Graph Introduction

8/6/2019 Graph Introduction

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CSC2105: AlgorithmsCSC2105: AlgorithmsGraphGraph -- IntroductionIntroduction

Mashiour RahmanMashiour [email protected]@aiub.edu

American International University BangladeshAmerican International University Bangladesh


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Mashiour Rahman AIUB::CSC2105::Algorithms Graph Introduction 2

GraphGraph -- IntroductionIntroductionReference: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 Dijkstras shortest path algorithms (Greedy)

c. Floyds algorithm (Dynamic Programming)d. Network flows


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MotivationMotivation

StateState--space search in Artificial Intelligencespace search in Artificial Intelligence

Geographical information systems, electronicGeographical 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


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Review: GraphsReview: GraphsGraphGraph GG = (= (VV,, EE))

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

Undirectedgraph:

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

no self-loops

Directedgraph (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 thevertices

Degree of a vertex v:

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


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Examples of GraphsExamples of Graphs

Directed & Undirected graphs.Directed & Undirected graphs.

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

loop.loop.b)b) An undirected graphAn undirected graph G = (V,E)G = (V,E), where, whereVV = {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 {1,2,3,6}The subgraph of the graph in part (a) induced by vertex set {1,2,3,6}


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Review: GraphsReview: Graphs

Acyclic

Acyclic: if a graph contains no cycles: if a graph contains no cycles

DAGDAG: Directed acyclic graphs: Directed acyclic graphs

ConnectedConnected: if every vertex of a graph can: if every vertex of a graph can reachreach

every other vertex, i.e., every pair of vertices isevery other vertex, i.e., every pair of vertices isconnected by a pathconnected by a path

Connected ComponentsConnected Components : equivalence classes of: equivalence classes of

vertices under is reachable from relationvertices under is reachable from relation

Strongly connectedStrongly connected: every 2 vertices are reachable: every 2 vertices are reachable

from each other (in a digraph)from each other (in a digraph)


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Forests, DAG, ComponentsForests, DAG, Components


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Review:Representing GraphsReview:Representing GraphsAdjacency matrixAdjacency matrix: represents a graph as: represents a graph as nn xx nn matrixmatrix 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 representationBut, can be very efficient for small graphs

Especially if store just one bit/edge

Undirected graph: only need half of matrix

Adjacency list

Adjacency 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)


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Review:Representing GraphsReview:Representing Graphs

Two representations of an undirected graph.

a) An undirected graph G having five vertices and sevenedges.

b) An adjacency-list representation ofG.

c) The adjacency-matrix representation ofG.


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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 graphon the graph

Pick a vertex as the root

Choose certain edges to produce a tree

Note: might also build a forestif graph is not connected

BreadthBreadth--firstsearchfirstsearch

DepthDepth--firstsearchfirstsearchOther variants: bestOther variants: best--first, iterated deepening search,first, iterated deepening search,etc.etc.


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DepthDepth--First Search (DFS)First Search (DFS)Explore deeper in the graph whenever possibleExplore deeper in the graph whenever possible

Edges are exploredout of the mostrecentlydiscoveredvertex vthat

still has unexplored edges (LIFO)

When all ofvs edges have been explored, backtrack to the vertex from

which vwas discovered

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

builds one or more depth-firsttree(s) (depth-firstforest)

Algorithm colors each vertexAlgorithm colors each vertex

WHITE: undiscovered

GRAY: discovered,inprocess

BLACK: finished,alladjacentverticeshavebeendiscovered


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DepthDepth--First Search: The CodeFirst Search: The CodeDFS(G)DFS(G)

{{

foreachvertex u V

color[u] = WHITE;

time = 0;

foreachvertex 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 u

if (color[v] == WHITE)if (color[v] == WHITE)p[v]= up[v]= u // build tree// build treeDFS_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[]

}}


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DFS AnalysisDFS Analysis

Running time ofRunning time ofDFSDFS=

=

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 ofv is scanned once

sum of lengths of adjacency lists =O(e)

This type of aggregate analysis is an informalThis type of aggregate analysis is an informalexample ofexample ofamortizedanalysisamortizedanalysis


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DFS Classification of EdgesDFS Classification of Edges

DFS can be used to classify edges ofDFS can be used to classify edges ofGG::1. Tree edges: edges in the depth-first forest.

2. Back edges: edges (u,v) connecting a vertex u to an

ancestorv 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 theDFS yields valuable information about the structurestructureof a graph.of a graph.


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Operations of DFSOperations of DFS

x zy

wvu

Undiscovered

Discovered,

On Process

FinishedFinished, allall

adjacent verticesadjacent vertices

have beenhave been

discovereddiscovered

Discover time/ Finish time


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x zy

wvu

x zy

wv

1/

u

Undiscovered

Discovered,

On Process



have beenhave been




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x zy

wvu

x zy

wv

1/

u

x zy

w

2/

v

1/

u

Tree edge

Undiscovered

Discovered,

On Process



have beenhave been




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x zy

wvu

x zy

wv

1/

u

x zy

w

2/

v

1/

u

Tree edge

x z

3/

y

w

2/

v

1/

u

Undiscovered

Discovered,

On Process



have beenhave been




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x zy

wvu

x zy

wv

1/

u

x zy

w

2/

v

1/

u

Tree edge

x z

3/

y

w

2/

v

1/

u

4/

x z

3/

y

w

2/

v

1/

u

Undiscovered

Discovered,

On Process



have beenhave been




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x zy

wvu

x zy

wv

1/

u

x zy

w

2/

v

1/

u

Tree edge

x z

3/

y

w

2/

v

1/

u

4/

x z

3/

y

w

2/

v

1/

u

4/

x z

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w

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v

1/

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Back Edge

Undiscovered

Discovered,

On Process



have beenhave been




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x zy

wvu

x zy

wv

1/

u

x zy

w

2/

v

1/

u

Tree edge

x z

3/

y

w

2/

v

1/

u

4/

x z

3/

y

w

2/

v

1/

u

4/

x z

3/

y

w

2/

v

1/

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Back Edge

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x z

3/

y

w

2/

v

1/

u

Undiscovered

Discovered,

On Process



have beenhave been




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Cycle DetectionCycle DetectionTheorem: An undirected graph isTheorem: An undirected graph is acyclicacycliciff a DFSiff a DFSyields noyields no backedgesbackedges

ProofProof

If acyclic, no back edges by definition (because a back edgeimplies 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 aThus, can run DFS to find whether a graph has acycle.cycle. HowwouldyoumodifythecodetodetectHowwouldyoumodifythecodetodetectcycles?cycles?


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Directed Acyclic Graph (DAG)Directed Acyclic Graph (DAG)

Arises in many applications where there areArises in many applications where there areprecedence or ordering constraints (e.g. schedulingprecedence 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 thatin 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


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Topological SortTopological SortFind a linear ordering of all vertices of the DAG suchFind a linear ordering of all vertices of the DAG suchthat if G contains an edge (u, v), u appears before vthat if G contains an edge (u, v), u appears before vin the ordering.in the ordering.

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

IdeaIdea::

1. CallDFS(G) tocomputefinishingtimef[ ]

2. Insertverticesontoalinkedlistaccordingtodecreasingorderoff[ ]

HowtomodifyDFStoperform TopologicalSortin O(n+e)HowtomodifyDFStoperform TopologicalSortin O(n+e)time?time?


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Strongly Connected ComponentsStrongly Connected Components

DigraphsDigraphs are often used to model communication andare often used to model communication and

transportation networkstransportation networks

People want to know that the networks are complete in thePeople want to know that the networks are complete in the

sense that from any location it is possible to reach anothersense that from any location it is possible to reach anotherlocation in the digraphlocation in the digraph

How to find strongly connected components (SCC) of aHow to find strongly connected components (SCC) of a

digraph?digraph?


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AlgorithmAlgorithm (SCC)(SCC)

StronglyStrongly--ConnectedConnected--Components(G)Components(G)1. callDFS(G) tocomputefinishtimesf[]

2. computeGT

3. callDFS(GT) andconsiderverticesindecreasingf[]

computedinstep1

4. outputverticesofeachtreeinDFS(GT) asseparate

stronglyconnectedcomponent

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


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BreadthBreadth--First Search (BFS)First Search (BFS)Given source vertexGiven 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-firsttree rooted at s

AlgorithmAlgorithm

colors each vertex:

WHITE : undiscovered

GRAY: discovered, in process

BLACK: finished, all adjacent vertices have been discovered


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BFS (Intuition)BFS (Intuition)


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

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

while(Qnotempty) {while(Qnotempty) {

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

foreachvadjacenttoudo {foreachvadjacenttoudo {

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

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

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

p[v] = u;//buildBFStreep[v] = u;//buildBFStree

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

}}

}}

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

}}

}}


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BFS ExampleBFS Example


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BFS AnalysisBFS Analysisinitializeinitialize :: 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 oncesum of lengths of adjacency lists =O(e)

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


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BreadthBreadth--First Search: PropertiesFirst Search: Properties

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

1.1. d[v]d[v] ==shortestshortest--pathdistancepathdistance fromfrom ss toto vv, i.e., i.e.

minimum number of edges fromminimum number of edges from ss toto vv, or, or ififvvnotnot

reachable fromreachable from ss

Proof : referCLRS

2.2. aa breadthbreadth--firstfirsttreetree, in which path from root, in which path from root ss toto

any vertexany vertex vv represent a shortest pathrepresent a shortest path

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

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Graph Introduction

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