Luby s Alg. for Maximal Independent Sets using Pairwise Independence


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1 Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, Maxmal Independent Sets For a graph G = (V, E), an ndependent set s a set S V whch contans no edges of G,.e., for all (u, v) E ether u S and/or v S. The ndependent set S s a maxmal ndependent set f for all v V, ether v S or N(v) S where N(v) denotes the neghbors of v. It s easy to fnd a maxmal ndependent set. For example, the followng algorthm works:. I =, V = V.. Whle (V ) do (a) Choose any v V. (b) Set I = I v. (c) Set V = V \ (v N(v)). 3. Output I. Our focus s fndng an ndependent set usng a parallel algorthm. The dea s that n every round we fnd a set S whch s an ndependent set. Then we add S to our current ndependent set I, and we remove S N(S) from the current graph V. If S N(S) s a constant fracton of V, then we wll only need O(log V ) rounds. We wll nstead ensure that by removng S N(S) from the graph, we remove a constant fracton of the edges. To choose S n parallel, each vertex v ndependently adds themselves to S wth a well chosen probablty p(v). We want to avod addng adjacent vertces to S. Hence, we wll prefer to add low degree vertces. But, f for some edge (u, v), both endponts were added to S, then we keep the hgher degree vertex. Here s the algorthm: The Algorthm Problem : Gven a graph fnd a maxmal ndependent set.. I =, G = G.. Whle (G s not the empty graph) do (a) Choose a random set of vertces S G by selectng each vertex v ndependently wth probablty /(d(v)). 8
2 8 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence (b) For every edge (u, v) E(G ) f both endponts are n S then remove the vertex of lower degree from S (Break tes arbtrarly). Call ths new set S. (c) I = I S. G = G \ (S N(S )),.e., G s the nduced subgraph on V \ (S N(S )) where V s the prevous vertex set. 3. output I Fg : The algorthm Correctness : We see that at each stage the set S that s added s an ndependent set. Moreover snce we remove, at each stage, S N(S ) the set I remans an ndependent set. Also note that all the vertces removed from G at a partcular stage are ether vertces n I or neghbours of some vertex n I. So the algorthm always outputs a maxmal ndependent set. We also note that t can be easly parallelzed on a CREW PRAM. 8. Expected Runnng Tme In ths secton we bound the expected runnng tme of the algorthm and n the next secton we derandomze t. Let G j = (V j, E j ) denote the graph after stage j. Man Lemma : For some c <, E ( E j E j ) < c E j. Hence, n expectaton, only O(log m) rounds wll be requred, where m = E 0. We say vertex v s BAD f more than /3 of the neghbors of v are of hgher degree than v. We say an edge s BAD f both of ts endponts are bad; otherwse the edge s GOOD. The key clams are that at least half the edges are GOOD, and each GOOD edge s deleted wth a constant probablty. The man lemma then follows mmedately. Lemma 8. At least half the edges are GOOD. Proof: Denote the set of bad edges by E B. We wll defne f : E B ( E ) so that for all e e E B, f(e ) f(e ) =. Ths proves E B E /, and we re done. The functon f s defned as follows. For each (u, v) E, drect t to the hgher degree vertex. Break tes as n the algorthm. Now, suppose (u, v) E B, and s drected towards v. Snce v s BAD, t has at least twce as many edges out as n. Hence we can par two edges out of v wth every edge nto v. Ths gves our mappng. Lemma 8. If v s GOOD then Pr ( N(v) S ) α, where α := ( e /6 ). Proof: Defne L(v) := {w N(v) d(w) d(v)}.
3 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence 83 By defnton, L(v) d(v) 3 f v s a GOOD vertex. Pr ( N(v) S ) = Pr ( N(v) S = ) = Pr ( w S ) w N(v) w L(v) = w L(v) w L(v) Pr ( w S ) ( ) d(w) ( ) d(v) exp( L(v) /d(v)) exp( /6), usng full ndependence Note, the above lemma s usng full ndependence n ts proof. Lemma 8.3 Pr ( w / S w S ) /. Proof: Let H(w) = N(w) \ L(w) = {z N(w) : d(z) > d(w)}. Pr ( w / S w S ) = Pr ( H(w) S w S ) Pr ( z S w S ) = = = Pr ( z S, w S ) Pr ( w S ) Pr ( z S ) Pr ( w S ) Pr ( w S ) Pr ( z S ) d(z) d(v) usng parwse ndependence. Lemma 8.4 If v s GOOD then Pr ( v N(S ) ) α
4 84 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Proof: Let V G denote the GOOD vertces. We have Pr ( v N(S ) v V G ) = Pr ( N(v) S v V G ) = Pr ( N(v) S N(v) S, v V G ) Pr ( N(v) S v V G ) Pr ( w S w N(v) S, v V G ) Pr ( N(v) S v V G ) (/)(α) = α Corollary 8.5 If v s GOOD then the probablty that v gets deleted s at least α. Corollary 8.6 If an edge e s GOOD then the probablty that t gets deleted s at least α. Proof: Pr ( e = (u, v) E j \ E j ) Pr ( v gets deleted ). We now restate the man lemma : Man Lemma : E ( E j E j ) E j ( α/). Proof: E ( E j E j ) = Pr ( e gets deleted ) e E j E j α GOOD edges E j ( α/). The constant α s approxmately Thus, E ( E j ) E 0 ( α )j m exp( jα/) <, for j > α log m. Therefore, the expected number of rounds requred s 4m = O(log m). 8.3 Derandomzng MIS The only step where we use full ndependence s n Lemma 8. for lower boundng the probablty that a GOOD vertex gets pcked.the argument we used was essentally the followng : Lemma 8.7 Let X, n be {0, } random varables and p := Pr ( X = ). ndependent then ( n ) n Pr X > 0 ( p ) If the X are fully
5 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence 85 Here s the correspondng bound f the varables are parwse ndependent Lemma 8.8 Let X, n be {0, } random varables and p := Pr ( X = ). If the X are parwse ndependent then ( n ) Pr X > 0 mn{, p } Proof: Suppose p /. Then we have the followng, (the condton p / wll only come nto some algebra at the end) ( ) Pr X > 0 Pr ( X = ) Pr ( X =, X j = ) j = p p p j j ( ) p p = ( ) p p p when p. If p > /, then we restrct our ndex of summaton to a set S [n] such that / S p, and the same basc argument works. Note, f p >, there always must exst a subset S [n] where / S p Usng the constructon descrbed earler we can now derandomze to get an algorthm that runs n O(mn/α) tme (ths s asymptotcally as good as the sequental algorthm). The advantage of ths method s that t can be easly parallelzed to gve an NC algorthm (usng O(m) processors). 8.4 Hstory and Open Questons The k wse ndependence derandomzaton approach was developed n [CG], [ABI] and [L85]. The maxmal ndependence problem was frst shown to be n NC n [KW].They showed that MIS n s NC 4. Subsequently, mprovements on ths were found by [ABI] and [L85]. The verson descrbed n these notes s the latter. The queston of whether MIS s n NC s stll open. References [] N Alon, L. Baba, A. Ita, A fast and smple randomzed parallel algorthm for the Maxmal Independent Set Problem, Journal of Algorthms, Vol. 7, 986, pp [] B. Chor, O. Goldrech, On the power of twopont samplng, Journal of Complexty, Vol. 5, 989, pp
6 86 Lecture 8: Luby s Alg. for Maxmal Independent Sets usng Parwse Independence [3] R.M. Karp, A. Wgderson, A fast parallel algorthm for the maxmal ndependent set problem, Proc. 6th ACM Symposum on Theory of Computng, 984, pp [4] M. Luby, A smple parallel algorthm for the maxmal ndependent set problem, Proc, 7th ACM Symposum on Theory of Computng, 985, pp. 0.
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