View
7
Download
0
Category
Preview:
Citation preview
Graphs-I CS2110,Spring2017Leonhard Euler. 1736 7 Bridges of Konigsberg
Thesearen’tthegraphswe’reinterestedin
Thisis
V.J.WedeenandL.L.Wald,MarAnosCenterforBiomedicalImagingatMGH
Andsoisthis
ThiscarriesInternettrafficacrosstheoceans
Anoldersocialgraph
Anoldersocialgraph
VoltaireandBenjaminFranklin
AficAonalsocialgraph
Atransportgraph:NYsubwaysystem
Anothertransportgraph
Acircuitgraph(flip-flop)
Acircuitgraph(Intel4004)
Thisisnotagraph,thisisacat
Thisisagraph(icalmodel)thathaslearnedtorecognizecats
Someabstractgraphs
K5
K3,3
ApplicaAonsofGraphs• CommunicaAonnetworks• Socialnetworks• RouAngandshortestpathproblems• CommoditydistribuAon(networkflow)• Trafficcontrol• ResourceallocaAon• Numericallinearalgebra(sparsematrices)• Geometricmodeling(meshes,topology,…)• Imageprocessing(e.g.graphcuts)• ComputeranimaAon(e.g.moAongraphs)• Systemsbiology• DigitalhumaniAes(e.g.RepublicofLe[ers)• …
Directedgraphs
• Adirectedgraph(digraph)isapair(V, E)where– Visa(finite)set– Eisasetoforderedpairs(u, v)whereu,v ∈ V
• O]enrequireu ≠ v (i.e.noself-loops)
• AnelementofViscalledavertexornode• AnelementofEiscalledanedgeorarc
• |V|=sizeofV,o]endenotedbyn • |E|=sizeofE,o]endenotedbym
A
B C
DE
V = {A, B, C, D, E} E = {(A,C), (B,A), (B,C),
(C,D), (D,C)} |V| = 5 |E| = 5
UndirectedGraphs
• Anundirectedgraphisjustlikeadirectedgraph!– …exceptthatEisnowasetofunordered
pairs{u, v}whereu,v ∈ V
• EveryundirectedgraphcanbeeasilyconvertedtoanequivalentdirectedgraphviaasimpletransformaAon:– ReplaceeveryundirectededgewithtwodirectededgesinoppositedirecAons
• …butnotviceversa
A
B C
DE
V = {A, B, C, D, E} E = {{A,C}, {B,A},
{B,C}, {C,D}} |V| = 5 |E| = 4
Graphterminology
• VerAcesuandvarecalled– thesourceandsinkofthedirectededge(u, v),respecAvely
– theendpointsof(u, v) or {u, v}• TwoverAcesareadjacentiftheyare
connectedbyanedge• Theoutdegreeofavertexuinadirected
graphisthenumberofedgesforwhichuisthesource
• Theindegreeofavertexvinadirectedgraphisthenumberofedgesforwhichvisthesink
• Thedegreeofavertexuinanundirectedgraphisthenumberofedgesofwhichuisanendpoint
A
B C
DE
A
B C
DE
Moregraphterminology
• Apathisasequencev0,v1,v2,...,vpofverAcessuchthatfor0 ≤ i < p,– (vi, vi+1)∈Eifthegraphisdirected – {vi, vi+1}∈Eifthegraphisundirected
• Thelengthofapathisitsnumberofedges• Apathissimpleifitdoesn’trepeatanyverAces• Acycleisapathv0, v1, v2, ..., vp suchthatv0 = vp
• AcycleissimpleifitdoesnotrepeatanyverAcesexceptthefirstandlast
• Agraphisacyclicifithasnocycles• AdirectedacyclicgraphiscalledaDAG
A
B C
DE
A
B C
DE
DAG
NotaDAG
PathA,C,D
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
A
B
C
D
E
F
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
B
C
D
E
F
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
C
D
E
F
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
D
E
F
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
E
F
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
F
IsthisaDAG?
• IntuiAon:– Ifit’saDAG,theremustbeavertexwithindegreezero
• Thisidealeadstoanalgorithm– AdigraphisaDAGifandonlyifwecaniteraAvelydeleteindegree-0verAcesunAlthegraphdisappears
YES!
Topologicalsort
• WejustcomputedatopologicalsortoftheDAG– ThisisanumberingoftheverAcessuchthatalledgesgofromlower-tohigher-numberedverAces
– Usefulinjobschedulingwithprecedenceconstraints
1
2
3
4
5
6
Graphcoloring
• AcoloringofanundirectedgraphisanassignmentofacolortoeachnodesuchthatnotwoadjacentverAcesgetthesamecolor
• Howmanycolorsareneededtocolorthisgraph?
A
B
C
D
E
F
Graphcoloring
• AcoloringofanundirectedgraphisanassignmentofacolortoeachnodesuchthatnotwoadjacentverAcesgetthesamecolor
• Howmanycolorsareneededtocolorthisgraph?
A
B
C
D
E
F
AnapplicaAonofcoloring
• VerAcesaretasks• Edge(u, v)ispresentiftasksuandveachrequireaccessto
thesamesharedresource,andthuscannotexecutesimultaneously
• Colorsare9meslotstoschedulethetasks• Minimumnumberofcolorsneededtocolorthegraph=
minimumnumberofAmeslotsrequired
A
B
C
D
EF
Planarity
• Agraphisplanarifitcanbedrawnintheplanewithoutanyedgescrossing
• Isthisgraphplanar?
A
B
C
D
E
F
Planarity
• Agraphisplanarifitcanbedrawnintheplanewithoutanyedgescrossing
• Isthisgraphplanar?
– Yes!
A
B
C
D
E
F
Planarity
• Agraphisplanarifitcanbedrawnintheplanewithoutanyedgescrossing
• Isthisgraphplanar?
– Yes!
A
B
C
D
E
F
DetecAngPlanarity
Kuratowski'sTheorem:
• AgraphisplanarifandonlyifitdoesnotcontainacopyofK5orK3,3(possiblywithothernodesalongtheedgesshown)
K5 K3,3
DetecAngPlanarityIn the early 1970’s, Cornell Prof John Hopcroft spent a sabbatical at Stanford and worked with PhD student Bob Tarjan. They developed the first linear-time algorithm for testing whether a graph was planar. They later received the ACM Turing Award for their work on algorithms.
Tarjan was hired at one point in the 1970’s into our department, but the Ithaca weather was too depressing for him and he left for Princeton.
Coloring a graph: How many colors are needed to color the countries so that no two adjacent countries have the same color? Question asked as early as 1852.
1879. Kemp publishes a theorem that only 4 colors are needed!
1880. Julius Peterson finds a flaw in the Kemp’s proof!
Four-ColorTheorem:Everyplanargraphis4-colorable[Appel&Haken,1976]Theproofrestedoncheckingthat1,936specialgraphshadacertainproperty.
Theyusedacomputertocheckthatthose1,936graphshadthatproperty!
BasicallythefirstAmeacomputerwasneededtochecksomething.Causedalotofcontroversy.
Grieslookedattheircomputerprogram,arecursiveprogramwri[enintheassemblylanguageoftheIBM7090computer,andfoundanerror,whichwassafe(itsaidsomethingdidn’thavethepropertywhenitdid)andcouldbefixed.Othersdidthesame.
Sincethen,therehavebeenimprovements.AndaformalproofhasevenbeendoneintheCoqproofsystem
Another4-coloredplanargraph
h[p://www.cs.cmu.edu/~bryant/boolean/maps.html
BiparAtegraphs
• AdirectedorundirectedgraphisbiparAteiftheverAcescanbeparAAonedintotwosetssuchthatnoedgeconnectstwoverAcesinthesameset
• Thefollowingareequivalent– GisbiparAte– Gis2-colorable– Ghasnocyclesofoddlength
1
2
3
A
B
C
D
Someabstractgraphs
K5
K3,3
=
Travelingsalesperson
Findapathofminimumdistancethatvisitseverycity
Amsterdam
Rome
Boston
Atlanta
London Paris
Copenhagen
Munich
Ithaca
New York
Washington
1202 1380
1214
1322
1356
1002
512 216
441
189 160
1556 1323
419
210
224 132
660 505
1078
RepresentaAonsofgraphs
2 3
2 4
3
1
2
3
4
AdjacencyList AdjacencyMatrix
1 2
3 4
12341234
0101001000000110
AdjacencymatrixoradjacencyList?– n=numberofverAces– m=numberofedges– d(u)=degreeofu=no.ofedgesleavingu
• AdjacencyMatrix– UsesspaceO(n2) – EnumeratealledgesinAmeO(n2)– Answer“Isthereanedgefromutov?”inO(1)Ame– Be[erfordensegraphs(lotsofedges)
12341234
0101001000000110
– n=numberofverAces– e=numberofedges– d(u)=degreeofu=no.edgesleavingu
• AdjacencyList– UsesspaceO(e + n)– EnumeratealledgesinAmeO(e + n)– Answer“Isthereanedgefromutov?”inO(d(u))Ame– Be[erforsparsegraphs(feweredges)
2 3
2 4
3
1
2
3
4
Adjacencymatrixoradjacencylist?
Graphalgorithms
• Search– Depth-firstsearch– Breadth-firstsearch
• Shortestpaths– Dijkstra'salgorithm
• Minimumspanningtrees– Jarnik/Prim/Dijkstraalgorithm– Kruskal'salgorithm
Recommended