Spectral Graph Theory and its Applications. Lillian Dai Oct. 20, 2004

Transcription

1 Spectral raph Theory ad its Applicatios Lillia Dai Oct. 0, 004

2 Outlie Basic spectral graph theory raph partitioig usig spectral methods D. Spielma ad S. Teg, Spectral Partitioig Works: Plaar raphs ad Fiite Elemet Meshes, 996

3 raph ad Associated Matrices 4 3 V E = ( V, E) = = = m= 4 5 Laplacia matrix L = D A = B B T Adjacecy matrix Degree matrix Icidecy matrix A = D = B =

4 Properties of the Laplacia Matrix L = λ = 0 0 { 0,,4,4} 3 0 Symmetric -> real eigevalues; eigespaces are mutually orthogoal Orthogoally diagoalizable -> a eigevalue with multiplicity k has k-dimesioal eigespace 4

5 More Properties of the Laplacia Matrix λ = { 0,,4,4} Positive semidefiite -> oegative eigevalues (,,..., ) x= x x x Row sum = 0 -> sigular -> at least oe eigevalue = 0, uity eigevector (sice row sum = ) Orthogoal eigespaces u = eigevector of o-zero eigevalue u = 0 i= i T T T T T T i j ( i, j) E m m ( )( ) ( ) x L x= x B B x= x B x B = x x 0 5

6 , Spectrum of Some raphs Complete Lie Rig Star Eigevalues { ( ) 0, } ( π ) cos k k =,..., cos k k =,..., { 0, ( ), } ( π ) Which graphs are determied by their spectrum? Complete raphs raphs with oe edge raphs missig edge Regular graphs with degree Regular graphs of degree - 3 6

7 raph Coectedess λ λ... λ λ = { 0,,4,4} For coected graphs, λ > Recall ( i, j) ( ) T x L x = x x i j E If is eigevector for eigevalue 0 x x x 0 i j L x = = ( i, j) E λ v Fiedler Value Fiedler Vector Multiplicity of the 0 eigevalue idicates # of coected compoets 7

8 Oto raph Partitioig 8

9 raph Partitioig Remove as little of the graph as possible to separate out a subset of vertices of some desired size Size may mea the umber of vertices, umber of edges, etc. Typical case is to remove as few edges as possible to discoect the graph ito two parts of almost equal size Diagram from Berkeley CS 67 lecture otes Isoperimetric problem Oe of the earliest problems i geometry cosidered by the aciet reeks: Fid, amog all closed curves of a give legth, the oe which ecloses the maximum area Stei, 84 9

10 Applicatios Load balacig while miimizig commuicatio Sparse matrix-vector multiplicatio Optimizig VLSI layout Commuicatio etwork desig 0

11 Bisectio ad Ratio-Partitio Divide vertices ito two disjoit subsets ad S S Cut Size Cut Ratio E( S, S) ( S ) φ = (, ) E S S ( S S ) mi, Isoperimetric Number φ = miφ S V ( S ) (, ) Bisectio Miimize E S S subject to # of odes i each partitio differ by at most. Ratio-Partitio Miimize φ ( S ) NP-Complete

12 Spectral Partitioig Fid Fiedler vector of the Laplacia matrix map to vertices Choose some real umber s Partitio vertices give by V = i: v s { } Bisectio, s = media of v,... v Ratio partitio, s is chose to give the best cut ratio L L i { : } V = i v > s { } i

13 Example Fiedler vector [- - - ] 3

14 Spectral Partitioig For Plaar raphs uattery ad Miller Performace of Spectral raph Partitioig, 995 Spielma ad Teg, Spectral Partitioig Works o Plaar raphs, 996 Keler, Spectral Partitioig Works o raphs with Bouded eus, 004 4

15 Simple Spectral Bisectio May Fail (uattery & Miller) The simple spectral bisectio method produces cut size of Θ( ) for, for ay k k 5

16 Optimal Bisector for raphs with Bouded eus (Keler) eus g of a graph : smallest iteger such that ca be embedded o a surface of geus g without ay of its edges crossig oe aother. Eg. Plaar graphs have geus 0 Sphere, disc, ad aulus has geus 0 Torus has geus There is a spectral algorithm that produces bisector of size O( g) For every g, there is a class of bouded degree graphs that have O g o bisectors smaller tha ( ) 6

17 Improved Bisectio Algorithm o Plaar raphs (Spielma ad Teg) Bisector of size O( ) Why does the spectral method work? Why does it work well o plaar graphs? Why does simple bisectio fail eve o plaar graphs? 7

18 Aother Look at Fiedler Value T Recall x L x= x x where Rayleigh quotiet: ( ) x ( x, x,..., x ) i j E ( i, j) T x L x ( i, j) E φx = = T x x x = ( x ) i xj λ = Fiedler value satisfies mi φx x (,...,) with the miimum occurrig oly whe x is a Fiedler vector. φ T T x L x x λx x = = = T T x x x x λ i 8

19 Coectio Betwee Fiedler Value ad Isoperimetric Number Recall Isoperimetric Number is the best ratio-partitio possible Theorem (Mihail 89) Let be a graph o odes of maximum degree. For ay vector such that T x Lx T x x φ Moreover, there is a so that the cut ratio at most φ ( ) s φ (, ) E S S = mi S V mi, x φ ( S S ) i= x i = 0 ood ratio-partitio λ ca be achieved if { i v s} { i v > s} : i Fiedler value is small : i has 9

20 0 Upperboud o the Fiedler Value for Plaar raphs Theorem (Spielma & Teg 96) For all plaar graphs with vertices ad maximum degree 8 λ O 8 φ λ 4 φ O By boudig Fiedler value of plaar graphs, ratio-partitioig method is show to work well What about bisectio?

21 Relatioship Betwee Ratio-Partitioig ad Bisectio Lemma 3 ive a algorithm that will fid a cut ratio of at most φ ( k ) i every k-ode subgraph of, for some mootoically decreasig fuctio φ. The repeated applicatio of this algorithm ca be used to fid a bisectio of of size at most x= ( ) φ xdx φ ( x) = φ ( xdx ) = ( ) x x= O( ) Bisectio ca be obtaied by repeated applicatio of ratiopartitioig

22 Theorem φ x x... x (, ) E S S = mi S V mi, ( S S ) T x L x T x x T x Lx T x x Map graph vertices to a lie x x φ ( x ) i, j E i x j = = xi i= x i = 0 ( ) sum ( legth of edge) sum ( legth away from 0) x 4 3 If i At least φ i edges must cross over xi

23 Proof of Theorem λ 8 Theorem 4 (Koebe-Adreev-Thursto). Let be a plaar graph. The, there exist a set of disks { D,..., D} i the plae with disjoit iteriors such that Di touches D j iff ( i, j) E. Kissig disks 3

24 Proof of Theorem cot. Stereographic Projectio { π( D ),..., π( D )} Circles i the plae -> circular caps o the sphere 4

25 Proof of Theorem cot. Let x i be the ceter of π ( D i ) i x i i= x = = o the sphere. Let ri be the radius of the cap π ( ) D i ( ) ( ) x x r + r r + r ( ) i j i j i j π r i 4π ( ) xi xj ri rj diri i, j E i, j E i λ ( ) ( i, j) + 8 x ( i, j) E i xi x j 8 E 5

26 Coclusio Why does the spectral method work? - close relatioship betwee Fiedler value ad Isoperimetric umber λ φ Why does it work well o plaar graphs? - plaar graphs have ice collectio of spherical cap embeddigs λ 8 Why does simple bisectio fail eve o plaar graphs? - eve though good ratio-partitios ca be foud, the result may be ubalaced i the size of the partitios 6