Introduction to Graph TheoryIntroduction to Graph Theory

HKOI Training (Intermediate)HKOI Training (Intermediate)Kelly ChoiKelly Choi

28 Mar 200928 Mar 2009

OverviewOverview

Introduction to GraphsIntroduction to Graphs Graph RepresentationGraph Representation Visiting a graphVisiting a graph

BFS (Breadth First Search)BFS (Breadth First Search) DFS (Depth First Search)DFS (Depth First Search)

Topological SortTopological Sort Flood FillFlood Fill Other TopicsOther Topics

A Classical ProblemA Classical Problem

MazeMaze Find a path Find a path

from S to Efrom S to E

S

E

RumoursRumours

In a group of people, a rumour start In a group of people, a rumour start from from SamSam to others. to others.

Each person can spread the rumour Each person can spread the rumour to his/her friends. to his/her friends.

Through how many persons will the Through how many persons will the rumour reach rumour reach EmilyEmily??

SamSam

EmilyEmily

TimTim

KateKate

JohnJohn

Aren’t the two problems similar?Aren’t the two problems similar?

Essentially we have some Essentially we have some verticesvertices which are linked together by some which are linked together by some edgesedges..

In both problems, we want to find a In both problems, we want to find a path to get from one vertex to path to get from one vertex to another.another. In other cases, it could be some other In other cases, it could be some other

problems. problems.

WHAT ARE GRAPHS?

GraphGraph

In a graph, we have some In a graph, we have some verticesvertices and and edgesedges. An edge links two . An edge links two vertices together, with or without a vertices together, with or without a direction. direction.

1

2

4

3

vertex

edge

GraphGraph

Mathematically, a Mathematically, a graphgraph is defined is defined as G=(V,E), as G=(V,E), V is the set of vertices (singular: vertex)V is the set of vertices (singular: vertex) E is the set of edges that connect some E is the set of edges that connect some

of the verticesof the vertices For convenience,For convenience,

Label vertices with 1, 2, 3, …Label vertices with 1, 2, 3, … Edges can be represented by their two Edges can be represented by their two

endpoints.endpoints.

GraphGraph

Directed/Undirected GraphDirected/Undirected Graph Weighted/Unweighted GraphWeighted/Unweighted Graph Simple GraphSimple Graph ConnectivityConnectivity

Graph Modelling

S

E

SS

EE

GRAPH REPRESENTATION

Representation of GraphRepresentation of Graph

How do we store a graph in a How do we store a graph in a program?program? Adjacency MatrixAdjacency Matrix Adjacency linked listAdjacency linked list Edge listEdge list

Adjacency Matrix

1 2 3 4 5 6

1 0 0 1 0 1 1

2 0 0 0 0 0 0

3 1 0 0 0 0 1

4 0 0 0 0 0 0

5 0 1 0 0 0 0

6 0 0 0 1 0 0

11

2233

44

66

55

Adjacency Linked List

1 3 5 6

2

3 1 6

4

5 2

6 4

11

2233

44

66

55

Edge List

i e[i][1] e[i][2]

1 1 3

2 1 5

3 1 6

4 3 1

5 3 6

6 5 2

7 6 4

11

2233

44

66

55

i s[i]

1 1

2 4

3 4

4 6

5 6

6 7

Representation of GraphRepresentation of Graph

Adjacency Adjacency MatrixMatrix

Adjacency Adjacency Linked ListLinked List

Edge ListEdge List

Memory Memory StorageStorage

O(VO(V22)) O(V+E)O(V+E) O(V+E)O(V+E)

Check Check whether whether ((uu,,vv) is an ) is an edgeedge

O(1)O(1) O(deg(u))O(deg(u)) O(deg(u))O(deg(u))

Find all Find all adjacent adjacent vertices of a vertices of a vertex vertex uu

O(V)O(V) O(deg(u))O(deg(u)) O(deg(u))O(deg(u))

deg(u): the number of edges connecting vertex deg(u): the number of edges connecting vertex uu

VISITING A GRAPH

Visiting a Graph

To scan an array: use iterations (for-loops)

How to scan the vertices of a graph? e.g. to find a path from S to E Usually we visit a vertex only after

discovering an edge to it from its neighbour

one of the ways: recursion

Using RecursionUsing Recursion

Strategy: Go as far as you can (if Strategy: Go as far as you can (if you have not visit there), otherwise, you have not visit there), otherwise, go back and try another waygo back and try another way

ImplementationImplementation

This is known as Depth-First Search This is known as Depth-First Search (DFS).(DFS).

DFS (vertex DFS (vertex uu) {) {

mark mark uu as as visitedvisited

for each vertex for each vertex vv directly reachable from directly reachable from uu

if if vv is is unvisitedunvisited

DFS (DFS (vv))

}}

Initially all vertices are marked as Initially all vertices are marked as unvisitedunvisited

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Note the color of the vertices

1. Vertices fall into 3 categories: Unvisited (White) Discovered (Grey) Dead (All reachable vertices from

these vertices are discovered) (Black)

Use of DFS

DFS is very useful When you have to do something with

every vertices When you want to know whether one

vertex is connected to another vertices Drawbacks

Stack overflow Finding a shortest path is difficult

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

BFS tries to find the target from BFS tries to find the target from nearestnearest vertices first. vertices first.

BFSBFS uses a queue to store discovered uses a queue to store discovered

verticesvertices expand the path from the earliest expand the path from the earliest

discovered vertices.discovered vertices.

1

4

3

25

6

Simulation of BFSSimulation of BFS

Queue:Queue: 1 4 3 5 2 6

Question

Why does BFS work to find the shortest path?

ImplementationImplementation

while queue Q not emptywhile queue Q not emptydequeue the first vertex dequeue the first vertex uu from Q from Qfor each vertex for each vertex vv directly reachable from directly reachable from uu

if if vv is is unvisitedunvisitedenqueue enqueue vv to Q to Qmark mark vv as as visitedvisited

Initially all vertices except the start Initially all vertices except the start vertex are marked as vertex are marked as unvisitedunvisited and and the queue contains the start vertex the queue contains the start vertex onlyonly

AdvantagesAdvantages

Guarantee shortest paths for Guarantee shortest paths for unweighted graphsunweighted graphs

Use queue instead of recursive Use queue instead of recursive functions – Avoiding stack overflowfunctions – Avoiding stack overflow

MORE GRAPH PROBLEMS

Finding the shortest paths isn’t the only graph problem…

Teacher’s Problem (HKOI 2004 Senior)Teacher’s Problem (HKOI 2004 Senior)

Emily wants to distribute candies to N students one by one, with a rule that if student A is teased by B, A can receive candy before B.

Given lists of students teased by each students, find a possible sequence to give the candies

Topological SortTopological Sort

In short, in a In short, in a directeddirected graph, graph, We want to give a label to the verticesWe want to give a label to the vertices So that if there is an edge from u to v, So that if there is an edge from u to v,

then u<vthen u<v Is it always possible?Is it always possible?

Topological Sort: to find such orderTopological Sort: to find such order

ObservationObservation

The vertex numbered 1 must have no incoming edge.

The vertex numbered 2 must have no incoming edges other than (possibly) one from vertex 1.

And so on…

How to solve the problem?

Find a vertex with no incoming edges. Number it and remove all outgoing edges from it.

Repeat the process.

Problem: How to implement the above process? Iteration Recursion

Finding area

Find the area that is reachable from A. A

Flood FillFlood Fill

Starting from one vertex, visit (fill) Starting from one vertex, visit (fill) all vertices that are connected, in all vertices that are connected, in order to get some information, e.g. order to get some information, e.g. areaarea

We can use BFS/DFSWe can use BFS/DFS

Example: Largest Continuous Example: Largest Continuous Region (HKOI2003 Senior Q4)Region (HKOI2003 Senior Q4)

Remark on representation of graphs

In BFS/DFS, we perform operations on all neighbours of some vertices. Use adjacency linked list / edge list

In some other applications, we may check whether there is an edge between two vertices. Adjacency matrix may be better in

some cases.

Summary

You should know What a graph is What it means to model a problem with

a graph Basic ideas and implementation of

DFS/BFS to find shortest paths / scan all vertices

Some ideas of Topological Sort and Floodfilling

Miscellaneous TopicsMiscellaneous Topics

Euler Path / CircuitEuler Path / Circuit A path/circuit that goes through every A path/circuit that goes through every

edge edge exactly onceexactly once Diameter and RadiusDiameter and Radius TreesTrees More advanced topicsMore advanced topics

Finding Strongly Connected Finding Strongly Connected Components (SCC)Components (SCC)

Preview of Other Advanced Problems

More on DFS and BFSMore on DFS and BFS Shortest path in a weighted graphShortest path in a weighted graph

Dijkstra’s Algorithm: Using priority Dijkstra’s Algorithm: Using priority queuequeue

Disjoint set and minimum spanning Disjoint set and minimum spanning treestrees

Preview of Other Advanced Problems

Searching techniques: Searching techniques: Bidirectional search (BDS) Bidirectional search (BDS) Iterative deepening search (IDS)Iterative deepening search (IDS)

Network Flow (not required in IOI)Network Flow (not required in IOI)

1067 Maze1067 Maze 2045 Teacher’s Problem2045 Teacher’s Problem 2037 Largest Continuous Region2037 Largest Continuous Region

(HKOI 2003 Senior Q4)(HKOI 2003 Senior Q4) 2066 Squareland2066 Squareland 3021 Bomber Man3021 Bomber Man

Practice Problems