On the Reliability of Fat-Trees S. Nikoletseas
1
G. Pantziou
1
P. Psycharis
1;2
P. Spirakis
1
(1) Computer Technology Institute, Patras, Greece (2) Intrasoft, Greece email: fnikole,pantziou,
[email protected] CTI Technical Report TR96.06.18
Abstract
In this paper we examine the reliability properties of the ideal fat-tree, a general model used to capture both distance and bandwidth constraints of various classes of fat-tree networks. We use the notion (see [NPSY94]) of a random graph of type?G obtained by selecting edges of a given undirected graph G independently and with probability p, thus representing a network in which links fail independently and with probability f = 1 ? p. In addition, we may allow vertices of the graph to fail independently with probability f . We show that: 1. At least half of the edge disjoint paths connecting any subsets of terminal nodes of a fat tree (whose fat-nodes do not have internal edges) are preserved with probability tending to 1 as the number of nodes in the tree tends to in nity, even in the case of constant vertex and edge failure probabilities f < 0:25. Thus, and by also showing the preservation of the minimum cut (and thus of the maximum ow), we prove that the ideal fat-tree remains, with high probability, \fat-enough", in the sense of the \ ?fat" de nition given in [BB94]. 2. In the case of the fat-nodes being random regular graphs of degree d (and thus expanders) we prove that at least (1 ? f ) times the initial number of edge disjoint paths survive despite the edge faults with high probability, even for degrees d as small as 1256 ?f . 3. There is a critical probability pc such that for p > pc the pruned butter y Pd =p of type?Pd has a connected component of linear size which includes at least a constant fraction of its terminal nodes, with high probability, while for p < pc such a component does not exist almost certainly.
1 Introduction There are many networks referred to as fat-trees in the literature, including the concentrator fat-tree, the pruned-butter y fat-tree and the sorting fat-tree. In all cases, a fat-tree can be viewed as a tree-like graph whose leaves act as input/output terminals, whose internal \nodes" are subnetworks (with switching capability) and whose \edges" are channels of appropriate capacity. Fat-trees play an important role in modern multiprocessor architectures because of the hardware-ecient (with respect to VLSI design) universality properties they have ([Lei85], This research was partially funded by the EU Long Term Research Projects GEPPCOM (Contract No. 9072) and ALCOM-IT (Contract No. 20244).
1
[BB93], [Gre94]) and, also, because they provide a natural basis for the development of ecient routing algorithms ([GL89], [LMR88], [BB95]) and for the implementation of interconnection networks in parallel supercomputers ([LAD+92]). Aiming at a general study of various existing fat-tree networks, we use the ideal fat-tree model introduced by Bilardi et al ([BCC+96]), which captures both distance and bandwidth constraints and provides a framework for designing algorithms portable among di erent fat-tree networks. An abstract de nition of the ideal fat-tree network is the following: The N terminal nodes (processors) are placed at the leaves of a complete binary tree whose internal nodes are switching modules building a routing network among the processors. The nodes of the tree are connected via channels of appropriate capacity, which is equal to the number of edges joining each node (except from the root) to its parent. The communication over the network is subject to certain distance and capacity constraints: 1) A message arrives at its destination 2 h steps after it is sent, where h 2 f1; : : :; log N g is the height of the lowest common ancestor of source and destination and 2) The number of messages traversing any channel should not exceed the the channel's capacity. This model is ideal in the sense that the routing time is the minimum satisfying distance and capacity constraints, while most existing fat-tree networks would incur a polylogarithmic (with respect to the ideal fat-tree) slowdown in routing time. Also, a genaral capacity function w(n) usually obeys the following natural conditions: 1. w(2n) w(n), that is, the outgoing bandwidth of a subtree does not decrease with its size 2. w(2n) 2 w(n), that is, the bandwidth toward the parent does not exceed the total bandwidth toward the children. Here n 2 f1; : : :; N=2g is the number of leaves of the subtree rooted at a given terminal node and ? 1=2 w(n) denotes the capacity of the fat-edge to the parent. Usually, w(n) = n is assumed since it satis es the above conditions and also ensures area universality (see [BB94]) of the ideal fat-tree. In this work, we analyze the reliability of communication in the presence of faults, which may lead to unavailability of fat-nodes or edges of the fat-tree (interconnection) network of the multiprocessor architecture. Let G be an undirected graph. A random graph G of type?G is obtained by selecting edges of G independently and with probability p. We can thus represent a communication network in which links fail independently and with probability f = 1 ? p. In [NPSY94], [NS95] we have de ned and used this model to prove multiconnectivity and expander properties in various graphs of type?G. Let G be the tree-structured undirected graph representing the initial fat-tree before the faults. The corresponding random graph G of type?G represents the faulty fat-tree network obtained by allowing edges to fail independently and with probability f . We show that various reliability properties in type?G fat-tree networks are preserved for a wide range of failure probabilities, including the constant failure probability f . In particular, we show that: 1. At least half of the edge disjoint paths connecting any subsets of terminal nodes of a fat tree (whose fat-nodes do not have internal edges) are preserved with probability tending to 1 as the number of nodes in the tree tends to in nity, even in the case of constant vertex and edge failure probabilities f < 0:25. Thus, and by also showing the preservation of the minimum cut (and thus of the maximum ow), we prove that the ideal fat-tree remains, with high probability, \fat-enough", in the sense of the \ ?fat" de nition given in [BB94]. 2
2. In the case of the fat-nodes being random regular graphs of degree d (and thus expanders) we prove that at least (1 ? f ) times the initial number of edge disjoint paths survive despite the edge faults with high probability, even for degrees d as small as 1256 ?f . We also consider the pruned butter y, which is an interesting special case of a fat-tree with fat-nodes having no internal edges. For this special case we study (in Section 5) the critical probability p for the existence of a linear sized component surviving the failures and including a large fraction of terminal nodes. Note that usually the assumption f = o(1) (e.g. f < 1=n) captures many realistic permanent failure patterns. A value of f = (1) (independent of n) is considered to be a worst-case assumption.
2 Faulty fat-trees remain \fat-enough" even for constant failure probabilities Let G be the undirected graph representing the tree-structured fat-tree network and TG be the corresponding tree taken by viewing each fat-node as a single (super)vertex. Bilardi and Bay ([BB94]) have suggested the following de nition of \ ?fat" tree-networks, which is based on the relation between the number of edge disjoint paths connecting two subsets A, B of terminal nodes and the maximum ow F (TG ; A; B ) that can be pushed from A to B over the network. Recall from [Tar83] that a ow on a graph G is a real-valued function f on vertex pairs having the following three properties: (i) skew symmetry (ii) capacity constraint and (iii) ow conservation. In the case where we consider ows between pairs of sets A, B of terminal nodes, the third condition becomes X For any u 2= A [ B : f (u; v) = 0 v
Also, recall the max- ow min-cut theorem stating that the value of the maximum ow is equal to the size of a minimum cut in the graph. The \ ?fat" de nition of Bilardi and Bay ([BB94]) is as follows: De nition 1 (Bilardi and Bay, [BB94]) A tree-structured graph G is \ ?fat" if for every disjoint sets of leaves A, B there exist at least F (TG ; A; B ) edge disjoint paths from A to B in G, where TG is the tree corresponding to graph G and 2 (0; 1) is a constant. 2 We refer to as the \fatness" constant. In this section, we rst show the preservation of the minimum cut (and thus of the maximum
ow). Then, we prove that at least half of the edge disjoint paths connecting any subsets A, B of terminal nodes of the fat-tree are preserved with high probability for all f < 0:25. Thus, we show that faulty fat-trees remain \fat-enough" almost certainly and for all f < 0:25 in the sense that ' 2 (1 ? ) where and are the fatness constants before and after the failures respectively and 2 (0; 1) an arbitrarily chosen constant. In the rst two sections we will consider the fat-tree partitioned into two parts, a higher (GH ) and a lower (GL) part. We will temporarily assume that at least a constant fraction of the terminal nodes connects to the higher part of the fat-tree on the average. We then show that GH is robust. Based on our assumption we thus prove robust properties even for the terminal nodes that remain connected. The assumption is proved in section 4. 3
2.1 Preservation of maximum ow in fat-trees with edge failures
Let Hp be such that all pchannels in the upper H levels of the graph G have capacity at least pn log n, where 2 a constant. Denote by GH this upper part of the graph G. Remark that a cut in GH with respect to any two subsets A, B of terminal nodes is a set of edges intersecting any path between nodes of A and B . Now let C (GH ) be the size of minimum (in cardinality) cut in GH and C (GH ) be the minimum cut value in the graph of type?GH .
Theorem 1
E (C (GH )) = (1 ? f ) C (GH ) Proof: Consider a random indicator variable Xe for every edge e in a minimum cut of GH , taken value 1 (0) when e remains in GH (or not, respectively). The theorem follows easily by linearity of expectation. 2 Now sort arbitrarily the edges of GH and let Ij be a random indicator variable showing whether edge ej remains in GH or not, that is Ij =
(
1 if ej 2 GH 0 otherwise
Consider the sequence X0 ; X1; : : :; Xt of random variables such that
Xk = E (C (GH )jI1; : : :; Ik) X0 = E (C (GH )) Xt = the value of minimum cut in GH
Lemma 1 X0; X1; : : :; Xt is a (Doob) martingale (in fact, an edge exposure martingale). Proof: Just de ne the lter Fk ; Fk?1; : : :; F0 where Fk is the - eld generated by the events corresponding to I1; : : :; Ik . Clearly
E (Xk+1jFk ) = Xk
2
and the martingale criterion is met.
Lemma 2
jXk ? Xk?1j 1
Proof: Remark that taking into account the indicator variable Ik, corresponding to whether ej 2 GH is holding or not, can only decrease the expectation of the minimum cut of the \so-far exposed" graph by at most 1 (this may happen when ej 2= GH , otherwise Xk = Xk?1). 2
Thus, the Lipschitz condition holds, so we can employ the powerful method of bounded di erences and get, by Azuma's inequality (see for example [MR95]), that, 8 > 0 p (1) PrfjXt ? X0 j tg 2 e? =2 We are now ready to prove the basic theorem of this section of the paper: 2
Theorem 2
p
for any 2
p p PrfjC (GH ) ? E (C (GH ))j > n log ng n2 4
Proof: By the de nition of X0; X1; : : :; Xt we get that p pp PrfjC (GH ) ? E (C (GH ))j > n log ng = PrfjXt ? X0j > ng p p where now = log n, 2. Then by equation 1 we get p n = 2 n? 2 PrfjXt ? X0j > ng 2 e? n 2 log 2
2 2
2
Thus, the minimum cut of H is around (1 )E (min cut H i.e. (by theorem 1) around (1 )(1 ? f )(min cut of GH ) with probability 1 ? n2 for some 2 (0; 1). By choosing a close to zero, we get Corollary 1 The maximum ow of GH is only fractionally reduced (by (1?f )) almost certainly.
G
of G ),
2.2 Preservation of edge disjoint paths under constant vertex and edge failures Recall that in this rst part of the paper, we consider the (worst, as far as connectivity issues are concerned) case where the fat-nodes of the tree-structured graph G have no internal edges at all. Now let H 0 be such that each fat-node of the part GH 0 consisting of the upper H 0 levels of graph G, has at least k log n (internal) vertices (k 2 a constant). The next theorem shows that, for f constant, a constant fraction of the number of vertices in each fat-node survive the vertex faults with high probability. De nition 2 For a particular fat-node V in GH0 let EV be the event "at least (1 ? )(1 ? f )jV j vertices of V remain in GH 0 ", where 2 (0; 1) a constant. Also, let E = \V EV .
Theorem 3
0
PrfE g 1 ? n?(k ?1)
where k0 2 a constant.
Proof: Let V be a particular fat-node in GH0 . Remark that the construction of GH0 consists in performing jV j Bernoulli trials with success probability p = 1 ? f at each such node. Thus, by Cherno bounds,
PrfEV g 1 ? e?
2 (1?f )jV j 2
1 ? e?
2 2
(1?f )k log n = 1 ? n?k0
where k0 = 2 (1 ? f )k is a constant that can be made at least 2 by choosing appropriate values for contants , k. Thus, 2
n
o
PrfE g = Pr [V EV PrfE g
1 ? n?(k0 ?1)
X
V
n
o
Pr EV n n?k0 = n?(k0 ?1) ,
2
The above theorem implies the survival of a constant fraction of vertices after the vertex faults within each fat-node. We will now prove that the edges connecting the fat-nodes tolerate the edge faults well-enough to guarantee the preservation of at least half of the edge disjoint paths when f < 0:25, almost certainly. 5
Theorem 4 At least half of the edge disjoint pathsconnecting any sets A, B of terminal nodes remain in GH 0 with probability at least 1 ? log1 n , for all failure probabilities f < 0:25. Proof: We assume here that at least a constant fraction of terminal nodes connect to the
upper part GH . We will show this holding (on the average) in the following section. >From Theorem 3, at least (1 ? )(1 ? f )jV j vertices within each fat-node survive the vertex faults. Remark that when (1 ? )(1 ? f ) > 3=4, then for any two children V1, V2 of the same parent V in GH 0 , more than half of the nodes of V get edges from both V1, V2. Now, for a fat-node V of GH0 at level k + 1, let p(k + 1) be the (bad-event) probability that in GH0 the remaining edge disjoint paths connecting any sets A, B of terminal nodes that arrive at the node V are less than half of the corresponding paths in GH 0 . Since, by our previous remark, more than half of the vertices of V get edges from both V1, V2 this bad event holds if the remaining edge disjoint paths arriving from either V1 or V2 are less than half of the ones before the failures, thus: 2 p (k + 1) = 1 ? (1 ? 0:5 p(k))2 = p(k) ? p 4(k) (2) By choosing q (k) such that p (k) = q(k1)+1 we get from equation 2 that 1 ? 1 4q (k) + 3 , 1 = = 2 q(k + 1) + 1 q(k) + 1 4(q(k) + 1) 4(q(k) + 1)2 q(k + 1) q(k) + 2 + q(1k) By induction, it follows that
k < q(k) < k + Hk?1 + 3
where Hk is the k?th harmonic number, implying that
1 q(k) = (log n) , p(k) = log n To complete the proof, remark that we get the best possible f satisfying (1 ? )(1 ? f ) > 3=4 by setting ' 0, i.e. 1 ? f > 3=4 , f < 1=4 = 0:25. 2
3 The case where the fat-nodes are random regular graphs In this section, we study the reliability properties of fat-trees whose fat-nodes, instead of having no edges at all (as in the previous section), now have quite a lot of \built-in" connectivity. To be more speci c, consider the case where the fat-nodes of the tree-structured graph G are random regular graphs of degree d (hence they are also expanders with high probability). Let H 0 be such that all channels in the upper H 0 levels of the graph G have capacity at least k log n, where k > 0 a constant. Denote by GH 0 this upper part of the graph (including the random regular fat-nodes) and by TGH 0 the corresponding tree-structured graph (where we view each fat-node as a single supervertex). Let C be a capacity in TGH 0 and C the corresponding capacity in TG H 0 (the type-TGH 0 random graph representing what remains from TGH 0 after the edge faults). Note that the capacity of a channel connecting two fat-nodes is equal to the number of edges joining these nodes. De nition 3 Let EC be the event \ C is within (1 )(1 ? f )C ", where 2 (0; 1) is a constant and f the edge failure probability. 2 6
The following theorem shows that at least a constant fraction of the capacity of any channel of this part of the fat-tree (and thus the number of edges connecting the corresponding endvertices of the channel) is preserved with high probability. Theorem 5 There is a constant k0 2 such that Prf[C 2TGH 0 EC g n?(k0 ?1)
Proof: As mentioned earlier, a channel capacity of size C in GH0 implies the existence of C
edges connecting the endpoints of the channel. But the edges in GH 0 fail independently and with probability f . Thus, GH 0 is the result of performing on each channel C Bernoulli trials with success probability p = 1 ? f . By Cherno bounds we thus have: PrfEC g = PrfC 2 (1 )(1 ? f )C g 1 ? e? 1 ? e? (1?f )k log n = 1 ? n?k0
2 (1?f )C 2
2 2
where k0 = 2 (1 ? f )k can be made at least 2 by choosing appropriate values for , k. Thus, 2
Prf9C in TGH 0 : EC in TG H 0 g n n?k0 = n?(k0?1)
2
Note that the above theorem implies the preservation of a constant fraction of the edges joining any vertex pair, by choosing a small (but even constant) failure probability f and close to 0. In order to investigate what happens in the interior of the fat-nodes we remark that each fat-node is a member of Gdn;p, the class of all random regular graphs of degree d, whose edges fail independently and with probability p. In [NPSY94], [NS95] we have shown: Fact 1 Gdn;p is highly disconnected when f = 1?p is constant and d < 21 plog n, almost certainly.
Fact 2 When Gdn;p is disconnected, it still has a giant connected component for any f < 1 ? 32d with high probability, for any d 64. Fact 3 The giant connected component of a random member of Gdn;p remains, with high probability, a certi able ecient expander, despite the link faults, provided that f < 1 ? 256 d . >From fact 3, we conclude that a constant degree d > d0, d0 = 1256 ?f suces to guarantee that each fat-node remains, with high probability, an ecient expander despite the constant edge failure probability f . By theorem 5 and fact 3 we get: Corollary 2 Let E be the number of edge disjoint paths in fat-trees whose fat-nodes are random regular graphs of degree d. At least (1 ? f )E edge disjoint paths survive the constant failure probability edge faults almost certainly, provided that d > 1256 2 ?f . The following special case is indicative of the power of corollary 2: Corollary 3 Let E be the number of edge disjoint paths in fat-trees whose fat-nodes are random regular graphs of degree d. At least (3=4)E edge disjoint paths survive the failures almost certainly, provided that f < 0:25 and d > d0, where d0 > (4=3) 256 i.e. d0 > 340. 2 7
4 The connectivity of terminal nodes Let T be the set of terminal nodes of a fat tree R. Let GL , GH be the lower (respectively, higher) part of the \tree" graph as previously. Let F (GL; T; GH ) be the maximum ow from T to the nodes of the GH through GL . Assuming that R is ?channel-sucient we know that Fact 4 (Bilardi and Bay, [BB95]) There exist at least F (GL; T; GH) edge disjoint paths from T to GH in R. If jT j = N , then the minimum cut between T and the nodes of GH is clearly at least N (0 < < 1 a constant) due to the channel capacities. Thus the nodes in T are connected to the upper part via N 0 = N disjoint paths. The survival probability of each such path is q = (1 ? f )(log log N ) . For f constant q becomes at least log1c n where c > 1 a constant. Then, from the Bernoulli of N 0 trials and success probability q we get (via Cherno bounds) that: Lemma 3 At least (1 )qN terminal nodes are connected to GH via edge disjoint paths with probability at least 1 ? e? qN for any 2 (0; 1). For example, when f is constant then the expected number of terminal nodes that connect to GH via edge disjoint paths is lognc N . Let X0; X1; : : : be the sequence where Xj is the expected number of terminal nodes in T that connect (possibly through non-edge-disjoint paths) to level j of the lower part (level 0 is T ). Assume that the fat-tree is 1-channel sucient (i.e. an ideal fat tree or a pruned butter y etc). Then, because = 1 and f < 0:5, it is E (Xj +1) = Xj , thus we get (for a proof see full paper) that: Lemma 4 fXj g is a martingale, for any f < 0:5. 2 By using Azuma's inequality and sums of Poisson trials we then get: Theorem 6 For any f < 0:5 and = 1 the expected number of terminal nodes that connect to GH (possibly via overlapping paths) is (N ). 2 Thus, we can suggest that robust fat-trees can be built in the following way: 1. The lower part should consist of fat-nodes guaranteeing = 1 (e.g. a pruned butter y or cliques or tree of meshes). 2. The upper part should be built via fat-nodes being concentrators (e.g. regular random graphs) because of our results in the previous sections. 2 2
5 On the robustness of the pruned butter y
5.1 Introductory comments
Given a pruned butter y Pd , let Pd =p denote the random (of type-Pd ) pruned subbutter y obtained by considering each edge independently and including it in the pruned subbuter y with probability p, and excluding it with probability 1 ? p. In this section we show that there 8
is a critical probability pc such that for p > pc the pruned butter y Pd =p has a connected component of linear size which includes at least a constant fraction of its terminal nodes, with high probability, while for p < pc such a component does not exist with high probability. Similar results for other special networks have been also presented in the literature. We refer here to the works of Karlin, Nelson and Tamaki ([KNT94]) for the butter y network, Kesten ([Kes81]) for the d by d two-dimensional mesh, Erdos and Renyi ([ER60]) for the complete graph, and Ajtai, Komlos and Szemeredi ([AKS82]) for the hypercube of dimension d. The existence of a linear-sized component in the faulted version of a network is an important measure of its robustness. If such a component does not exist, then the computation in the faulted version of the network is bound to be more than a constant factor slower than the computation in the fault-free version. An interesting question is how this critical probability of a network relates to other fault tolerance measures of the network, such as the ability to embed the fault-free version in the faulted one.
5.2 The critical probability
The de nition of the pruned butter y Pd is given in the Appendix (part 1) (see also [BB94]). Note that in our case the number of terminal nodes in Pd is 2d , while the total number of nodes (both terminal and nonterminal) is nd 4 2d . Given a pruned butter y Pd , let Pd =p denote the random pruned subbutter y of type-Pd obtained by considering each edge independently and including (excluding) it in the pruned subbutter y with probability p (respectively, f = 1 ? p). In this section we show that there is a critical probability pc such that for p > pc the pruned butter y Pd =p has a connected component of linear size which includes at least h 2d terminal nodes, where 0 < h < 1 a constant, with high probability, while for p < pc such a component does not exist almost certainly. The approach that one may use for showing that Pd =p has a linear-sized component is to compute an upper bound on the probability that Pd is separated into two smaller components, by multiplying the number of separations (which is 2c2d , c 4), with the probability that any separation survives, i.e., none of the edges of Pd =p connects the two components. If many edges of Pd bridge the separation, then with high probability the separation will not survive. Since only 2d0 =2 edges connect a pruned subbutter y Pd0 to the main pruned butter y Pd (d > d0), the above approach is dicult to apply. We follow closely, but judisiously modify, the approach of Karlin, Nelson and Tamaki in [KNT94] for the case of the butter y network, and the approach of Ajtai, Komlos and Szemeredi in [AKS82] for the hypercube network. We rst choose p1 > pc and p2 > 0 such that (1 ? p1 )(1 ? p2) = 1 ? p and therefore Pd =p1 [ Pd =p2 = Pd =p. Then, we prove that a constant fraction of the nodes of Pd =p1 are in connected components of an appropriate size, which are called atoms. Finally, we show that the additional edges in Pd =p2 connect the atoms into a linear-sized component.
5.3 Existence of a linear-sized component
5.3.1 Preliminaries
Let u be a node at level 0 of Pd and let Yd (p) be the random variable denoting the number of nodes at level d of Pd which are connected to u in Pd =p. Note that from the structure of Pd , the distribution of the above random variable does not depend on the choice of u. De ne
pc = inf fp j E (Yd(p)) > 1 for some dg 9
By theorems 7, 8 we show that pc is the critical probability for the existence of a linear-sized component in Pd that contains a constant fraction of the terminal nodes. We consider a discretetime branching process with an initial population of size 1. Each individual in the process has a number of immediate descendants given by the random variable Z1 . The ith generation of the process is composed of descendants of generation (i ? 1). If Zn is a random variable denoting the population size of the nth generation, then E (Zn) = (E (Z1))n. An important fact for our analysis is the following: Fact 5 (Karlin et al, [KNT94]) Suppose that E (Z1) = > 1. Then, for any constant 0 < < 1, lim inf n!1Pr(Zn > n ) > 0
5.3.2 Above the threshold (p > pc)
We show that when p > pc the pruned butter y Pd =p has a connected component of linear size which includes at least h 2d terminal nodes, where 0 < h < 1 a constant, with high probability. Choose p1 and p2 with p1 > pc and p2 > 0, so that (1 ? p1)(1 ? p2) = 1 ? p. We rst show that a constant fraction of the nodes of Pd =p1 are in connected components of size (nd ) which are called atoms, and that with high probability, each node in Pd =p1 is within distance d of an atom (see Appendix, part 2: Lemmata 6, 7). We then show that the edges of Pd =p2 with high probability connect the atoms of Pd =p1 into a linear sized component, that includes at least h 2d, terminal nodes, where 0 < h < 1 is a constant (see Appendix, part 2: De nition 4, Fact 6, Lemma 8). Therefore, we have the following theorem: Theorem 7 For any constant p > pc, Pd =p has a linear-sized component with high probability.
5.3.3 Below the threshold (p < pc) and a lower bound
We show that when p < pc , then the pruned butter y Pd =p does not contain a linear-sized connected component with high probability.
Theorem 8 For any constant p < pc, Pd =p has a linear-sized component with probability tending to 0.
Proof: See full paper.
2
The characterization of the critical probability given above provides an algorithm to compute it. In the full version of this paper we will give the details of the computation of the critical probability. The following lemma gives a lower bound on the value of the critical probability pc . Lemma 5 pc 0:42 Proof: See Appendix, part 3. 2
Acknowledgement: We wish to thank G. Bilardi for his encouragement and fruitful
discussions.
10
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[NS95] [Tar83]
S. Nikoletseas and P. Spirakis, \Expander Properties in Random Regular Graphs with Edge Faults", 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Munchen, pp. 421 { 432, 1995. R. Tarjan, \Data structures and Network Algorithms", 1985.
R.2
APPENDIX Part 1: The de nition of the pruned butter y Pd A pruned butter y Pd is a graph G(V; E ) de ned, for d 2, as follows: the vertex set is
V = f(i; j; k) : 0 i d; 0 j < 2i; 0 k < 2d=2 2?bi=2cg Vertices (i; j; k) with a given value of i and j form node Vij of a tree representation of Pd . There are no edges within a node. The channel from node Vij to its parent is de ned as: for odd i, and
Cij = f((i; j; k); (i ? 1; bj=2c; k)) : 0 k < 2d=2 2?bi=2cg;
Cij = f((i; j; k); (i ? 1; bj=2c; k)); ((i; j; k); (i ? 1; bj=2c; k + 2d=22b?i=2c )) : 0 k < 2d=22?bi=2c g for even i. The rst argument in the node de nition represents the level of a node in Pd . Therefore, a node v with representation (i; j; k) is of level i. We call the nodes of level d terminal nodes (processors), and the nodes of level k, 0 k < d nonterminal (or internal) nodes. Note that in our case the number of terminal nodes in Pd is 2d , while the total number of nodes (both terminal and nonterminal) is nd 4 2d .
Part 2: Above the threshold (p > pc) Lemma 6 Let u be a nonterminal node on level 0 of Pd. Then there exist constants e and such that, for every sucient large d,
Prfu belongs to a connected component of Pd =p1 with at least (nd )e nodesg
Proof: Let d0 be a constant such that E (Yd (p1)) = > 1. In a way similar to the one in [KNT94] for the butter y, we consider the branching process with initial population being the node u, and generation r individuals being nodes on level rd0 of Pd =p1. The process is de ned up to the kth generation for k = bd=d0c. Note that for any two distinct nodes in the same generation, the pruned subbutter ies de ning their immediate descendants do not overlap. Therefore, we have a branching process with Z1 = Yd (p1) and from the de nition of d0, E (Z1) = > 1. Thus, fact 5 applies and, for < 1, there exists a constant such that Pr(Zk > k ) > . Since there is a constant d00 such that k = d00d, there exists a constant e such that 0
0
Pr(Zk > nd e) >
2
Lemma 7 There exist constants c > 0 and c0 > 0 such that:
0
(i) The probability that fewer than c 2d terminal nodes in Pd =p1 are in atoms is at most 2?ncd . 0 Also, the probability that fewer than cnd nodes in Pd =p1 are in atoms is at most 2?ncd . (ii) The probability that there 0 exists a node in Pd whose distance from an atom is more than d, for 0 < < 1 is at most 2?ncd .
A.1
Proof: (i) Consider the set A of the Pd=4 pruned subbutter ies of Pd whose 0-level nonterminal nodes are level 3d=4 nonterminal nodes of Pd . Note that A has 23d=4 elements. Pick exactly one node from each pruned subbutter y in A and collect them into a set S of 23d=4 nodes. For each v 2 S , let P (v) denote the pruned subbuter y v is picked from. By the previous lemma applied to P (v ), for each v , we have that the probability that v belongs to an atom with (nd=4 )e (nd ) nodes is at least . These events are independent. Letd= c = =2. Then the probability that less than cjS j nodes in S are in atoms is 2 ? (jS j) = 2? (2 ) . Note that the set of terminal nodes of Pd can be partitioned into 2d=4 disjoint subsets each of which is constructed as S above. Applying the same argument to each subset and then summing up the failure probability we have that the probability that less than c 2d terminal nodes of Pd =p1 d= are in atoms is 2? (2 ) . Now, by applying the same argument to the nonterminal nodes of the pruned butter y we prove that the probability that less than cnd nodes of Pd =p1 are in atoms is 2? (2 d ) , where > 0 is a constant. (ii) Let v be an arbitrary node in Pd . Consider the P d=2 pruned subbutter y that contains v . Let T be the set of d=2-level nodes of P d=2 (jT j = 2 d=2 ). T has the same property as S above. Therefore, by a similar argument, the probability that no node in T is in an atom, is at most 2? (2 d= ) . Since the distance from v to any node in T is at most d, this bounds the probability that a given node violates the statement of the lemma (part (ii)). We complete the proof by summing this failure probability over all nodes of the network. 2 We then show that the edges of Pd =p2 with high probability connect the atoms of Pd =p1 into a linear sized component, that includes at least h 2d , terminal nodes, where 0 < h < 1 is a constant. As in [KNT94], we assume that the edges of Pd =p2 connect the atoms into clumps. Then we can show that if no clump contains at least c 2d=3 terminal nodes and at least c(nd ? 2d )=3 nonterminal nodes, then the clumps can be partitioned into two groups, such that no edge of Pd =p2 connects the two groups, and each group has at least c2d =3 terminal nodes and at least c(nd ? 2d )=3 nonterminal nodes. 3
3
4
4
2
De nition 4 A pair of sets of nodes (C; D) is called a separation if each one of C and D has
size at least cnd =3, each atom is contained in either C or D, and C and D are disjoint. The separation survives if no path of Pd =p2 connects C and D.
Note that there are at most (nd )1? atoms and therefore, at most 2(nd ) ? separations. Since the probability that some separation survives is bounded by the number of separations times the maximum probability that a separation survives, our proof that the edges of Pd =p2 connect the atoms into a linear-sized component, will be completed by showing the following lemma. We will use the following fact, which can be easily seen: 1
Fact 6 Let S be a set of nodes in Pd such that both S and its complement are linear-sized. Let F (S ) be the set of edges with one endpoint in S and the other in the complement of S . Then jF (S )j (jS j1=2). Lemma 8 Let (C; D) be a separation. The probability that the separation survives is at most ? ( n ) d 2 , where > 1 ? . Proof: Our proof is similar to the one in [KNT94]. Let (C; D) be a separation as it is described above. Let C (D ) be the set of nodes whose distance from C (respectively, D) is at most d. Note that C [ D contains all the nodes. Let E = F (C nD). >From fact 6, jE j (n1d=2). Each edge in E has one endpoint in C and the other in D , and the endpoint A.2
in C (D ) is at distance at most d from some node in C (respectively, D). Thus, a path of length at most 2 d + 1 that can connect C and D is associated to each edge of E . But each edge can participate in at most0 2 d 3 d (2 d + 1) such paths. Therefore, we can choose at least jE j=(2 d + 1)22 d3 d n1d=2? ?o(1) edge-disjoint paths from the set, where 0 > 0 is a small constant. For suciently small, the probability that the separation survives is at most 0
(1 ? p22 d+1 )(nd) = ? ?o 2?(nd ) 1 2
(1)
Part 3: Below the threshold (p < pc) and a lower bound
2
Lemma 5 pc 0:42 Proof: Let kd be the number of simple paths of length d in Pd starting from a particular level-0 node v . Then, starting from v , the expected number of simple paths of length d in Pd =p is kd pd . Note that kd pd = (1) for p > pc . The bound kd 3d2d implies pc 0:4. By observing that for half of the steps, one out of the 27 possible ways of continuing a path from each intermediate node v in 3 steps leads to the predecessor of v forming a cycle, and for the other half steps, one out of the 8 possible ways leads to a cycle, we have that kd 26d=67d=6 . Therefore, pc 0:42. 2
A.3
