aredirected (bottom fig.).

What is a graph?

Applications

• Network representation: traffic, aerial routing, internet…• Automata: languages, discrete state systems• Dynamic system modeling• Probabilistic model: Bayesian network, neural network…

Algorithms

• Shortest path• Optimal flow• Optimal tour: traveling salesman• Clustering: k-neighboring• Complexity of a network

• The end-vertices of an edge are the vertices connected by that edge.

• Adjacent vertices: directly linked by an edge

• Adjacent edges: share a common end-vertex

• An edge is incident to a vertex if it connects that vertex to another vertex.

• The degree of a vertex is the number of edges that are incident to that vertex.

Terminology

Properties

Let G be a graph with n vertices and m edges.

Property 1

Proof: each edge is counted twice.

Property 2

If G is undirected with no self-loops and no multiple edges:

m ≤ n(n − 1)/2

Proof: the maximum number of edges is obtained when each vertex is connected to all the other vertices of the graph. We have,

(n − 1) + (n − 2) + (n − 3) + . . . + 2 + 1 = n(n − 1)/2

• Path: sequence of vertices v1 ,v2 ,. . .vk such that all consecutive vertices are adjacent.

Simple path : no repeated vertex

Cycle: simple path, excepted that the last vertex is the same as the first one.

Terminology

b e c

a c d a

• Connex graph: each pair of vertices is linked by a path

• Sub-graph: sub-set of vertices and edges forming a graph

• Connex component: sub-graph connex

example: le graph bellow has 3 connex components

Terminology

• Tree – connex graph without cycle

• Forest - collection of trees

Terminology

Connectivity

Let n = # vertices

m = # edges• Complete graph (clique) – each pair of vertices are

adjacent

• Each of the n vertices are incident to n-1 edges, but each edge is summed two time! Therefore, m = n(n-1)/2.

• So iff a graph is not complete then m < n(n-1)/2

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1)deg(

2

1

Vv Vv

nnnvm

n 5m(5

• Clique: A subgraph in which each pair of vertices are adjacent: a complete subgraph.

• Search for the maximum clique: a naïve algorithm.

• Examine each set of k vertices to determine if it is a clique.

• But the number of possible cliques of size k in a graph of size V

• Lot of research on heuristic algorithms to find good non-exact solutions

• In case of a tree m = n - 1

• if m < n - 1, G is not connex

Connectivity

n 5m 4

n 5m 3

A spanning tree of G is a sub-graph which is a tree and which contains all vertices of G

G Spanning tree of G

Spanning tree

Curiosity

Euler and the bridges of Koenigsberg: the first problem of graph theory?

• Is it possible to make a walk crossing each bridge one and only one time and to come back to the starting point?

• The graph model

• Eulerian circuit: path which use each edge exactly once and come back to the initial vertex. • Euler’s theorem: a connected graph has an Eulerian circuit iff it has no vertex of odd degree

No, it is not possible!

Curiosity

More definitions

• Oriented graph: each edge go only in one direction

• Acyclic oriented graph

Without cycle With cycle

Accessibility

A tree rooted in v contains all accessible vertices from v using oriented path

strongly connex

each vertex is accessible from each other using an oriented path

Strongly connex component

Transitive closure

It is the graph G* obtained from the graph G after applying the following rule:

If there exists an oriented path from a to b in G then add an oriented edge from a to b in G*.

{ a , c , g }

{ f , d , e , b }

Graph representation

Adjacency list

Definition• The adjacency list of a graph with n vertices is an array of n lists of vertices.

• The list i contains vertex j if there is an edge from vertex i tovertex j.

Example 2 in an oriented graph

Adjacency matrix

Definition

• Let adjacency matrix of a graph with n vertices is a n × n matrix A where:

Remark

The adjacency matrix of an undirected graph must be symmetric.

Example 1

Example 2

• They explore the vertices that are reachable starting from a source vertex.

• Depth-First Search

• Breadth-First Search (level-order search)

Exploration algorithms

Breath-First Search

• Algorithm that explores the vertices that are reachable starting from a source vertex s and constructs a breadth-first search tree (spanning tree).

• Compute a distance from each vertices to the source vertex.

Color terminology

• WHITE vertices are unexplored.

• BLACK vertices are fully explored vertices.

• GRAY vertices are being explored: these vertices define the ”frontier” between explored and unexplored vertices.

BSF algorithm

Example

The resulting BSF spanning tree

Analysis of BFS

• Let n be the number of vertices and m the number of edges in a graph.

• Each vertex is enqueued once in the queue: to enqueue all vertices it takes O(n).

• Each edge is visited at most once: visiting all edges takes O(m).

• Complexity of BFS

As a result, BFS takes O(n + m) time.

• Start from a vertex s.

• Set s as the current vertex u. Mark u as ‘visited’.

• Select arbitrarily one adjacent vertex v of u.

• If v is ‘visited’ go back to u

• Else mark v ‘visited’. V become the current vertex. Repeat the previous steps

• When all vertices adjacent to the current vertex are ‘visited’ backtrack to a previous ‘visited’ vertex. Repeat the previous steps.

• When backtrack leads to vertex s and if all the adjacent vertices of s are ‘visited’, the algorithm stop.

Depth-First Search

Algorithme DFS(u);Input: a vertex u of GOutput: a graph with all vertices labeled ‘visited’

for each edge e incident to u dolet v be the other extremity of eif vertex v is not ‘visited’ then

mark v ‘visited’recursively call DFS(v)

Example

1) 2)

3) 4)

5) 6)

Definition of a weighted graph

• A graph, in which each edge has an associated numerical value, is calleda weighted graph.

• The numerical value associated to an edge is the weight of the edge.

• The weight of an edge can represent a distance, a cost. . . etc.

Applications

• Weighted graphs find their application in various problems such as communication or transportation networks.

Definition of the MST (Minimum Spanning Tree)

The MST is a spanning tree of a connex, weighted and undirected graph with minimum total weight.

Example

The weight of the MST:W(MST) = 8 + 2 + 4 + 7 + 4 + 2 + 9 + 1 = 37 is minimal.

Formal definition of a MST

Given a connex, weighted and undirected graph G = (V,E), find an acyclic subset T E connecting all vertices in V such that:

weight(u, v) is the weight the edge (u, v).

Safe edges

Definition

• Let A be a subset of edges of a MST of a graph G.

• An edge (u, v) of G is safe for A if A {(u, v)} is also a subset of a MST.

• We can deduce from the above definition that finding a MST can be done by greedily grow a set of safe edges:

More definitions...

The cut of a graph

• A cut of a graph G = (V,E) is a partition of the vertices of the graph into 2 sets: S and V − S.

• The cut is denoted: (S,V − S).

An edge crossing the cut

• An edge crosses the cut (S,V − S) if one of its end-vertices is in S and the other one in V − S.

• The edges (b, c), (c, d), (d, f ), (a, h), (e, f) and (b, h) cross the cut (S,V-S).

A cut respects a set...

• A cut respects a set A of edges if no edge in A crosses the cut.

Light edges

An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

Characterization of a safe edge

Theorem

• Let G = (V,E) and A E included in some MST of G.

• Let (S,V − S) be a cut of G that respects A.

• Let e = (u, v) be a light edge crossing (S,V − S).

• Then, edge e is safe for A, which mean that e is in the MST of G

Proof by contradiction

• Suppose you have a MST T not containing e; then we want to show that T is not the MST.

• Let e=(u,v), with u in S and v in V - S.

• Then because T is a spanning tree it contains a unique path from u to v, which together with e forms a cycle in G.

• This path has to include another edge f connecting S to V - S.

• T+e-f is another spanning tree (it has the same number of edges, and remains connected since you can replace any path containing f by one going the other way around the cycle).

• It has smaller weight than T since e has smaller weight than f.

• So T was not minimum, which is what we wanted to prove.

Prim’s algorithm for finding a MST

Minimum-Spanning-Tree-by-Prim(G, weight-function, source)

for each vertex u in graph G

set key of u to ∞

set parent of u to nil

set key of source vertex to zero

enqueue to minimum-heap Q all vertices in graph G.

while Q is not empty

extract vertex u from Q // u is the vertex with the lowest key that is in Q

for each adjacent vertex v of u do

if (v is still in Q) and (weight-function(u, v) < key of v) then

set u to be parent of v // in minimum-spanning-tree

update v's key to equal weight-function(u, v)

Complexity: O(nlogn + mlogn) using a heap

Prim’s algorithm on an example...

Shortest path

Weight (or length) of a path• In a weighted graph, the weight (or length) of a path is the sum of the weights of its edges.

Shortest path• Given a weighted graph and two vertices u and v, find the path of minimum weight between u and v.

Property• A subpath of a shortest path is itself a shortest path. (Proof: by contradiction.)

Applications• Networks,• Driving directions,• Flights. . .

Djikstra’s algorithm for shortest path

Definition

• Djikstra’s algorithm for shortest path incrementally constructs a set of vertices (or cloud) to which the shortest path is known.

• At each iteration, the algorithm adds to the cloud a vertex v (not in the cloud) whose the distance to the source is the shortest of the remaining vertices that are not in the cloud.

Assumption

• Djikstra’s algorithm assumes that the weights are non-negative.

Relaxation

• Let v.distance be the shortest ”known” path between vertex v (not in the cloud) and the source vertex.

• When u is added to the cloud, we discover a new path (that contains u) from the source to v. In this case, v.distance may (or may not) change:

v.distance = min(v.distance, u.distance + weight(u, v))

Note: The values in the vertices represent the distance from the vertex to the source.

Dijkstra’s algorithm on an example...

Initialize all distances to infinity except for the source vertex which distance is zero (from itself).

Goal: incrementally construct a cloud of vertices whose finalshortest path weights is determined.

Djikstra’s formal algorithm

Djikstra(G,s)input: G = (V,E), s is the source vertexoutput: Shortest pathsfor each v V

v.distance = ;v.parent = null;s.distance = 0;

Q = V; //Q is a priority queueCloud = ; //cloud is emptywhile(!Q.isEmpty())

u = Q.extract minimum()Cloud = Cloud {u};for each v not in cloud adjacent to u

relax(u,v,Q);Relax(u,v,Q)

if (v.distance > u.distance + weight(u, v))v.distance = u.distance + weight(u, v);v.parent = u;update Q

Complexity (using a heap):

O(mlogn + nlogn)