9 Graph Cutting Procedures

Transcription

1 9 Grph Cutting Procedures Lst clss we begn looking t how to embed rbitrry metrics into distributions of trees, nd proved the following theorem due to Brtl (1996): Theorem 9.1 (Brtl (1996)) Given metric (X, d) with dimeter, let DT be the set of ll tree metrics tht dominte d. Then, O(log n log ) (X, d) distr(dt ). In the lst clss, we proved Theorem 9.1 ssuming the following theorem on grph decompositions: Theorem 9.2 Given grph G = (V, E) with edge lengths, nd prmeter, there exists procedure tht deletes edges E such tht: 1. Ech connected component C in (V, E E ) hs (wek) dimeter smller thn. 2. P r[edge e is cut] 8 log n (d(e)/). In this clss, we will describe two rndomized lgorithms tht ccomplish the required grph decomposition, which will implicitly define the distribution of trees tht re used in the embedding. Let us mke two ssumptions bout the edge lengths d(e): 1. The edge lengths re integers; i.e., d(e) Z +. This ssumption cn be dischrged by tking rtionl pproximtions nd scling. Hence, the smllest edge length is We cn ssume for the bove theorem tht d(e) 8. Indeed, if edge e hs d(e) > 8, then we cn just delete the edge. Since 8d(e) 1, the quntity 8 log n (d(e)/) 1 for n 2, nd hence the theorem still holds. This susmption will be convenient lter in the clss. 9.1 Brtls Cutting Scheme The first grph cutting procedure is essentilly due to Brtl (1996). The lgorithm is simple: it picks n rbitrry vertex, nd constructs component with sufficiently smll dimeter by cutting ll edges t certin (rndom) distnce wy from the chosen vertex. The distnce is essentilly chosen from distribution tht flls off exponentilly with distnce from the vertex. It then removes the component from the grph, nd itertes on the remining grph until the remining grph hs smll dimeter. We formlize this s follows: Algorithm Cut-1(G, ) let G = G. while G do Pick n rbitrry vertex r G. Construct shortest pth tree in G from r. V-1

2 1 2 r L R 1 R Figure 9.1: Illustrting the procedure Cut-1. Let R be drwn from the geometric distribution with prmeter p = 4 log n. Cut ll edges in G crossing [R 1, R] in the shortest pth tree. (I.e., edges hving one endpoint t distnce < R nd the other t distnce R from r.) Let L be the resulting component tht contins the vertex r. let G G \ L. end Recll tht smpling from geometric distribution with prmeter p is the sme s flipping coin with heds probbility p, nd returning the number of flips until we see heds. Hence, we cn imgine the process s setting counter R 1, nd deciding the fte of the edges crossing from R 1 to R by flipping coin with heds probbility p: if the coin turns up tils, then it increments R (nd hence sves the edges crossing from R 1 to R). If the coins turns up heds, it cuts those edges, removes the component thus creted, nd continues on the remining grph. The procedure is illustrted in Figure 9.1. Let r be the vertex chosen s the root from which we construct shortest-pth tree, nd R be the (rndom) distnce t which the procedure will cut the grph. The edges ( 1, 2 ), ( 3, 4 ), ( 5, 4 ) nd ( 5, 6 ) (mong others) re cut, nd L is the connected component tht is removed from the grph. Now we verify tht the procedure Cut-1 stisfies the two conditions required by Theorem 9.2. Fct 9.3 In the procedure Cut-1, for ny edge e with weight d(e) Pr[edge e is cut] 4 log n (d(e)/). Proof. For this proof, it is more instructive to consider cutting or sving view of the lgorithm: the chnce tht e is cut is precisely the chnce tht we get heds for some coin flip when the edge e is crossing the current intervl [R 1, R]. Since there re t most d(e) V-2

3 coin flips, nd ech one cn come up heds with probbility p, we get tht Pr[edge e is cut] = 1 (1 p) d(e) 1 (1 pd(e)) = 4 log n d(e). Lemm 9.4 The probbility tht some component C creted in the lgorithm hs dimeter dim(c) > is t most 1 n. Proof. Consider ny fixed component L: note tht if the dimeter of the component L exceeds, then R must hve been > /2. Hence, Pr[dim(L) > ] P r[r > /2] = (1 p) 2 exp{ p log n } = exp{ } = 1 n 2. Since there re t most n components creted by the lgorithm, the trivil bound implies the lemm. Therefore, the procedure Cut-1 stisfies the conditions of Theorem 9.2 with probbility t lest (1 1 n ). To stisfy Theorem 9.2, we cn mplify the probbility of success in one of two wys: we cn either run the lgorithm on ny lrge components L (i.e., those with dim(l) > ) remining until ll components re smll enough; or, we cn truncte the distribution of R t 2. Let us exmine the trunction ide more closely. Suppose we truncte the distribution s follows: we follow the regulr cutting procedure bove; if we pick vlue of R so tht R > 2, we simply set R to be 2. Let the cuts t distnce /2 be sid to be specil cuts. Since specil cut is chosen with probbility 1/n 2 (due to the nlysis in Lemm 9.4), nd there re t most n such cuts Pr[edge e is cut due to specil cut] 1 n. If e.g. 1 n 2 d(e), then the clims of Theorem 9.2 re stisfied (lbeit with Pr[edge e is cut] 4 log n+2 being d(e)). If not, though, the probbility of cutting n edge is too high. To fix this, we do the following: Algorithm Preprocessing 1. Contrct ll edges e E with d(e) < 2n. 2. Run the procedure Cut-1 on the resulting grph with ˆ = Expnd ll the edges tht were contrcted in Step 1. This procedure will llow us to hve components no lrger thn in the originl grph, since expnding the contrcted edges in Step 3 will increse the dimeter by t most 2n n = 2, nd the cutting procedure cretes grphs of dimeter ˆ = 2. (This, however, comes t slightly 8 log n+2 higher price, since the bound on P r[edge e is cut] hs now incresed to ( d(e), which is slightly higher thn promised). V-3

4 σ 4 σ 2 σ 3 σ The FRT-Cutting Scheme Figure 9.2: Illustrting the construction of ˆB i. While the procedure given bove cn be tightened further, we will move on to nother prtitioning procedure. This procedure first ppered in pper of Clinescu, Krloff nd Rbni (2001), nd ws lter used by Fkchroenphol, Ro nd Tlwr (2003) to chieve the O(log n) bound on the distortion for embedding rbitrry grphs into distributions of trees. Algorithm Cut-2(G, ) Pick rdius R uniformly t rndom from [/4, /2]. Pick rndom permuttion σ S n, which defines n order < σ on the vertices. For every vertex v i, define bll B i = B(v i, R). Assign ech vertex to the first bll it lies in. Formlly, define ˆB i = B i j< σi B j. Delete ll edges in the cut (B i, V \ B i ) for ll i. An exmple of this procedure is given in Figure 9.2. Blls of rdius R re drwn round the first four vertices in permuttion σ, which re denoted by σ 1, σ 2, σ 3, nd σ 4 respectively. The dotted lines indicte the blls B i ; ˆB1 is the unshded region, while ˆB 2, ˆB 3 nd ˆB 4 re the shded regions from lighest to drkest. Note tht B 1 B 2 (nd thus vertex σ 2 ) belongs to ˆB 1, nd B 2 B 4 belongs to ˆB 2, nd so on. We now show tht this procedure lso results in prtition tht stisfies the two conditions of the theorem. The first condition holds by construction the wek dimeter of ech L i is no more thn the dimeter of ˆB i. But the wek dimeter of ˆB i is no more thn tht of B i ; since the rdius of ech B i is no more thn 2, the dimeter is no more thn. We will prove something slightly stronger thn the second condition: Clim 9.5 Given vertex v nd rdius ρ, the probbility tht the procedure Cut-2 cuts the bll B(v, ρ) is t most 8 log n (ρ/). As cn be expected, set S is cut by the prtitioning procedure if there re two components C 1 nd C 2 in the prtition such tht vertices from S lie in both these components. V-4

5 It is esy to see tht Clim 9.5 implies the second condition in Theorem 9.2: given n edge e = {u, v}, consider the bll of rdius d(e) round u. Any prtition tht cuts the edge e lso cuts the bll B(u, d(e)), nd hence Clim 9.5 gives us the desired condition. Let us now prove Clim 9.5. Since the nmes of the vertices do not mtter in the lgorithm, let us ssume tht they re numbered in order of their distnce from u, i.e., d(u, v 1 ) d(u, v 2 )... d(u, v n ). Now for every vertex v i in the grph, let us consider the following events. We sy node v i intersects the Bll B(u, ρ) if the rndom rdius R is chosen such tht R [d(v i, u) ρ, d(v i, u) + ρ] (this mens it cn possibly hppen tht nodes from B(u, ρ) lie inside ˆ(B) i nd outside of ˆ(B) i ). We sy tht node protects the bll if R > d(v i, u) + ρ, becuse then the bll is contined in the cluster generted by v i nd cnnot be cut nymore by nodes tht come fter v i in the permuttion σ. Finlly, we sy node v i cuts the bll first if v i intersects B(u, ρ) nd if no node prior to v i in the permuttion σ intersects or protects the bll. We cn mke the following observtion. Observtion 9.6 If the bll B(u, ρ) is cut then there must exist node v i tht cuts it first with respect to the definition bove. This mens Pr[B(u, ρ) is cut] i Pr[v i cuts B(u, ρ) first], nd it suffices to bound the probbility tht node cuts the bll first. Note tht for the i-th node to cut bll first, it must hppen tht R flls into the right rnge (i.e., R [d(v i, u) ρ, d(v i, u) + ρ]) nd further tht ll nodes v j, j < i come fter v i in the permution σ, s ll these nodes would either cut the bll or protect it (recll tht we ordered the nodes ccording to the distnce from u). The event thtt v i precedes v j, j < i in σ holdes with probbility 1/i nd is independent from the choice of the rdius R. Altogether we get By summing over ll i we get Pr[v i cuts B(u, ρ) first] Pr[R [d(v i, u) ρ, d(v i, u) + ρ]] 1 i = 2ρ /4 1 i = 1 i 8ρ. Pr[B(u, ρ) is cut] n Pr[v i cuts B(u, ρ) first] i=1 n i 1 i 8ρ = 8ρ H n, (9.1) where H n = n = O(log n) denotes the n-th hromnic number. Thus, the rndom prtitions generted by procedure Cut-2 lso meet the second condition of Theorem 9.2, nd this proves the theorem. V-5

6 9.2.1 Improved Embedding into Rndom Trees We cn slightly improve Eqution 9.1 by observing tht node tht is closer to u thn /4 ρ or frther thn /2 + ρ cnnot cut the bll B(u, ρ) t ll. Furthermore, we cn ssume ρ /8 s otherwise Clim 9.5 trivilly holds. This mens we cn restrict the sum in Eqution 9.1 to nodes v i tht re inside the 5 8 -bll round u, but outside the 1 8-bll. This gives Pr[B(u, ρ) is cut] B(u,) i= B(u,/8) Pr[v i cuts B(u, ρ) first] 8ρ ( ( B(u, ) )) O log B(u, /8) This improved bound on the probbility of cutting n edge cn be used to get n improved distortion for embedding into distributions over dominting trees. Recll tht in the lst lecture we showed tht the expected length of n edge e = (x, y) in the rndom tree when using the recursive prtitioning pproch is E[d T (x, y)] = 4 Pr[x nd y re cut in level 0] + 2 Pr[x nd y re cut in level 1 they re in the sme level 1 grph] j Pr[x nd y re cut in level j they re in sme level j grph ] , where the sum hs log terms. Now, using the prtitioning scheme with = /2 j+1 in level j gives Pr[x, y cut in level j they re in sme level j grph] 8 1 ( ( B(u, /2 j+1 /2 j+1 O ) )) log B(u, /2 j 2 ) nd we cn bound the sum by E[d T (x, y)] log j=1 ( 64 O log ( B(u, /2 j+1 ) )) B(u, /2 j 2 O(log n), ) since the sum telescopes. This gives n embedding into distribution over dominting trees with distortion O(log n). References [Br96] Yir Brtl. Probbilistic pproximtions of metric spces nd its lgorithmic pplictions. In Proceedings of the 37th IEEE Symposium on Foundtions of Computer Science (FOCS), pges , , V-6

7 [CKR01] Grui Călinescu, Howrd Krloff, nd Yuvl Rbni. Approximtion lgorithms for the 0-extension problem. In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA), pges 8 16, [FRT03] Jittt Fkchroenphol, Stish B. Ro, nd Kunl Tlwr. A tight bound on pproximting rbitrry metrics by tree metrics. In Proceedings of the 35th ACM Symposium on Theory of Computing (STOC), pges , V-7