CSC2105 Graph Introduction

CSC2105: AlgorithmsCSC2105: AlgorithmsGraph - IntroductionGraph - Introduction

Mashiour RahmanMashiour [email protected]@aiub.edu

American International University BangladeshAmerican International University Bangladesh

Mashiour Rahman

AIUB::CSC2105::Algorithms Graph Introduction 2

Graph - IntroductionGraph - IntroductionReference:Reference:

• CLRS Appendix B.4, Chapters 22, 24-26

Objectives:Objectives:

• To learn several significant graph algorithms and their applications

a. Graph search (BFS, DFS)

b. Bellman/Ford and Dijkstra’s shortest path algorithms (Greedy)

c. Floyd’s algorithm (Dynamic Programming)

d. Network flows

Mashiour Rahman


MotivationMotivationState-space search in Artificial IntelligenceState-space search in Artificial Intelligence

Geographical information systems, electronic Geographical information systems, electronic street directorystreet directory

Logistics and supply chain managementLogistics and supply chain management

Telecommunications network designTelecommunications network design

Many more industry applicationsMany more industry applications

Mashiour Rahman


Review: GraphsReview: GraphsGraph Graph GG = ( = (VV, , EE))

V = {1,…n} = set of vertices, E = set of e edges

Undirected graph:

edge (u, v) = edge (v, u)

no self-loops

Directed graph (digraph):

edge (u, v) goes from vertex u to vertex v

Sparse graph:

e = O(n), dense otherwise

A weighted graph associates weights with either the edges or the vertices

Degree of a vertex v:

deg(v) = number of edges adjacent on v (in-degree and out-degree for directed graphs)

Mashiour Rahman


Examples of GraphsExamples of Graphs

Directed & Undirected graphs.Directed & Undirected graphs.

a)a) A directed graph A directed graph G= (V, E)G= (V, E), where , where VV = {1,2,3,4,5,6} and = {1,2,3,4,5,6} and EE={(1,2),(2,2),(2,4),(2,5),(4,1),(4,5),(5,4),(3,6)}. The edge (2,2) ={(1,2),(2,2),(2,4),(2,5),(4,1),(4,5),(5,4),(3,6)}. The edge (2,2) is self loop.is self loop.

b)b) An undirected graph An undirected graph G = (V,E)G = (V,E), where , where VV = {1,2,3,4,5,6} and = {1,2,3,4,5,6} and EE = {(1,2),(1,5),(2,5),(3,6)}. The vertex 4 is isolated.= {(1,2),(1,5),(2,5),(3,6)}. The vertex 4 is isolated.

c)c) The subgraph of the graph in part (a) induced by vertex set The subgraph of the graph in part (a) induced by vertex set {1,2,3,6}{1,2,3,6}

Mashiour Rahman


Review: GraphsReview: GraphsAcyclic Acyclic : if a graph contains no cycles: if a graph contains no cycles

DAG DAG : Directed acyclic graphs: Directed acyclic graphs

Connected Connected : if every vertex of a graph can : if every vertex of a graph can reachreach every other vertex, i.e., every pair of vertices is every other vertex, i.e., every pair of vertices is connected by a pathconnected by a path

Connected Components Connected Components : equivalence classes of : equivalence classes of vertices under “is reachable from” relationvertices under “is reachable from” relation

Strongly connected Strongly connected : every 2 vertices are reachable : every 2 vertices are reachable from each other (in a digraph)from each other (in a digraph)

Mashiour Rahman


Forests, DAG, ComponentsForests, DAG, Components

Mashiour Rahman


Review: Representing GraphsReview: Representing GraphsAdjacency matrixAdjacency matrix: represents a graph as : represents a graph as n n x x n n matrix matrix AA::

A[i, j] = 1 if edge (i, j) E (or weight of edge)

= 0 if edge (i, j) E

Storage requirements: O(n2)

A dense representation

But, can be very efficient for small graphs

Especially if store just one bit/edge

Undirected graph: only need half of matrix

Adjacency listAdjacency list: list of adjacent vertices: list of adjacent vertices

For each vertex v V, store a list of vertices adjacent to v

Storage requirements: O(n+e)

Good for large, sparse graphs (e.g., planar maps)

Mashiour Rahman


Review: Representing GraphsReview: Representing Graphs

Two representations of an undirected graph.

a) An undirected graph G having five vertices and seven edges.

b) An adjacency-list representation of G.

c) The adjacency-matrix representation of G.

Mashiour Rahman


Graph SearchingGraph SearchingGivenGiven: a graph : a graph GG = = (V, E)(V, E), directed or undirected, directed or undirected

GoalGoal: methodically explore every vertex and edge: methodically explore every vertex and edge

UltimatelyUltimately: build a : build a treetree on the graph on the graph

Pick a vertex as the root

Choose certain edges to produce a tree

Note: might also build a forest if graph is not connected

Breadth-first searchBreadth-first search

Depth-first searchDepth-first search

Other variants: best-first, iterated deepening search, Other variants: best-first, iterated deepening search, etc.etc.

Mashiour Rahman


Depth-First Search (DFS)Depth-First Search (DFS)Explore “deeper” in the graph whenever possibleExplore “deeper” in the graph whenever possible

Edges are explored out of the most recently discovered vertex v that

still has unexplored edges (LIFO)

When all of v’s edges have been explored, backtrack to the vertex from

which v was discovered

computes 2 timestamps: d[ ] (discovered) and f[ ] (finished)

builds one or more depth-first tree(s) (depth-first forest)

Algorithm colors each vertexAlgorithm colors each vertex

WHITE: undiscovered

GRAY: discovered, in process

BLACK: finished, all adjacent vertices have been discovered

Mashiour Rahman


Depth-First Search: The CodeDepth-First Search: The Code DFS(G)DFS(G)

{{

for each vertex u V

color[u] = WHITE;

time = 0;

for each vertex u V

if (color[u] == WHITE)

DFS_Visit(u);

}}

DFS_Visit(u)DFS_Visit(u){{

color[u] = GREY;color[u] = GREY;time = time+1;time = time+1;d[u] = time; d[u] = time; // compute d[]// compute d[]

for each v adjacent to ufor each v adjacent to uif (color[v] == WHITE)if (color[v] == WHITE)

p[v]= u p[v]= u // build tree// build tree

DFS_Visit(v);DFS_Visit(v);color[u] = BLACK;color[u] = BLACK;time = time+1;time = time+1;f[u] = time; f[u] = time; // compute f[]// compute f[]

}}

Mashiour Rahman


DFS AnalysisDFS AnalysisRunning time of Running time of DFSDFS = = O(n+e)O(n+e)

DFS (excluding DFS_Visit) takes O(DFS (excluding DFS_Visit) takes O(nn) time) time

DFS_Visit:DFS_Visit:

DFS_Visit( v ) is called exactly once for each vertex v

During DFS_Visit( v ), adjacency list of v is scanned once

sum of lengths of adjacency lists = O(e)

This type of aggregate analysis is an informal This type of aggregate analysis is an informal example of example of amortized analysisamortized analysis

Mashiour Rahman


DFS Classification of EdgesDFS Classification of EdgesDFS can be used to classify edges of DFS can be used to classify edges of GG::

1. Tree edges: edges in the depth-first forest.

2. Back edges: edges (u, v) connecting a vertex u to an ancestor v in a depth-first tree.

3. Forward edges: non-tree edges (u, v) connecting a vertex u to a descendant v in a depth-first tree.

4. Cross edges: all other edges.

DFS yields valuable information about the DFS yields valuable information about the structurestructure of a graph.of a graph.

Mashiour Rahman


Operations of DFSOperations of DFS

x z y

w v u

Undiscovered

Discovered, On Process

FinishedFinished, all all adjacent vertices adjacent vertices have been have been discovereddiscovered

Discover time/ Finish time

Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/54/5

x z3/

y

w

2/

v

1/

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/54/5

x z3/63/6

y

w

2/

v

1/

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/54/5

x z3/63/6

y

w

2/72/7

v

1/

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/54/5

x z3/63/6

y

w

2/72/7

v

1/

u

Forward Edge

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/54/5

x z3/63/6

y

w

2/72/7

v

1/81/8

u

Undiscovered




4/54/5

x z3/63/6

y

w

2/72/7

v

1/81/8

u

Forward Edge

Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/5

x z3/6

y

w

2/7

v

1/8

u

Forward Edge

4/54/5

x z3/63/6

y

9/

w

2/72/7

v

1/81/8

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/5

x z3/6

y

w

2/7

v

1/8

u

Forward Edge

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

4/54/5

x z3/63/6

y

9/

w

2/72/7

v

1/81/8

u

Cross Edge

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/5

x z3/6

y

w

2/7

v

1/8

u

Forward Edge

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

Cross Edge

4/54/5

x10/

z3/63/6

y

9/

w

2/72/7

v

1/81/8

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/5

x z3/6

y

w

2/7

v

1/8

u

Forward Edge

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

Cross Edge

4/5

x10/

z3/6

y

9/

w

2/7

v

1/8

u

4/54/5

x10/

z3/63/6

y

9/

w

2/72/7

v

1/81/8

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/5

x z3/6

y

w

2/7

v

1/8

u

Forward Edge

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

Cross Edge

4/5

x10/

z3/6

y

9/

w

2/7

v

1/8

u

4/5

x10/

z3/6

y

9/

w

2/7

v

1/8

u

4/54/5

x10/1110/11

z3/63/6

y

9/

w

2/72/7

v

1/81/8

u

Undiscovered




Mashiour Rahman



x z y

w v u

x z y

w v

1/

u

x z y

w

2/

v

1/

u

Tree edge

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

4/

x z3/

y

w

2/

v

1/

u

Back Edge

4/5

x z3/

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/

u

4/5

x z3/6

y

w

2/7

v

1/8

u

4/5

x z3/6

y

w

2/7

v

1/8

u

Forward Edge

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

4/5

x z3/6

y

9/

w

2/7

v

1/8

u

Cross Edge

4/5

x10/

z3/6

y

9/

w

2/7

v

1/8

u

4/5

x10/

z3/6

y

9/

w

2/7

v

1/8

u

4/5

x10/11

z3/6

y

9/

w

2/7

v

1/8

u

4/54/5

x10/1110/11

z3/63/6

y

9/129/12

w

2/72/7

v

1/81/8

u

Undiscovered




Mashiour Rahman


Cycle DetectionCycle DetectionTheorem: An undirected graph is Theorem: An undirected graph is acyclic acyclic iff a DFS iff a DFS yields no yields no back edgesback edges

ProofProof

If acyclic, no back edges by definition (because a back edge implies a cycle)

If no back edges, acyclic

No back edges implies only tree edges (Why?)

Only tree edges implies we have a tree or a forest

Which by definition is acyclic

Thus, can run DFS to find whether a graph has a Thus, can run DFS to find whether a graph has a cycle. cycle. How would you modify the code to detect How would you modify the code to detect cycles?cycles?

Mashiour Rahman


Directed Acyclic Graph (DAG)Directed Acyclic Graph (DAG)Arises in many applications where there are Arises in many applications where there are precedence or ordering constraints (e.g. scheduling precedence or ordering constraints (e.g. scheduling problems)problems)

if there are a series of tasks to be performed, and certain tasks must precede other tasks

In general, a precedence constraint graph is a DAG, In general, a precedence constraint graph is a DAG, in which vertices are tasks and edge (u, v) means that in which vertices are tasks and edge (u, v) means that task u must be completed before task v beginstask u must be completed before task v begins

Mashiour Rahman


Topological SortTopological SortFind a linear ordering of all vertices of the DAG such Find a linear ordering of all vertices of the DAG such that if G contains an edge (u, v), u appears before v that if G contains an edge (u, v), u appears before v in the ordering.in the ordering.

In general, there may be many legal topological In general, there may be many legal topological orders for a given DAG.orders for a given DAG.

IdeaIdea::

1. Call DFS(G) to compute finishing time f[ ]

2. Insert vertices onto a linked list according to decreasing order of f[ ]

How to modify DFS to perform Topological Sort in O(n+e) How to modify DFS to perform Topological Sort in O(n+e) time?time?

Mashiour Rahman


Strongly Connected ComponentsStrongly Connected ComponentsDigraphsDigraphs are often used to model communication and are often used to model communication and transportation networks transportation networks

People want to know that the networks are complete in the People want to know that the networks are complete in the sense that from any location it is possible to reach another sense that from any location it is possible to reach another location in the digraphlocation in the digraph

How to find strongly connected components (SCC) of a How to find strongly connected components (SCC) of a digraph?digraph?

Mashiour Rahman


AlgorithmAlgorithm (SCC) (SCC)Strongly-Connected-Components(G)Strongly-Connected-Components(G)

1. call DFS(G) to compute finish times f[]

2. compute GT

3. call DFS(GT) and consider vertices in decreasing f[] computed in step 1

4. output vertices of each tree in DFS(GT) as separate strongly connected component

Total running time Total running time ΘΘT(n + e)T(n + e)

Mashiour Rahman


Breadth-First Search (BFS)Breadth-First Search (BFS)

Given source vertex Given source vertex ss,,

systematically explore the breadth of the frontier to

discover every vertex reachable from s

computes the distance d[ ] from s to all reachable vertices builds a breadth-first tree rooted at s

AlgorithmAlgorithm

colors each vertex:

WHITE : undiscovered

GRAY: discovered, in process

BLACK: finished, all adjacent vertices have been discovered

Mashiour Rahman


BFS (Intuition)BFS (Intuition)

Mashiour Rahman


BFS: The CodeBFS: The CodeBFS(G, s) {BFS(G, s) {

initialize vertices;initialize vertices;

Q = {s};Q = {s};

while (Q not empty) {while (Q not empty) {

u = Dequeue(Q);u = Dequeue(Q);

for each v adjacent to u do {for each v adjacent to u do {

if (color[v] == WHITE) {if (color[v] == WHITE) {

color[v] = GRAY;color[v] = GRAY;

d[v] = d[u] + 1;// compute d[]d[v] = d[u] + 1;// compute d[]

p[v] = u; // build BFS treep[v] = u; // build BFS tree

Enqueue(Q, v);Enqueue(Q, v);

}}

}}

color[u] = BLACK;color[u] = BLACK;

}}

}}

Mashiour Rahman


BFS ExampleBFS Example

Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman



Mashiour Rahman


BFS Analysis BFS Analysis initializeinitialize : : O(n)O(n)

LoopLoop: Queue operations and Adjacency checks: Queue operations and Adjacency checks

Queue operationsQueue operations

each vertex is enqueued/dequeued at most once. Why?

each operation takes O(1) time, hence O(n)

Adjacency checksAdjacency checks

adjacency list of each vertex is scanned at most once

sum of lengths of adjacency lists = O(e)

Total run time of BFS = Total run time of BFS = O(n+e)O(n+e)

Mashiour Rahman


Breadth-First Search: Breadth-First Search: PropertiesProperties

What do we get the end of BFS?What do we get the end of BFS?

1.1. d[v]d[v] = = shortest-path distanceshortest-path distance from from ss to to vv, i.e. , i.e. minimum number of edges from minimum number of edges from ss to to vv, or , or ∞∞ if if v v not not reachable from reachable from ss

Proof : refer CLRS

2.2. a a breadth-firstbreadth-first tree tree, in which path from root , in which path from root ss to to any vertex any vertex vv represent a shortest path represent a shortest path

Thus can use BFS to calculate shortest path from one vertex to another in O(n+e) time, for unweighted graphs.

Date post:	01-Jan-2016
Category:	Documents
Upload:	sharowar-jahan
View:	21 times
Download:	0 times

CSC2105 Graph Introduction

Documents