Lecture 21 Nov 18, 2015

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1 CS 388R: Randoized Algoriths Fall 05 Prof. Eric Price Lecture Nov 8, 05 Scribe: Chad Voegele, Arun Sai Overview In the last class, we defined the ters cut sparsifier and spectral sparsifier and introduced Roan Vershynin (RV) Lea which will help us analyse Laplacians of rando graphs. In this lecture, we show how to produce spectral sparsifiers with O(n log n/ɛ ) edges where n is the nuber of nodes in the graph and ɛ is a easure of the quality of the sparsifier. Background Definition. For A and B syetric atrices, A B if x, x Ax x Bx Definition. raph Laplacian The Laplacian atrix of a weighted graph = (V, E, w), where w (u,v) is the weight of edge (u, v) is defined by { L (u, v) = w (u,v) z w (u,z) if u v if u = v The Laplacian can be expressed in ters of differences of standard basis vectors. L = w e (e u e v )(e u e v ) e=(u,v) = e=(u,v) w e u e u e where e i is the standard basis vector such that (e i ) j = δ ij and for edge e = (u, v), u e = e u e v. Definition 3. Spectral Sparsifier A graph H = (V, E, w ) is an ɛ spectral approxiation of a graph = (V, E, w) if ( ɛ)l L H ( + ɛ)l where L, L H are the Laplacians of graphs, H respectively. Note that x L x = e=(u,v) w e(x u x v ) is shift invariant. So in the analysis below, we restrict ourselves to x such that x = 0. Lea 4. Roan Vershynin Lea Let {X i } be i.i.d rando vectors in Rn, such that each X i is uniforly bounded X i κ, E[X i X i ] i []

2 Then E [ ] X i X log n i E[XX ] κ Last class, we proposed the following randoized algorith for coputing a spectral sparsifier. Algorith enerates spectral sparsifier Input: = (V, E, w). Output: H = (V, E, w ), a spectral sparsifier of : for ties do : Choose each edge e E with soe probability 3: Add edge e to E with w (e) = we In expectation, the Laplacian of the graph H output by the above algorith is equal to the Laplacian of. Let Y e = we u e and let {Z i } be independent rando variables where Z i = Y e with probability. Note that L H = Z iz i. [ ] E[L H ] = E Z i Z i = E[Z Z ] = e E Y e Y e = e E w e u e u e = L In the next section we discuss how to choose, a probability distribution over edges in, that gives us a good spectral sparsifier. 3 Spectral Sparsifiers We start with the siple case of coplete graphs, which have a spherical Laplacian, and ove to non-coplete graphs in Section 3.. To keep the analysis siple we only consider unweighted graphs. 3. Coplete raphs When is a coplete graph, the Laplacian L is given by: n n L =..... = ni. n

3 where is a vector of all s. Fro Definition 3, for H to be a spectral sparsifier, we need that ( ɛ)l L H ( + ɛ)l ( ɛ)x L x x L H x ( + ɛ)x L x x s.t. x = 0 x (L H L )x ɛx L x x s.t. x = 0 L H L ɛn where the last step follows fro the assuption that x = 0 and x L x = x (ni ) x = n x. We now show that when ( n log n ) and p ɛ e is unifor over edges, Algorith outputs an ɛ spectral approxiatior of. We have: = ( ( ) n = Θ ) n It is easy to see that the rando variables {Z i } are uniforly bounded: Z i = Y e (for soe edge e) we = u e u e = = Θ(n) Also, fro before we have that E [Z i Z i ] = L = n Applying RV Lea on rando variables X i = Z i n, we get Thus ( ) n log n E n L H L n log n E ( L H L ) n ( ) n log n ɛn if So for ( n log n ɛ ), we get an ɛ approxiate spectral sparsifier of. ɛ 3. Non-coplete raphs For non-coplete graphs, there are two issues that we need to deal with. 3

4 . The Laplacian of a non-coplete graph need not be spherical. So in order to apply RV lea, rather than looking at rando variables {Z i }, we look at transfored rando variables {AZ i } for soe atrix A.. Need to find a better sapling distribution,. For exaple, in the case of a barbell graph, we need to return the iddle edge to get a good sparsifier. Consider an unweighted graph. Let U R E n be atrix representing E edges where U = u u u E Then the Laplacian for can be represented as L = U U L is syetric since L = L. Also we have that x, x L x = x U Ux = Ux 0 and thus L is positive sei-definite. This iplies that all eigenvalues of L are non-negative. Using the eigenvalue decoposition of L, we can express L as L = n λ i b i b i where {b i } n are orthonoral eigenvectors and λ i 0. Positive powers of L can be calculated by L p = n λ p i b ib i The Moore-Penrose pseudo-inverse of L is given by L = n λ i 0 λ i b i b i and thus (L ) = n λ i 0 λ / i b i b i Using these, we can express the projector onto the span of L as Π L = b i b i = (L ) L = L (L ) λ i 0 4

5 Note that Π L Π L = Π L. For H to be a spectral sparsifier of, we need that x L H x = ( ± ɛ)x L x, x x L H x = ( ± ɛ)x L x, x s.t. x T = 0 x L H x = ( ± ɛ)x L x, x span(l ) where the last stateent holds when is a connected graph (because L has rank n for a connected graph and L = 0). Thus for any x span(l ): where y = L x. Then the condition becoes: x = Π L x = (L ) L x = (L ) y x L H x = ( ± ɛ)x L x, x span(l ) y L L HL y = ( ± ɛ)y L L L y L = ( ± ɛ)y L L L = ( ± ɛ)y Π L Π L y = ( ± ɛ)y Π L y, y y Subtracting y Π L y fro both sides, we get y (L L HL Π L )y ɛy Π L y = ɛy y L L HL Π L ɛ We now apply RV Lea on rando variables A i = L Z i. Let κ = ax A i and we have: Applying RV Lea we get: E[A i A i ] = L E[Z iz i ]L = L L L = Π L Ai A i E[A ia i ] = L L HL Π L κ log n So if (κ log n/ɛ ), we get a ɛ approxiate sparsifier. Note that we haven t yet defined the probability distribution. κ will depend on the choice of. To pick a good probability distribution and to copute κ, we appeal to physical intution. Consider the graph to represent nodes on a circuit and let x R n denote the voltages on each 5

6 node. The current flow along edge e, denoted by I e, fro u to v is related to the voltage drop. Thus I e = x u x v = u e x. The flow along all edges is given by I = Ux where I R E. iven a battery on the circuit, with x s and x t fixed at soe voltages, we can calculate the rest of the internal voltages x v using Kirchoff s Laws: current into vertex current out of vertex = external flow We know that the external flow is: ( at s, at t, 0 elsewhere ) For node v, (I ext ) v = I e e=(u,v) e=(v,u) I e = e I e (U e ) v = (U I) v = (U Ux) v = (L x) v We know the external flow and want voltages so we copute x = L I ext where L is the pseudoinverse. If we set I ext to u e for soe edge (u, v) to indicate that unit of current is pushed fro u to v, then L u e is a vector of all voltages in the circuit and thus u el u e is the voltage drop fro u to v. Fro Oh s Law, we know that V = IR eff where R eff is the effective resistance. Since we have unit of current, we conclude that R eff = u el u e. We use this fact in our calculation of A i. A i = A i A i = Z i L L Z i = Z i L Z i = u el u e = R eff This suggests to set R eff and after noralizing = R eff (e) Reff (e) Thus κ = R eff (e) and we need (( R eff (e)) log n/ɛ ) to get a ɛ approxiate sparsifier. And finally to copute R eff (e), we use Foster s Theore. 6

7 Theore 5. Foster s Theore Let R eff (e) denote the effective resistance along edge e on a connected graph of n nodes. Then R eff (e) = n e E Proof. Define P = UL U. Then P = UL L L U = P Thus P is a projection atrix, and all its eigenvalues λ i {0, }. Since L has rank n and P has the sae rank as L, n eigenvalues of P are equal to and the rest are 0. Fro the definition of effective resistance we have: R eff (e) = u el u e = P e,e R eff (e) = tr(p ) = λ i = n Finally, we conclude that we need (n log n/ɛ ) and coplete the proof for non-coplete graphs. References [] R. Motwani and P. Raghavan. Randoized Algoriths. Cabridge University Press, 995. [] D. A. Spielan and S. Teng. Spectral sparsification of graphs. CoRR, abs/ , 008. [3] R. Vershynin. Introduction to the non-asyptotic analysis of rando atrices, 00. [4] E. W. Weisstein. Foster s theores. htl. 7

Quantum algorithms (CO 781, Winter 2008) Prof. Andrew Childs, University of Waterloo LECTURE 15: Unstructured search and spatial search

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