Analysis of Random. Processes via And-Or Tree Evaluation. Michael Mitzenmachert M. Amin Shokrollahiz

Transcription

1 Anaysis of Random Processes via And-Or Tree Evauation Michae G. Luby* Michae Mitzenmachert M. Amin Shokroahiz Abstract distributions not considered by previous anayses. We introduce a new set of probabiistic anaysis toos Our main too is a simpe anaysis of the probabiity based on the anaysis of And-Or trees with random in- an And-Or tree formua evauates to 1. The simpe puts. These toos provide a unifying, intuitive, and pow- version of this And-Or tree evauation probem is the erfu framework for carrying out the anaysis of severa foowing. Let TL be a tree of depth 2e with each eaf previousy studied random processes, incuding random node abeed with either 0 or 1. (The root of the tree is oss-resiient codes, soving random L-SAT formuae using the pure itera rue, the greedy agorithm for matchat depth 0, and the eaves are at depth 2f!.) Each node at depth 0,2,4,..., 2t! - 2 is abeed as an OR gate ings in random graphs. In addition, these toos aow (and it evauates to the OR of its chidren), and each generaizations of these probems not previousy ana- node at depth 1,3,5,..., 21-1 is abeed ss an AND yzed to be anayzed in a straightforward manner. We gate (and it evauates to the AND of its chidren). We iustrate our methodoogy on the three probems isted say the tree is (do&&-reguar if each OR node has above. do, chidren and each AND node has d,d chidren. Our anaysis is reated to the study of ampification, 1 Introduction initiated by Moore and Shannon [15], and continued in We introduce a new set of probabiistic anaysis toos reated to the ampification method introduced by [15] and further deveoped and used in [17, 61. These toos provide a unifying, intuitive, and powerfu framework for carrying out the anaysis of severa previousy studied random processes, incuding the random ossresiient codes introduced in [], the greedy agorithm for matchings in random graphs studied in [9], the threshod for soving random k-sat formuae using the pure itera rue [5], the emergence of a giant k-core [16], and error-correcting codes introduced in [8, 121. In addition, generaizations of these probems not previousy anayzed can now be anayzed in a straightforward man- ner. For exampe, we can anayze generaizations of the oss-resiient codes considered in [] where the goa is to recover a certain fraction of the message packets. As another exampe, we can anayze the behavior of the pure itera rue on random SAT formuae chosen from *Internationa Computer Science Institute, Berkeey, CA. Parts of this research were done whie sti at the Digita Equipment Corporation Systems Research Center, Pao Ato, CA. Research partiay supported by NSF operating grant NCR tdigita1 Equipment Corporation, Systems Research Center, Pao Ato, CA. tinternationa Computer Science Institute Berkeey, and Institut fiir Informatik der Universitgt Bonn, Germany. Research sup ported by a Habiitationsstipendium of the Deutsche Forschungsgemeinscbaft, Grant Sh 57/1-. severa works [17, 61. Consider the probabiity that the root of T evauates to 0 when the vaue of each eaf is independenty chosen to be 0 with probabiity p. Let us denote this probabiity aa y. One typica ampification question with respect to And-Or trees is whether or not there is a threshod phenomenon, i.e., is there a critica vaue 4 such that if p > 4 then ye goes to 1 as.fy goes to infinity and if p < 4 then as y goes to 0 as 1 goes to infinity. Of primary interest in these studies is the rate of ampification, i.e., the rate at which y goes to either 0 or 1 as a function of e. One work that uses exacty this type of anaysis is the eegant randomized construction, given in [17], of a poynomia size monotone booean formua that computes the majority function. The basic idea behind the construction and proof expoits the fact that a (2,2)-reguar And-O r t ree has a critica vaue of 4 = (3 - &)/2, and that if ye-1 = 4 + c then y > 4 + cc for a constant c > 1. (Anaogousy, if y-i = 4 - e, then y < 4 - cc.) In further work, [4] and [S] provide beautifu proofs that the construction size of [17] is 364 optima, this time using ampification anaysis to prove a ower bound. This type of anaysis has aso been noted se a possibe attack for specific random graph probems, incuding the greedy matching agorithm of [9] and the emergence of the k-core [16]. Our work has a simiar spirit to this previous work, but it differs in severa ways. We generaize to aow the

2 number of chidren of each node to vary in the foowing the random process and that of the poynomia. One way. Let (ao,q,..., a~) be a probabiity vector, i.e., of the ingredients acking in the previous anaysis of ai > 0 for a i E (0,...,A} and ct=,ai = 1. these codes was a simpe intuitive connection between Simiary, et (PO,&,...,/3~) be a probabiity vector. the poynomia soution and the origina process. With Starting at the root and working down the tree, each the new anaysis, this intuitive connection is direct and OR node chooses to have i chidren with probabiity compeing. In addition, the new toos can be easiy (pi independent of any other node, and simiary each used to anayze important generaizations of the origi- AND node choose to have j chidren with probabiity na process, which woud have been much more difficut pj independent of any other node. (Some previous using the previous anaysis. Finay, the simpicity and research, e.g., [6], introduced a variant of this form generaity of the new anaysis wi undoubtaby ead to and used it in a imited way in their construction.) A a number of other appications. We do note, however, further generaization is to aow two positive vaues a that the differentia equations approach can ead to a and b such that each OR? node is independenty short much more sophisticated anaysis of the underying procircuited to produce the vaue 1 with probabiity a, and cesses (see, for exampe, [16] or [3]). Hence we beieve each AND node is independenty short circuited to our anaysis toos wi prove most usefu in conjunction produce the vaue 0 with probabiity b. with differentia equations. Our new anaysis is based on the foowing simpe In the next three sections, we appy Lemma 1 to definitions and emma. Define the anaysis of oss-resiient codes, the pure itera rue for random B-CNF formua, and the greedy matching agorithm for random graphs. i=o p(z) = &j * xi, i=o f(x) = (1 -a).a(1 - (1 -b)./3(1-z)). LEMMA 1. Define yo = p to be the probabiity node is abeed 0. Then for a e 2 1, yf = f(y-1). a eaf Athough this emmais simpe to prove (in fact,, we eave it as an exercise), as we sha see it is quite powerfu. The most significant difference between our work and previous work on ampification is our goas. For exampe, for the oss-resiient codes, our fina goa is to design ((~0,..., (YA) and (PO,..., /3~) so that the average number of chidren per node is not too arge, the average number of chidren for an OR gate and for an AND gate satisfies a certain ratio, and for as sma as possibe a vaue.of a and as arge as possibe a vaue of b the critica vaue of the tree is as cose to 1 as possibe. Some of the anayses we present in the paper were previousy done using different methodoogies. One common technique invoved modeing the random process using differentia equations. This type of approach was pioneered in the anaysis of agorithms domain in by Karp and Sipser, who used it to anayze a greedy agorithm for matchings in [9]. It has aso been used to anayze the pure itera on random k-sat formuae [14]. (See aso [13, 14, 161 f or references to other uses.) Simiary, the anaysis of the oss-resiient codes described in [] was done by modeing the random process using differentia equations, soving the equations to obtain a poynomia, and using a version of Kurtz s theorem [o] to make the connection between the behavior of 2 Loss-Resiient Code Anaysis 2.1 Essentias of the Codes 365 The codes described in [] consist of a cascading sequence of random bipartite graphs. Because the code requires the same properties from a of these bipartite graphs, it is enough to consider one generic bipartite graph in the sequence when describing the encoding and decoding process and its anaysis. Let G be a bipartite graph with n nodes on the eft side, m nodes on the right side, and e edges in tota between the nodes on the eft and the right. We associate one message bit with each eft node and one check bit with each right node. (This is for simpicity of description; in practice it is more efficient to associate severa bytes of information with each node.) The encoding process computes the check bits from the message bits in the obvious way: the check bit associated with right node w is computed as the excusive-or of a the message bits associated with the neighbors of w. The entire encoding is transmitted, and we woud ike to recover a the message bits from a random fraction of the entire encoding, where this fraction is as sma as possibe. Assume inductivey that a the check bits associated with the right nodes have aready been recovered. Labe the eft nodes with a 0 if the associated message bit is missing, and with a 1 if the associated message bit has been either received directy or recovered indirecty as described beow. The decoding process to recover the missing message bits invokes the foowing rue as ong as it is appicabe. Substitution Recovery Rue: The rue can be ap-

3 366 pied at any eft node v with abe 0 that has at east one right neighbor w such that a the eft neighbors of w excuding v are abeed with a 1. The vaue of v can be recovered by computing the excusive-or of the check bit associated with w and a the vaues associated with neighbors of w excuding v. Since the message bit associated with v has been recovered, the abe of v is changed to 1 at this point. In terms of a graph process, the substitution recovery rue can be written more succincty as foows: Graph Substitution Recovery Rue: A eft node v with abe 0 is aowed to change its abe to a 1 if it has at east one right neighbor w such that a eft neighbors of w except v have abe 1. The decoding process terminates successfuy with a message bits recovered if and ony if the graph substitution recovery rue ends with no remaining eft nodes with abe New Anaysis of the Origina Process The paper [] gave an anaysis of the decoding process described in the previous subsection using differentia equations to mode the process, and then soving these equations as poynomias. In this subsection, we obtain the same resut using Lemma 1. The advantage of the anaysis here is that it gives direct and intuitive insight into how the fina condition arises. In the foowing subsections we show how this new anaysis can be used to derive severa additiona resuts. Let (PO, PI,...,p~) and (qo, q,..., QR) be probabiity vectors. As in [], consider choosing a random bipartite graph with n eft nodes and m right nodes as foows: each node on the eft is chosen to have degree i with probabiity pi, and each node on the right is chosen to have degree j with probabiity pj, where a choices are made independenty. Counting the number e of edges using the eft and the right nodes gives e=n. L ipj = m. c c 5.i. i=o j=o A random permutation?r of { 1,..., e} is chosen, and then, for a i E (1,..., e}, the edge with index i out of the eft side is identified with the edge with index pi out of the right side. For fixed probabiity vectors (~0, p,..., pi) and (QO,Q,..., QR) and for a fixed constant c > 0, we are interested in properties of such a graph as n and m = cn grow to infinity. Consider the random subgraph GL of this graph obtained by the foowing process: choose an edge (v, w) uniformy at random from among a edges, and then R consider the subgraph Gt induced by the eft node v and a neighbors of v within distance 24 after deeting the edge (v, w). We caim that the probabiity that GL fais to be a tree is proportiona to /n., i.e., asymptoticay this probabiity goes to zero as n grows to infinity for a fixed vaue of fj. Furthermore, asymptoticay the distribution on the shape of GL can be described as foows. For a i = 1,..., L, Xi := ipi/ Efz jpj is the probabiity that a uniformy chosen edge is attached to a eft node of degree i. Simiary, for a i = 1,..., R, Pi = Gi ET=1 j qj is the probabiity that a uniformy chosen edge is attached to a right node of degree i. The asymptotic distribution on the shape of G as n goes to infinity is as described above for a randomy chosen And-Or tree with the foowing parameters: the number of chidren of an OR node is i - 1 with probabiity &, for a i = 1,..., L; the number of chidren of an AND node is i - 1 with probabiity pi, forai=,..., R. Hereafter, we make the assumption that G is a tree and that the distribution on G is the asymptotic distribution. This assumption can be shown to change the anaysis by asymptoticay sma quantities, and these changes can be deat with using an appropriate martingae argument, see e.g., Section 2.3. Consider a process where at the start each eft node in the graph is abeed with 0 initiay with probabiity S, and is abeed with 1 with probabiity 1 - S. This corresponds to missing a random fraction 6 of the message bits. The goa is to eiminate as many as possibe 0 abes according to the graph substitution recovery rue described in the previous subsection, i.e., to recover as many of the missing message bits as possibe using the simpe decoding process. Let us anayze the probabiity yf that the eft node v of a uniformy chosen edge (v, w) is abeed with 0 considering the process running ony on the subgraph G induced by v. (It is cear that v wi definitey change its abe to 1 in the process running on the entire graph if it does so with process running just on GL.) Note that v obtains the abe 1 with respect to G1 if it is either received directy (with probabiity -6)) or if for at east one of its right neighbors w (w # w ), a eft neighbors of w excuding v are received directy. Note that v has i - 1 right chidren excuding w with probabiity Xi, and that for any chid w of v, WI has i - 1 eft chidren excuding v with probabiity pi. Define the poynomias X(Z) = &Xi. &, and i= p(z) = ep.2-. i=

4 Then from Lemma 1, y = 6. X( 1 - p( 1 - y-i)). Using this equation, the probabiity that u has abe 0 with respect to G can be computed iterativey starting with the equation ys = 6. We woud ike that y goes to 0 as e grows. This condition wi be true if (2.1) 6 * X(1 - p( - 2)) < 2 for a z E (0, S]. (B ecaue X and p are continuous and the yt are decreasing, the imit of the ye is easiy shown to be 0 if this condition hods.) This turns out to equivaent to the condition given in [] for this process to end successfuy. 2.3 The Overa Anaysis It is not hard to argue that the process terminates with a message vaues successfuy recovered with high probabiity if the probabiity that a message bit is not directy received is S and if Condition (2.1) is fufied. However, the detais are somewhat tedious and thus we ony sketch the proof here. Suppose that X, p, and 6 satisfy Condition (2.1). Then for any constant 7 > 0 we can set L to a constant so that yf < 7. If 1 is a constant then the number of nodes in the graph G is aso a constant. We can use this fact and standard martingae arguments to show that the true fraction of bits not recovered after e rounds is highy concentrated around yf. We first utiize an edge exposure martingae to show that the number of trees of each shape is cose to its expectation with high probabiity. With this fact, a vertex exposure martingae that tracks whether a node is abeed 1 initiay or not can be used to show the strong concentration around yf. Hence the number of message bits not recovered at the end of the decoding process is greater than y/n with probabiity exponentiay sma in n, for 7 M 7. Then using the expansion properties of the random graph, which foows from standard combinatoria arguments as outined in [], it is not hard to argue that if at most 7 n message bits are eft recovered then the decoding process fais to recover more than O(n7 ) message bits with probabiity at most inverse poynomia in n, for some constant 7 < 1. Finay, by a sma sup pement to the graph (adding a few nodes on the right and having three additiona edges out of each node on the eft mapping randomy to these few additiona right nodes), one can see that if the process fais to recover at most O(n7 ) of the message bits in the origina graph, then in the suppemented graph a message bits fai to be recovered with probabiity at most inverse poynomia in n. From this it foows that, with high probabiity, when the decoding process terminates a message bits have been successfuy recovered. 2.4 The Dua Inequaity 367 In [] a procedure is described for finding (cose to) optima right probabiities pi,..., PR for a given set of eft probabiities Ai,..., XL using a inear programming approach. However, [] did not describe how to find the optima eft probabiities for a given set of right probabiities. Using Condition (2.1), it is easy to see how to use the methodoogy described in [] to do exacty this. In fact, Condition (2.1) is in some sense the dua of the corresponding condition described in [], which was (2.2) p( -8 * X(1 - z)) > t for a z E (O,]. It is from Condition (2.2) that [] shows how to find the optima right probabiities for a given set of eft probabiities. We eave it as an exercise how to use the And-Or tree anaysis to easiy derive Condition (2.2). It turns out that Condition (2.2) can aso be derived from Condition (2.1) using a few simpe agebraic manipuations. We eave this as an exercise as we. One advantage of being abe to sove for both the optima eft probabiities for a given set of right probabiities and the optima right probabiities for a given set of eft probabiities is that we can invoke a back and forth strategy to get a good pair of distributions. This strategy consists of starting with any given set of eft and right probabiities with a given average degree, and then iterativey invoking the find the best eft for the given right foowed by find the best right for the given eft. We have tried this strategy and it gives good resuts, athough at this point we have not proved anything about its convergence to a (possiby optima) pair of probabiity distributions. 2.5 Fraction of Left Nodes Unrecovered The new anaysis of the decoding process aso yieds extensions that hep to overcome other practica probems in the design of oss-resiient codes. In the origina ana- ysis it is assumed inductivey that a the check bits are received when trying to recover the message bits. The reason we made this assumption is that in the origina construction the cascading sequence of bipartite graphs is competed by adding a standard oss-resiient code at the ast eve. There are some practica probems with this. One annoyance is that it is inconvenient to combine two different types of codes. A more serious probem is that standard oss-resiient codes take quadratic time to encode and decode. Suppose the message is stretched to an encoding twice its ength. In order to have the combined code run in inear time, this impies that the ast graph in the cascading sequence has fi eft nodes,

5 368 where n is the number of nodes associated with the origina message, i.e., there are og(n)/2 graphs in the sequence. In the anaysis, we assume that an equa fraction of the nodes in each eve of the graph are received. However, there is variance in this fraction at each eve, with the worst expected fractiona variance at the ast eve of /i/51. Thus, if a message of ength 65,536 is stretched to an encoding of ength 131,072, then just because of the variance of /i/fi = 0.063, we expect to have to receive times the message ength of the encoding in order to recover the message. A soution to this probem is to use many fewer eves of graphs in the cascade, and at the ast eve aso use a random graph in pace of a standard oss resiient code. We have tried this idea, with the ast graph chosen from an appropriate distribution, and it works quite we. For exampe, using ony three eves of graphs we can reiaby recover a message of ength 65,536 from a random portion of ength 67,700 (i.e., times the optima of 65,536) of an encoding of ength 131,072. To design the graph for this soution, we need to anayze the decoding process when a random portion of both the message bits and the check bits are missing. With the And-Or tree anaysis, this is straightforward. Reca the terminoogy estabished in Subsection 2.2. Suppose that a random fraction 6 of the message bits are not received directy and a random fraction 6 of the check bits are not received directy. The generaization of Condition (2.1) for this case when there are osses on both sides is that the process terminates in a state where a uniformy chosen edge is adjacent to a eft node with a missing message bit with probabiity at most y if P-3) 6 * X(1 - (1-6 ). p( - z)) < 2 the graph unti it is in practice abe to recover a of the first (message) ayer. That is, the constant fraction eft unrecovered is so sma that in practice a nodes corresponding to the message are recovered. Given the fractions of eft nodes pi and right nodes qi of degree i for a i, X(z) and p(z) can be easiy derived, and then the argest vaue x* for which Condition (2.4) is vaid can be computed. We show here how to compute the fraction of unrecovered nodes on the eft at this fina vaue z*. The vaue of z* has a natura interpretation, i.e., it is the fraction of edges (v, w) for which a of the eft neighbors of UI, excuding u, have abe 1 at the end of the process. Thus; this is the fraction of edges (v, w) which coud cause v to receive the abe 1, assuming that w is directy received. Define L p(x) = Cp i=o. Xi. From this interpretation, it can be seen that the fraction of unrecovered eft nodes at the termination of the process is 6. p( - (1 - a )%*). This is because y = 1 - (-8)2* is the fraction of edges (v, w) which cannot hep to recover v. Thus, a eft node v of degree i is not recovered at the end with probabiity S (its origina missing probabiity) times y, and there is a pi fraction of such eft nodes. 3 Pure Litera Anaysis In this section, we consider a simpe heuristic, caed the pure itera rue, for finding a satisfying truth assignment to a booean formua. The behavior of the pure itera rue has been studied previousy with for a 2 E (-y,j]. respect to randomy chosen L-SAT formuae ([5, 141). The more genera version of Condition (2.2), when (See aso [7] for reated resuts using more sophisticated a fraction 6 of the right nodes are missing, is heuristics.) Here, we show how the tree anaysis gives a direct expanation of the behavior of the pure itera rue (24 p( - 6. X(1 - (1-6 )t)) > 2 for a randomy chosen k-sat formua with respect to the same distributions considered in [5] and [14]. With The Condition (2.4) is not possibe to satisfy for a this new anaysis, it is aso straightforward to anayze I: E (0, ] if 6 > 0, for any vaue of S. This is distributions that were not previousy considered and because there is a constant probabiity that a the right which woud be much harder to anayze using previous neighbors of a missing eft node are aso missing, e.g., techniques appied to this probem. if the eft node has degree d then the probabiity is Std. A B-SAT formua F with m causes on n variabes However, it turns out to be an interesting question to {X,...,Xn} consists of m causes Ci,..., C,,,, each see what fraction of the eft nodes can be recovered cause containing exacty k of the 2n possibe iteras when a fraction 6 of the right nodes are missing. The answer to this question can be used to design cascading A:={X,X1,...) Xn,Xn}. codes where the decoding process moves from right to eft bootstrapping up to recover a higher and higher Then the formua F is the AND of the m causes, and fraction of nodes at each successive decoded ayer of each cause is the OR of the k iteras it contains, i.e.,

6 for any O/ assignment to the variabes, F evauates to with the same abe with probabiity qi, and the distri- 1 if and ony if in each cause there is at east one itera bution is the same for both possibe abes. Counting that has vaue 1 with respect to the assignment. The the number e of edges using the eft and the right nodes most widey studied distribution on F is the uniform gives distribution. For fixed vaue of k, m, n the uniform R distribution on choosing a formua F can be described e=m -Cjpj =2n*~ig. as foows: for each i E { 1,..., m} and for each j E j=o i=o {,...,k, each of the 2n possibe iteras is chosen with equa probabiity to be the jth itera in cause Ci. The pure item rue heuristic for finding a satisfying assignment consists of repeated appication of the foowing: Pure Litera Rue: Whie there is a itera Z E A that appears in zero causes, remove a causes containing the negation.%? of 2, assign,?? the vaue 1 (and thus 2 is assigned 0), and remove both Z and.!? from A. The probem of interest is to study the asymptotic behavior of the pure itera rue with respect to uniformy chosen k-sat formuae for a fixed vaue of k, and for a fixed ratio c = m/n of the number of causes to the number of variabes, as the number of variabes n grows to infinity. The more particuar question is for which vaues of k and c wi the pure itera rue amost surey (with respect to a uniformy chosen formua F) find an assignment which makes F evauate to 1 as n goes to.infinity. Simiar to the oss-resiient codes, we can describe the structure of the formua F as a bipartite graph, ony in this case the edges are abeed. There are n right nodes in the graph corresponding to the variabes, and there are m eft nodes corresponding to the causes. There is an edge abeed + from variabe X to a causes that contain X, and there is an edge abeed _,, from variabe X to a causes that contain X. One can describe the behavior of the pure itera rue on this graph. The pure itera rue is equivaent to repeated appication of the foowing process on this graph: Graph Pure Litera Rue: Zf there is a variabe X such that a edges touching X have the same abe (either a + or a - ) then deete X, a neighboring causes of X, and a edges touching any of these nodes. The pure itera rue finds an assignment that satisfies the formua if and ony if there are no remaining right nodes after a possibe appications of this process have been made. We describe a genera way of choosing a random formua F in the terminoogy of bipartite graphs. Let (PO,P, * *.,PL) and (me,..., qn) be probabiity vectors. Suppose each cause chooses independenty to have degree j with probabiity pj. Suppose each variabe X chooses independenty to have i edges attached to it 369 A random permutation 1~ of { 1,..., e} is chosen, and then, for a i E (1,..., e}, the edge with index i out of the eft side is identified with the edge with index ni out of the right side. For the specia case of the uniform distribution on k- SAT, each cause has degree k, and the number of edges with the same abe out of each variabe (corresponding to the number of appearances of the corresponding itera in causes) is asymptoticay distributed according to the Poisson distribution with mean 0 = km/2n as n goes to infinity, i.e., the probabiity that a particuar itera appears in i causes is asymptoticay exp(f9). ~9 /6!. Consider the random subgraph GL of this graph ob- tained as foows: choose an edge uniformy at random from among a edges, and suppose it is an edge between cause C and variabe X with abe * E {+, -}. Consider the subgraph G obtained by the foowing search. Consider variabe X to be at the zeroth eve of the search. Foow a the edges out of X with the opposite abe of *. This eads to a first eve of cause nodes. Let C be one of the causes at the first eve. Foow a edges out of C except the edge that ed into Ct. This eads to a second eve of variabe nodes. Let * E {+, -} be the abe of an edge from C to some variabe X. Foow a the edges out of X with the opposite abe of *I. In a simiar pattern, continue this breadth first search out to eve 24. As was the case for the oss-resiient codes, G is a tree with high probabiity for a fixed vaue of e as n goes to infinity. Furthermore, asymptoticay the distribution on the shape of G can be described ss foows. For a i = 1,..., L, et Xi = ipi/ctz1 jpj be the probabiity that a uniformy chosen edge is attached to a cause node of degree i. For a i = 0,..., R, et pi = qi be the probabiity that a uniformy chosen edge is attached to a variabe node with i edges of the opposite abe attached. Then the distribution on the shape of G is as described above for a randomy chosen And-Or tree with the foowing parameters: the number of chidren of a cause node is i - 1 with probabiity Xi, forai=,..., L. The number of chidren of a variabe node is i with probabiity pi, for a i = 0,..., R. Consider the foowing abeing process of the nodes in the tree G that starts at the eaves at eve 21 and works up towards the root at eve 0. A eaf variabe node is abeed with a 1 if and ony if it woud have

7 370 no descendants if the tree were extended one additiona eve. An interna variabe node is abeed with a 1 if and ony if it either has no direct descendants or ese they are a abeed with a 1. A cause node is abeed with a 1 if and ony if at east one direct descendant is abeed with a 1. It can be checked that if the root node receives the abe 1 then the pure itera rue wi give that variabe an assignment. Let ye be the probabiity that the root node of Ge wi receive the abe 1 by the above abeing process. Define the poynomias L X(z) = CXi. xi- and i= p(z) = &.zi. i=o From Lemma 1 it foows that this can be expressed as Ye = p(1 - X(1 - Ye-)) In order for the pure itera rue to end with a compete assignment that satisfies the formua, we want a variabes to disappear from the formua, or equivaenty, receive the abe 1. This means that we want require using the differentia equations based approach, as our tree-based approach ony shows that there is a witness for the satisfiabiity of a random formua with high probabiity. (See, for exampe, the discussion in [16], or the proof of [9, Theorem 91.) Intuitivey, however, it is cear that as ong as the imiting behavior of ye behaves propery, this methodoogy finds the correct threshod. 4 Greedy Matching Anaysis In the paper [9], the anaysis of a simpe and fast heuristic for finding matchings was described and anayzed with respect to randomy chosen graphs. (This anaysis has since been extended in [3].) The tree approach provides an anaysis of what they ca Phase 1 ; in fact, Karp and Sipser provide an argument based on a simiar tree argument. The detais are simiar to, but somewhat different than, those presented above to anayze the oss-resiient codes. As mentioned above, the advantage of the tree anaysis is that it can be adopted to anayze a variety of distributions on the graph with itte additiona effort. The first phase of [9] is a greedy matching agorithm in a random graph. The basic step of the first phase of their agorithm is the foowing. Greedy Matching Step: Find a node v of degree one (34 PP - X - Y)) > Y in the graph; match it to its unique neighbor w; remove v, w, and a edges touching either v or w, from the for a y in the range [PO, 1) (Note that ys = ~0.) graph. For a uniformy chosen k-sat formua we have X(z) = P-1 and p(z) = exp(e(x - )), where 0 = kc/2. This matching step is appied iterativey unti there For a specific k we can easiy determine the threshod are no degree one nodes in the graph. The basic vaue c for which (3.5) is satisfied. In particuar, for quantity of interest is the expected number of edges k = 3 we obtain the vaue c This resut has in the matching produced by this phase in a random been found previousy by [5] and [14] using a different graph. approach. The advantage of the tree anaysis approach In the work of [9], this basic quantity is anayzed empoyed in this paper is that, with itte additiona with respect to a random graph with n nodes chosen difficuty, it is easiy possibe to anayze substantiay as foows: each edge is chosen to exist with probabiity different distributions for choosing the formua, merey 0/(n-1) independenty of a other edges. Using the tree by estabishing the proper functions p(x) and X(z). approach, we can easiy generaize this to the foowing. It requires some additiona technica work, as in Let (PO,PI,..., pi) be a probabiity vector. Each node Section 2.3, to prove that if (3.5) is satisfied then the is chosen to have degree i with probabiity pi. The pure itera rue finds a soution with high probabiity. distribution anayzed by [9] is the specia case where Specificay, we need an expansion based argument ike that given in [5, Lemma 4.41 to show that once a but pi = exp(-0)(0) /a!. Once the graph is fixed, the edges in the matching a constant fraction of the iteras are assigned vaues, produced by repeated appication of the greedy matchthe process must compete with high probabiity. Aso, ing step described above depend on the order in which technicay we have ony proven one direction, namey the nodes are chosen. Nevertheess, it is cear that the that the pure itera rue finds a soution is (3.5) is size of the matching produced by repeated appication satisfied. In fact we can aso show that if (3.5) is not of the greedy matching step does not depend on this orsatisfied, that is if c is chosen arger than the threshod, dering. To be abe to anayze the size of the matching, then with high probabiity the pure itera rue does we now describe an order invariant abeing process on not find a soution. Unfortunatey, doing so appears to the graph which wi be used to anayze the matching

8 371 process. For each edge (v, w) in the graph, we form directed edge (21, w) pointing from PI to w and directed edge (w, V) pointing from w to v. The process abes each directed edge with one of the three symbos {?,O, 1). Initiay, each directed edge is abeed with?. The abeing process consists of repeated appication of either one of the two cases in the greedy abeing step unti no more appications are possibe. Greedy Labeing Step: Labe directed edge (v, w) with a 1 if a (possiby zero) directed edges pointing out of w excuding directed edge (w, v) are abeed 0. Labe directed edge (v, w) with a 0 if at east one directed edge pointing out of w excuding directed edge (w, v) is abeed 1. The key point is that there is the foowing correspondence between directed edges with abe 1 and the matching produced by any execution of the greedy matching process. If directed edge (v, w) is abeed 1 and the reverse directed edge (w, v) is aso abeed 1 then edge (v, w) is in the matching for every execution of the greedy matching process. For a k 2 1, if there are k directed edges abeed 1 pointing out of a node then in every execution of the greedy matching process exacty one of these k edges (considered as undirected edges) is in the matching. For every execution of the greedy matching process, for each edge (v, w) in the matching produced by the execution, at east one the directed edges (v, w) and (w,~) is abeed 1. From this, we can cacuate the size of the matching produced by any execution of the greedy matching process as foows. Let M be the set of directed edges abeed 1 for which the reverse directed edge is aso abeed 1. For a k: 2 2, et Nk be the set of directed edges abeed 1 that are pointing out of a node w such that w has in tota k directed edges abeed 1 pointing out of it. Then the size of the matching is exacty We first justify the Cf=, INk/k terms in (4.6). Consider a node v with k 2 2 directed edges with abe 1 pointing out of it. Note that there cannot be any directed edges with abe 1 pointing into v. Thus the directed edges in M and Nk for k 2 2 are disjoint. Furthermore, the k directed edges abeed 1 pointing out of v are a in Nk, and there is exacty one corresponding matched edge in any execution of the greedy matching process. These k edges contribute k/k = 1 as required to these terms. We now justify the INi - IM1/2 terms in (4.6). Consider a node v with exacty one directed edge (v, w) with abe 1 pointing out of it. Then the edge (v, w) is in the matching produced by the greedy matching process, and the tota contribution to the size of the matching shoud be 1. There are two cases to consider, depending on whether directed edge (w, v) is abeed 1 or not. Suppose (w, v) is not abeed 1. Then neither (v, w) nor (w, v) is in M, and ony (v, w) is in Ni. These two directed edges contribute 1 - O/2 = 1 to (4.6), as required. Suppose (w, v) is abeed 1. Then both (v, w) and (w, v) are in both M and Ni. These two directed edges contribute 2-2/2 = 1 to (4.6), as required. The strategy now is to first estimate the probabiity that a uniformy chosen directed edge is abeed 1 by the greedy abeing process, and then to use this probabiity to estimate the size of the matching produced by the greedy matching agorithm based on the observations just made. As in the anaysis of oss-resiient codes, we are interested in properties of such a graph as n grows to infinity. Aso as before, it turns out that it is easiest to anayze the properties by considering a randomy chosen edge from the graph. Fix a directed edge (v, w) in the graph, and et GL(v, w) be the subgraph induced by the edge (v, w) and a neighbors of w reachabe within distance 2L, foowing the rue that if node w is reached using directed edge (v, w ) then the reverse edge (w, v ) is not used. As was true for the oss-resiient code anaysis, the probabiity that GL(v, w) fais to be a directed tree for a uniformy chosen edge (v, w) of the graph is proportiona to /n, i.e., asymptoticay this probabiity goes to zero as n grows to infinity for a fixed vaue of e. Furthermore, asymptoticay the distribution on the shape of GL(v, w) for uniformy chosen (v, w) can be described aa foows. For a i =,...,L, Xi := ipi/ xtz1 jpj is the probabiity that a uniformy chosen edge is attached to a node of degree i. Each node in the tree GL(v, w) at distance ess than 2e from the root has i - 1 chidren with probabiity Xi, for a i=, > L. Consider the foowing abeing process on the directed edges of tree GL(v, w). The tree abeing process starts with a nodes abeed?, and then works up from the eaves towards the root. Each directed edge (v, w ) pointing to a eaf node WI at distance 2& from w is abeed with a 1 if there is no directed edge other than

9 372 the reverse edge (w,v ) pointing out of w in the entire graph. The abeing of the rest of the tree, and the directed edge (v, w), works according to the greedy abeing step described previousy. The difference between the tree abeing and the greedy abeing process is that the tree abeing makes a decision to abe a eaf with a 1 ony in the case when the eaf has no other edges attached to it, whereas in the greedy abeing process the eaf may aso receive the abe 1 when the eaf has degree greater than 1 in the origina graph. It is not hard to verify that if the directed edge (v, w) receives the abe 1 in the tree abeing of GL(v, UJ) then it receives the abe 1 in the greedy abeing process. Athough the reverse is not true, the caim is that as e grows, the probabiity of these two events approach one another. More formay, et z be the probabiity that a uniformy chosen directed edge receives the abe 1 in the greedy abeing process on a randomy chosen graph. Let y be the probabiity that a uniformy chosen directed edge (v, w) receives the abe 1 in the tree abeing of GL(v, w). It is cear that z 2?/L for a 1. We caim that y approaches P as 1 grows; however, the ony justificrrtion we currenty know for this reies on the differentia equation approach. (See Theorem 9 of [9].) We assume that this is the case hereafter. Let, us anayze the probabiity yt that a uniformy chosen directed edge (v, TN) is abeed with 1 in the tree abeing of G(v, w). As before, define the poynomia A(z) = kxi.2--. i= Then from Lemma 1 and the description of the process, yf = X(1 - X(1 - ~1-1)). Using this equation, the y can be computed iterativey starting with the equation Yo = X. Let y be the asymptotic imit of yf as f! grows. We need a coupe of additiona observations in order to be abe to compute the expected size of the matching returned by the greedy matching process as a function of y. The observations are that asymptoticay, as n goes to infinity: The abes received by a given set of directed edges pointing out of a particuar node are independent of one another. The abes of a given directed edge and its reverse directed edge are independent of one another. From these observations, we can concude that the probabiity a randomy chosen directed edge is in Nk Furthermore, the probabiity a randomy chosen directed edge is in M is 9. Thus, the expected size of the matching produced by the greedy agorithm is equa to the expected number of directed edges in the graph times L c ak/k - Y2/2. k= 5 Acknowedgements The second author thanks Aan Frieze for the references to [16] and [3], and thanks Aan Frieze and Andrei Broder for severa usefu discussions. References PI N. AIon, J. Edmonds, M. Luby, Linear Time Erasure Codes With Neary Optima Recovery, Proc. of the 36th Annua Symp. on Foundations of Computer Science, 1995, pp PI N. Aon, M. Luby, A Linear Time Erasure-Resiient Code With Neary Optima Recovery, IEEE Tmnsactions on Information Theory (specia issue devoted to coding theory), Vo. 42, No. 6, November 1996, pp PI J. Aronson, A. %eze, B. G. Pitte, Maximum Matchings in Sparse Random Graphs: Karp-Sipser revisited, preprint. [41 R. Boppana, Ampification of probabiistic Booean formuas, Advances in Computer Research, Vo. 5: Randomness and Computation, JAI Press, Greenwich, CI, 1989, pp [51 A. Broder, A. Frieze, E. UpfaI, On the Satisfiabiity and Maximum SatisfiabiIity of Random 3-CNF Formuas, Proc. of the 4th ACM-SIAM Symp. on Discrete Agorithms, 1993, pp PI M. Dubiner, U. Zwick, Ampification by Read-Once Formuas, Siam J. on Computing, Vo. 26, No. 1, Feb. 1997, pp [71 A. tieze, S. Suen, Anaysis of Two Simpe Heuristics on a Random Instance of k-sat, J. of Agorithms, Vo. 20, 1996, pp PI R. G. Gaager, Low-Density Parity-Check Codes, MIT Press, PI R. Karp, M. Sipser, Maximum Matchings in Sparse Random Graphs, FOCS, 1981, pp T.G. Kurtz, Approximation of Popuation Processes, CBMS-NSF Regiona Conf. Series in Appied Math, SIAM, M. Luby, M. Mitzenmacher, M. A. ShokroIahi, D. Spieman, V. Stemann, Practica Loss-Resiient Codes, Proc. 2gth Symp. on Theory of Computing, 1997, pp M. Luby, M. Mitzenmacher, M. A. ShokroIIahi, D. Spieman, Anaysis of Low Density Codes and Improved Designs using Irreguar Graphs, submitted to STOC [W P WI

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