Philip Bille

Introduction to Graphs

• Undirected Graphs• Representation • Depth-First Search

• Connected Components• Breadth-First Search

• Bipartite Graphs

Introduction to Graphs

• Undirected Graphs• Representation • Depth-First Search

• Connected Components• Breadth-First Search

• Bipartite Graphs

• Undirected graph. Set of vertices pairwise joined by edges.

• Why graphs?• Models many natural problems from many different areas.• Thousands of practical applications.• Hundreds of well-known graph algorithms.

Undirected graphs

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vertex edge

Visualizing the Internet

http://en.wikipedia.org/wiki/Opte_Project

Visualizing Friendships on Facebook

"Visualizing friendships", Paul Butler

London Metro

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Putney Bridge

East Putney

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Woodside Park

West Finchley

Finchley CentralWoodford

South Woodford

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Newbury Park

East Finchley

Highgate

Archway

Devons Road

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Kentish Town

Neasden

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Willesden Green

South Tottenham

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ImperialWharf

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West Hampstead

Blackhorse Road

Acton Town

CanningTown

Finchley Road

Highbury &Islington

Canary Wharf

Stratford

StratfordInternational

FinsburyPark

Elephant & Castle

Stepney Green

Barking

East Ham

Plaistow

Upton Park

Poplar

West Ham

Upper Holloway

PuddingMill Lane

Kennington

Borough

Elm ParkDagenham

East

DagenhamHeathway

Becontree

Upney

Heathrow Terminal 5

Finchley Road& Frognal

CrouchHill

Northfields

Boston Manor

South Ealing

Osterley

Hounslow Central

Hounslow East

Clapham North

Clapham High Street

Oval

Arsenal

Holloway Road

Caledonian Road

West Croydon

HounslowWest

Hatton Cross

HeathrowTerminals 1, 2, 3

ClaphamJunction

WestHarrow

Brondesbury CaledonianRoad &

Barnsbury

TottenhamHale

WalthamstowCentral

HackneyWick

Homerton

WestActon

Limehouse EastIndia

Crystal Palace

ChiswickPark

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GrangeHill

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Redbridge

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Ickenham

TurnhamGreen

Uxbridge

Hillingdon Ruislip

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Mile End

Bow Road

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Upminster

Upminster Bridge

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Norwood Junction

Sydenham

Forest Hill

Anerley

Penge West

Honor Oak Park

Brockley

Harrow &Wealdstone

Cutty Sark for Maritime Greenwich

Ruislip Manor

Eastcote

Wapping

New Cross

Queens RoadPeckham

Peckham Rye

Denmark Hill

Surrey Quays

Whitechapel

Lewisham

Kilburn Park

Regent’s Park

KilburnHigh Road

EdgwareRoad

SouthHampstead

GoodgeStreet

Shepherd’s BushMarket

Goldhawk Road

Hammersmith

Bayswater

Warren Street

Aldgate

Euston

Farringdon

BarbicanRussellSquare

Kensington(Olympia)

MorningtonCrescent

High StreetKensington

Old Street

St. John’s Wood

Green Park

BakerStreet

NottingHill Gate

Victoria

AldgateEast

Mansion House

Temple

OxfordCircus

BondStreet

TowerHill

Westminster

PiccadillyCircus

CharingCross

Holborn

Tower Gateway

Monument

Moorgate

Leicester Square

London Bridge

St. Paul’s

Hyde Park Corner

Knightsbridge

StamfordBrook

RavenscourtPark

WestKensington

NorthActon

HollandPark

Marylebone

Angel

Queensway MarbleArch

SouthKensington

SloaneSquare

WandsworthRoad

Covent Garden

LiverpoolStreet

GreatPortland

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Bank

EastActon

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Warwick AvenueMaida Vale

Fenchurch Street

Paddington

BaronsCourt

GloucesterRoad St. James’s

Park

Latimer RoadLadbroke Grove

Royal Oak

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Bermondsey

Rotherhithe

ShoreditchHigh Street

Dalston Junction

Haggerston

Hoxton

Wood Lane

Shepherd’sBush

WhiteCity

King’s CrossSt. Pancras

EustonSquareEdgware

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Southwark

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Balham

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South Wimbledon

MordenThis diagram is an evolution of the original design conceived in 1931 by Harry Beck

Correct at time of going to print, December 2013

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Tube map

London metro, London Transport

Protein Interaction Networks

Protein-protein interaktionsnetværk, Jeong et al, Nature Review | Genetics

Applications of Graphs

Graph Vertices Edges

communication computers cables

transport intersections roads

transport airports flight routes

games position valid move

neural network neuron synapses

financial network stocks transactions

circuit logical gates connections

food chain species predator-prey

molecule atom bindings

• Undirected graph. G = (V, E)• V = set of vertices• E = set of edges (each edge is a pair of vertices)• n = |V|, m = |E|

• Path. Sequence of vertices connected by edges.• Cycle. Path starting and ending at the same vertex.• Degree. deg(v) = the number of neighbors of v, or edges incident to v. • Connectivity. A pair of vertices are connected if there is a path between them

Terminology

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path from 0 to 3

cycle

deg(2) = 3V = {0,1,2,…,12}E = {(0,1), (0,2) (0,4),(2,3),…, (11,12)}n = 13, m = 15

• Lemma. ∑v∈V

deg(v) = 2m.

• Proof. How many times is each edge counted in the sum?

Undirected Graphs

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• Path. Is there a path connecting s and t?• Shortest path. What is the shortest path connecting s and t?• Longest path. What is the longest path connecting s and t?

• Cycle. Is there a cycle in the graph?• Euler tour. Is there a cycle that uses each edge exactly once?• Hamilton cycle. Is there a cycle that uses each vertex exactly once?

• Connectivity. Are all pairs of vertices connected?• Minimum spanning tree. What is the best way of connecting all vertices?• Biconnectivity. Is there a vertex whose removal would cause the graph to be

disconnected?

• Planarity. Is it possible to draw the graph in the plane without edges crossing?• Graph isomorphism. Do these sets of vertices and edges represent the same graph?

Algoritmic Problems on Graphs

Introduction to Graphs

• Undirected Graphs• Representation • Depth-First Search

• Connected Components• Breadth-First Search

• Bipartite Graphs

• Graph G with n vertices and m edges.• Representation. We need the following operations on graphs.

• ADJACENT(v, u): determine if u and v are neighbors.• NEIGHBORS(v): return all neighbors of v.• INSERT(v, u): add the edge (v, u) to G (unless it is already there).

Representation

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• Graph G with n vertices and m edges.• Adjacency matrix.

• 2D n ⨉ n array A.• A[i,j] = 1 if i and j are neighbors, 0 otherwise

• Complexity?• Space. O(n2)• Time.

• ADJACENT and INSERT in O(1) time.• NEIGHBOURS in O(n) time.

Adjacency Matrix

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• Graph G with n vertices and m edges.• Adjacency list.

• Array A[0..n-1].• A[i] is a linked list of all neighbors of i.

• Complexity?

• Space. O(n + ∑v∈V

deg(v)) = O(n + m)

• Time.• ADJACENT, NEIGHBOURS, INSERT

O(deg(v)) time.

Adjacency List 0

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0123456789

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1 2 400 3 52 4 50 3 52 3 47 86610 11 129 129 129 10 11

• Real world graphs are often sparse.

Repræsentation

Data structure ADJACENT NEIGHBOURS INSERT space

adjacency matrix O(1) O(n) O(1) O(n2)

adjacency list O(deg(v)) O(deg(v)) O(deg(v)) O(n+m)

Introduction to Graphs

• Undirected Graphs• Representation • Depth-First Search

• Connected Components• Breadth-First Search

• Bipartite Graphs

• Algorithm for systematically visiting all vertices and edges.• Depth first search from vertex s.

• Unmark all vertices and visit s. • Visit vertex v:

• Mark v.• Visit all unmarked neighbours of v recursively.

• Intuition.• Explore from s in some direction, until we read dead end. • Backtrack to the last position with unexplored edges. • Repeat.

• Discovery time. First time a vertex is visited.• Finish time. Last time a vertex is visited.

Depth-First Search

s

• Time. (on adjacency list representation)• Recursion? once per vertex.• O(deg(v)) time spent on vertex v.

• ⟹ total O(n + ∑v∈V

deg(v)) = O(n + m) time.

• Only visits vertices connected to s.

Depth-First SearchDFS(s)

time = 0DFS-VISIT(s)

DFS-VISIT(v)v.d = time++mark vfor each unmarked neighbor u

DFS-VISIT(u) u.π = v

v.f = time++

s

• Flood fill. Chance the color of a connected area of green pixels.

• Algorithm. • Build a grid graph and run DFS. • Vertex: pixel.• Edge: between neighboring pixels of same color.• Area: connected component

Flood Fill

• Definition. A connected component is a maximal subset of connected vertices.

• How to find all connected components? • Algorithm.

• Unmark all vertices.• While there is an unmarked vertex:

• Chose an unmarked vertex v, run DFS from v. • Time. O(n + m).

Connected Components

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Introduction to Graphs

• Undirected Graphs• Representation • Depth-First Search

• Connected Components• Breadth-First Search

• Bipartite Graphs

• Breadth first search from s. • Unmark all vertices and initialize queue Q.• Mark s and Q.ENQUEUE(s). • While Q is not empty:

• v = Q.DEQUEUE().• For each unmarked neighbor u of v

• Mark u.• Q.ENQUEUE(u).

• Intuition.• Explore, starting from s, in all directions - in increasing distance from s.

• Shortest paths from s. • Distance to s in BFS tree = shortest distance to s in the original graph.

Breadth-First Search

s

• Lemma. BFS finds the length of the shortest path from s to all other vertices.• Intuition.

• BFS assigns vertices to layers. Layer number i contains all vertices of distance i to s.

• What does each layer contain?• L0 : {s}• L1 : all neighbours of L0.• L2 : all neighbours if L1 that are not neighbours of L0

• L3 : all neighbours of L2 that neither are neighbours of L0 nor L1.• …• Li : all neighbours til Li-1 not neighbouring Lj for j < i-1

• = all vertices of distance i from s.

Shortest Paths

s

L0

L1 L2

L3

• Time. (on adjacency list representation)• Each vertex is visited at most once.• O(deg(v)) time spent on vertex v.

• ⟹ total O(n + ∑v∈V

deg(v)) = O(n + m) time.

• Only vertices connected to s are visited.

Breadth-First SearchBFS(s)

mark ss.d = 0Q.ENQUEUE(s)repeat until Q.ISEMPTY()

v = Q.DEQUEUE()for each unmarked neighbor u

mark uu.d = v.d + 1 u.π = v Q.ENQUEUE(u)

s

• Definition. A graph is bipartite if and only if all vertices can be colored red and blue such that every edge has exactly one red endpoint and one blue endpoint.

• Equivalent definition. A graph is bipartite if and only if its vertices can be partitioned into two sets V1 and V2 such that all edges go between V1 and V2.

• Application. • Scheduling, matching, assigning clients to servers, assigning jobs to machines,

assigning students to advisors/labs, …• Many graph problems are easier on bipartite graphs.

Bipartite Graphs

• Challenge. Given a graph G, determine whether G is bipartite.

Bipartite Graphs

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• Lemma. A graph G is bipartite if and only if all cycles in G have even length.• Proof. ⟹

• If G is bipartite, all cycles start and end on the same side.

Bipartite Graphs

• Lemma. A graph G is bipartite if and only if all cycles in G have even length.• Proof. ⟸

• Choose a vertex v and consider BFS layers L0, L1, …, Lk.• All cycles have even length • ⟹ There is no edge between vertices of the same layer • ⟹ We can assign alternating (red, blue) colours to the layers • ⟹ G is bipartite.

Bipartite Graphs

• Algorithm.• Run BFS on G.• For each edge in G, check if it's

endpoints are in the same layer.

• Time.• O(n + m)

Bipartite Graphs

s

L0

L1 L2

L3

s

L0

L1 L2

L3

• All on the adjacency list representation.

Graph AlgorithmsAlgorithm Time Space

Depth first search O(n + m) O(n + m)

Breadth first search O(n + m) O(n + m)

Connected components O(n + m) O(n + m)

Bipartite O(n + m) O(n + m)

Introduction to Graphs

• Undirected Graphs• Representation • Depth-First Search

• Connected Components• Breadth-First Search

• Bipartite Graphs