Measures of Node Importance for Quorum Structures

Measures of Node Importance for Quorum Structures Mitchell L. Neilsen

Masaaki Mizuno

Computer Science Department Oklahoma State University Stillwater, Oklahoma 74078

Dept. of Comp. and Info. Sci. Kansas State University Manhattan, Kansas 66506 Abstract

Quorum-based protocols can be formalized in terms of the data structures they use. These data structures, called quorum structures, include quorum sets, coteries, and bicoteries. Traditionally, weighted voting has been used to construct quorum structures. In order to obtain better performance, several researchers have proposed methods that impose a logical structure on the nodes. We have shown that these methods can be generalized by using composition. Several measures have been proposed to analyze quorum structures, including availability, communication latency, expected number of operational nodes, quorum size, node and edge vulnerability, reliability, and success rate. However, none of these measures can be used to measure the relative importance of each node in a quorum structure. In this paper, we present such a measure, called the Banzhaf index. By knowing the relative importance of each node in the system, it is possible to improve the overall system reliability by investing more eort on improving the reliability of important nodes. Secondly, it is frequently not possible to compute these measures directly for a quorum structure in a large system. However, we present a simple recursive method to compute the importance of nodes in quorum sets constructed by using composition. Finally, we propose a new measure that can be used to evaluate the importance of each node in a bicoterie, and show that if the bicoterie is nondominated, this measure reduces to nding the importance of each node in a quorum set. We prove the correctness of our recursive methods and show that they are very ecient. Index Terms : Banzhaf index, coteries, distributed computing, fault tolerance, mutual exclusion, performance, quorums.  This work was supported in part by the National Science Foundation under Grant CCR-9201645

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1 Introduction In distributed systems, quorum-based protocols are an important class of protocols. They gracefully tolerate node and communication line failures, and can be used in a large number of applications, such as mutual exclusion, replica control, leader election, termination detection, and name serving [6]. Several authors have formalized quorum-based protocols in terms of the data structures that are used by the protocols [6, 11, 13, 16]. These data structures, which we call quorum structures, are a generalization of an idea that was rst proposed by Lamport [19]. There are many methods available to construct quorum structures. One very well-known method, called quorum consensus, uses weighted voting [14]. In order to improve on the performance exhibited by quorum consensus, several researchers have proposed methods that impose a logical structure on the nodes. These methods include: Kumar's hierarchical quorum consensus protocol [17], Agrawal and El Abbadi's tree protocol and hybrid replica control protocols [1, 2], Rangarajan, Setia, and Tripathi's protocol [25], and Kumar and Cheung's hierarchical grid protocol [18]. We have proposed a more general method, called composition [20]. Composition generalizes all of the above protocols. Furthermore, if a distributed system is constructed from several networks connected by gateways, composition provides a natural method to construct a coterie for the entire system. Recently, Ibaraki and Kameda showed that when quorum structures are viewed as boolean functions, composition corresponds to the classical boolean function decomposition, called disjunctive decomposition, due to Ashenhurst [4, 16]. The same idea has been studied in other areas under dierent titles, including: compound simple games in game theory [24, 27], compound clutters in set theory [9], and modules in reliability theory [10, 24]. Several measures have been proposed to analyze coteries [3, 7, 8, 12, 22, 23, 26, 28, 29]. These measures include availability, communication latency, expected number of operational nodes, quorum size, node and edge vulnerability, reliability, and success rate. None of these measures address the relative importance of each node in the system. In this paper, we present such a measure, used in game theory, called the Banzhaf index. By knowing the relative importance of each node in the system, it is possible to improve the overall system reliability by investing more eort on improving the reliability of important nodes. Secondly, it is frequently not possible to compute these measures directly for a quorum structure in a large system. We present a simple recursive method to compute the Banzhaf index of composite quorum sets; that is, quorum sets constructed by using composition. Finally, we propose a new measure that can be used to evaluate the importance of each node in a bicoterie, and show that if the bicoterie is nondominated, this measure reduces to nding the importance of each node in a quorum set. We prove the correctness of our recursive methods and show that they are very ecient. The organization of this paper is as follows: Previous work is presented in Section 2. Ecient methods of evaluating composite structures are presented in Section 3. Finally, Section 4 summarizes the results presented.

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2 De nitions and Properties In this section, we review the de nitions of quorum structures and composition.

2.1 Quorum Structures

Several authors have de ned quorum structures that can be used in a wide variety of distributed protocols [6, 11, 13, 16]. In this section, these structures are de ned. Let U denote a non-empty set of nodes. The term node may refer to a computer in a network or a copy of some data object in a replicated database. The nodes are fully connected. A collection of sets, Q, is a quorum set under U if 1. (8G 2 Q) [ G 6= ; and G  U ]. 2. (Minimality): (8G; H 2 Q) [ G 6 H ]. The sets G 2 Q are called quorums. For example, let U = fa; b; c; dg. Then, Q = ffa; bg; fb; cgg is a quorum set under U . Note that not all nodes must appear in some quorum; in particular, node d does not appear in either quorum of Q. Nodes that appear in some quorum are called used nodes. A quorum set Q can be represented by the set of all subsets of U containing a quorum of Q. We call such a set the acceptance set corresponding to Q, and denote it by A(Q). A(Q) = fH  U j G  H for some G 2 Qg In the above example, with Q = ffa; bg; fb; cgg under U = fa; b; c; dg, the corresponding acceptance set A(Q) is given by A(Q) = ffa; bg; fb; cg; fa; b; cg; fa; b; dg; fb; c; dg; fa; b; c; dgg There is a one-to-one correspondence between acceptance sets and quorum sets. A quorum set, Q, is a coterie under U if the intersection property is satis ed; that is, (8G; H 2 Q) [ G \ H 6= ; ]. A coterie Q = fGg, containing a single quorum, is called a singleton coterie. Let Q and Q be coteries under U . Then, Q dominates Q if 1. Q 6= Q . 2. (8H 2 Q ) [ 9G 2 Q such that G  H ]. A coterie, Q under U , is dominated if there is another coterie under U that dominates Q. If there is no such coterie, then Q is nondominated. If Q is a nondominated coteire under U , then jA(Q)j = 2jU j? . Let Q be a quorum set under U . Then, a complimentary quorum set, Qc, is another quorum set under U such that (8G 2 Q) (8H 2 Qc) [ G \ H 6= ; ]. The pair B = (Q; Qc) is called a bicoterie under U . If Q or Qc is a coterie, then the pair B is called a semicoterie. Suppose that B = (Q ; Qc ) and B = (Q ; Qc ) are bicoteries under U . Then, B dominates B if 1

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1. B 6= B ; that is, Q 6= Q or Qc 6= Qc . 2. (8H 2 Q ) [ 9G 2 Q such that G  H ]. 3. (8H 2 Qc ) [ 9G 2 Qc such that G  H ]. A bicoterie, B under U , is dominated if there is another bicoterie under U that dominates B. If there is no such bicoterie, then B is nondominated. Let Q be a quorum set under U . A transversal of Q is a subset of U that intersects with all quorums in Q; that is, H  U is a transversal of Q if G \ H 6= ; for any G 2 Q. A minimal transversal is a transversal, H , such that any proper subset of H is not a transversal. The set of all minimal transversals of Q, denoted by Tr(Q), is a complementary quorum set. In the literature, Tr(Q) is also called the antiquorum set of Q, and the pair (Q; Tr(Q)) is called a quorum agreement [6]. It is easy to show that quorum agreements are the same as nondominated bicoteries. 1

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2.2 Composition

Composition provides a simple way of combining non-empty structures to construct new, larger structures. First, we discuss the composition of quorum sets. Then, we present properties satis ed by composition. Let U be a non-empty set of nodes and x 2 U . Let U be another non-empty set of nodes such that U \ U = ;. Let U = (U ? fxg) [ U . Given a quorum set Q under U and a quorum set Q under U , a new quorum set Q under U can be constructed by replacing each occurrence of x in quorums of Q by nodes in a quorum of Q . More formally, let QU denote the set of all non-empty quorum sets under Ui for i = 1; 2; 3, and de ne a function, Tx : QU1  QU2 ! QU3 , by Tx(Q ; Q ) = fQTx(G ; G ) j G 2 Q ; G 2 Q g where ( if x 2 G QTx(G ; G ) = (GG ? fxg) [ G otherwise 1

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A quorum set constructed by using such a function is called a composite quorum set; that is, Q = Tx(Q ; Q ) is a composite quorum set. The node x is called a logical node and quorum sets Q and Q are called input quorum sets. For example, suppose that U = f1; 2; ag, x = a, and U = f3; 4; 5g. De ne the input quorum sets, Q under U and Q under U , as follows Q = ff1; 2g; f1; ag; f2; agg Q = ff3; 4g; f3; 5g; f4; 5gg Then, Ta (Q ; Q ) = Q , where Q is a quorum set under U = f1; 2; 3; 4; 5g, and Q is constructed by replacing each occurrence of x = a in quorums of Q by nodes in a quorum of Q ; that is Q = ff1; 2g; f1; 3; 4g; f1; 3; 5g; f1; 4; 5g; f2; 3; 4g; f2; 3; 5g; f2; 4; 5gg 3

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Note that the above quorum sets, Q ; Q , and Q , are all nondominated coteries. This is no accident. In particular, if the input quorum sets are nondominated coteries, then the resulting composite quorum set is also a nondominated coterie [20]. Composition can also be used to construct bicoteries. Suppose that B = (Q ; Qc ) is a bicoterie under U and B = (Q ; Qc ) is a bicoterie under U . Let Q = Tx(Q ; Q ) and Qc = Tx(Qc ; Qc ). Finally, let B = (Q ; Qc ). Then, B is a bicoterie under U . Furthermore, if B and B are both nondominated, then B is also nondominated. These properties are similar to some of the more general properties presented by Ibaraki and Kameda [15, 16]. 1

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3 Measures of Power In this section, we present measures that can be used to evaluate the relative importance of each node in the system. In the literature on reliability theory, this type of measure is called a measure of structural importance [24]. Intuitively, if an individual node failure will cause the system to fail, then that node is powerful. On the other hand, if the system can continue to operate, even if a given node fails, then that node is less powerful. We start with some de nitions. Let U be a non-empty set of nodes such that N = jU j. Let Q be a quorum set under U . A critical set of Q for node j 2 U is a set G 2 A(Q) such

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1. j 2 G. 2. G ? fj g 62 A(Q). The number of critical sets of Q of size r for node j is denoted by cs(Q; r; j ). The total number of critical sets of Q for node j is denoted by nc(Q; j ); that is N X nc(Q; j ) = cs(Q; r; j ) r N ? Note that 0  nc(Q; j )  2 . The extreme cases occur when node j is not used (nc(Q; j ) = 0) and when Q is a singleton coterie (Q = ffj gg ) nc(Q; j ) = 2N ? ). In reliability theory, a critical set is called a critical path vector and in game theory, a critical set is called a swing [24]. In 1965, Banzhaf proposed an index to help solve legal battles concerning constitutional fairness [5]. The index measures the relative power of each player in a simple game. Unlike other power indices, this index is based on the assumption of node (player) independence. Thus, it is well suited for measuring the power of each node in a distributed system if we assume that node failures are independent. Let Q be a quorum set under a non-empty set U with N = jU j. Let j 2 U . The Banzhaf index (Q; j ) of node j is de ned by (Q; j ) = nc(Q; j )=2N ? Thus, (Q; j ) is a measure of the relative importance of node j . Note that the denominator 2N ? is just a scaling factor. =1

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3.1 On quorum sets and coteries 3.1.1 Method

This section describes an ecient method to compute the Banzhaf index of a node in a composite quorum set. Let U be a non-empty set of nodes and let x 2 U . Let U be a non-empty set of nodes such that U \ U = ;. Let U = (U ? fxg) [ U . Let Q = Tx(Q ; Q ), where Qi is a quorum set under Ui for i = 1; 2; 3. Let Ax(Q ) = fG 2 A(Q )jx 2 Gg. Let ncx(Q ; j ) = jfG 2 Ax(Q ) j j 2 G; G ? fj g 62 Ax(Q )gj 1

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That is, ncx(Q ; j ) is the number of critical sets of Q for node j that contain node x. Let ncx(Q ; j ) = nc(Q ; j ) ? ncx(Q ; j ); that is, ncx(Q ; j ) is the number of critical sets of Q for node j that do not contain node x. Finally, let A(Q ) = P (U ) ? A(Q ). Then, the index (Q ; j ) of node j 2 U is computed as follows 1

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(Q ; x) (Q ; j ) if j 2 U (Q ; j ) = (nc   j U j? 3 ) otherwise x(Q ; j )jA(Q )j + ncx(Q ; j )jA (Q )j)=(2 1

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If Q is a nondominated coterie, the method simpli es as follows 2

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In the next section, we prove the correctness of this method.

3.1.2 Correctness

To show that this method is correct, we need the following lemmas. The rst lemma describes how to decompose the acceptance set.

Lemma 1 [21]: The resulting acceptance set A(Q ) is given by A(Q ) = Tx(Ax(Q ); A(Q )) [ (Ax(Q ) A(Q )) where Ax(Q ) = A(Q ) ? Ax(Q ) and (Ax(Q ) A(Q )) = fG [ G jG 2 Ax(Q ); G 2 A(Q )g. Furthermore, Tx(Ax(Q ); A(Q )) \ (Ax(Q ) A(Q )) = ;: 3

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The second lemma describes how to compute the number of critical sets.

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Lemma 2: The(number of critical sets of Q for node j is given by (Q ; x) nc(Q ; j ) if j 2 U nc(Q ; j ) = nc   ncx(Q ; j )jA(Q )j + nc (Q ; j )jA (Q )j otherwise 3

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Furthermore, if Q is a nondominated coterie under U , ( (Q ; x) nc(Q ; j ) if j 2 U nc(Q ; j ) = nc nc(Q ; j ) 2jU2 j? otherwise 2

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Proof: Recall that A(Q ) = Tx(Ax(Q ); A(Q )) [ (Ax(Q ) A(Q )) by Lemma 1. There are two cases to consider: either j 2 U or it is not. Suppose that j 2 U . First, we will show that nc(Q ; j )  nc(Q ; x) nc(Q ; j ). Let G be a critical set of Q for node j . We claim G 2 Tx(Ax(Q ); A(Q )). Assume not; then, G 2 (Ax(Q ) A(Q )). Since ; 2 A(Q ) and j 2 U , it follows that G ?fj g 2 (Ax(Q )

A(Q )). So, G ? fj g 2 A(Q ), and this is a contradiction. Thus, G 2 Tx(Ax(Q ); A(Q )). So G = (G ? fxg) [ G for some unique G 2 Ax(Q ) and some unique G 2 A(Q ). Furthermore, j 2 G . Now we will show that G and G are critical sets. 1. To show that G is a critical set of Q for node x, we only need to show that (G ?fxg) 62 A(Q ) because x 2 G and G 2 Ax(Q )  A(Q ). Assume that (G ? fxg) 2 A(Q ). Then, (G ?fxg) 2 A(Q ). Since (G ?fxg)  (G ?fj g), it follows that (G ?fj g) 2 3

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A(Q ), and this is a contradiction. Thus, G is a critical set of Q for node x. 2. To show that G is a critical set of Q for node j , we only need to show that (G ? fj g) 62 A(Q ). Assume that (G ? fj g) 2 A(Q ). Then, (G ? fj g) 2 A(Q ) because (G ? fj g)  (G ? fj g). This is a contradiction. Thus, G is a critical set of Q for node j . 3

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Therefore, nc(Q ; j )  nc(Q ; x) nc(Q ; j ). 3

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Next, we will show that nc(Q ; x) nc(Q ; j )  nc(Q ; j ). 1

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Let G be a critical set of Q for node x, and let G be a critical set of Q for node j . Then, there exists a unique G 2 A(Q ) such that G = (G ? fxg) [ G . Since j 2 G , it follows that j 2 G . So, we only need to show that (G ? fj g) 62 A(Q ). Assume that (G ?fj g) 2 A(Q ). Since (G ?fj g) 62 A(Q ), it follows that (G ?fj g) 2 (Ax(Q ) A(Q )). So, (G ? fxg) 2 Ax(Q )  A(Q ). This is a contradiction because G is a critical set of Q for node x. Thus, (G ? fj g) 62 A(Q ), and G is a critical set of Q for node j . 1

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Therefore, nc(Q ; x) nc(Q ; j )  nc(Q ; j ). 1

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By combining the above two results, we obtain nc(Q ; j ) = nc(Q ; x) nc(Q ; j ) if j 2 U . 3

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Now, we will consider the second case; that is, j 62 U . Then, j 2 (U ? fxg). Let G be a critical set of Q for node j . There are two cases to consider: 2

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1. Suppose that G 2 Tx(Ax(Q ); A(Q )). Then, G = (G ? fxg) [ G for some unique G 2 A(Q ) and some unique G 2 U . We want to show that G is a critical set of Q for node j . Assume that (G ? fj g) 2 A(Q ). Then, (G ? fj g) 2 A(Q ), and this is a contradiction. Therefore, G is a critical set of Q for node j . 2. Suppose that G 2 (Ax(Q ) A(Q )). Then, G = G [ G for some unique G 2 Ax(Q ) and some unique G 2 A(Q ). We want to show that G is a critical set of Q for node j . Assume that (G ? fj g) 2 A(Q ). Then, (G ? fj g) 2 Ax(Q ). So, (G ? fj g) 2 A(Q ), and this is a contradiction. Therefore, G is a critical set of Q for node j . 1

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Thus, nc(Q ; j )  ncx(Q ; j )jA(Q )j + ncx(Q ; j )jA(Q )j. 3

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On the other hand, let G be a critical set of Q for node j . There are two cases to consider: 1

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1. Suppose that x 2 G . Let G 2 A(Q ). Let G = (G ? fxg) [ G . We want to show that G is a critical set of Q for node j . Assume that (G ? fj g) 2 A(Q ). Then, (G ? fj g) 2 A(Q ), and this is a contradiction. Thus, G is a critical set of Q for node j . 2. Suppose that x 62 G . Let G 2 A(Q ), and let G = G [ G . We want to show that G is a critical set of Q for node j . Clearly, it follows that G 2 A(Q ) and j 2 G . Assume that (G ? fj g) 2 A(Q ). Then, (G ? fj g) 2 Ax(Q )  A(Q ). So, (G ? fj g) 2 A(Q ), and this is a contradiction. Thus, G is a critical set of Q for node j . 1

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Thus, ncx(Q ; j )jA(Q )j + ncx(Q ; j )jA(Q )j  nc(Q ; j ) if j 62 U . 1

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By combining the above two results, we obtain nc(Q ; j ) = ncx(Q ; j )jA(Q )j + ncx(Q ; j )jA(Q )j 3

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and the proof of the rst part is complete. If Q is a nondominated coterie under U , then jA(Q )j = jA(Q )j = 2jU2 j? . Since nc(Q ; j ) = ncx(Q ; j ) + ncx(Q ; j ), ncx(Q ; j )jA(Q )j + ncx(Q ; j )jA(Q )j = nc(Q ; j ) 2jU2 j? 2 2

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Theorem 1: Suppose that Q = Tx(Q ; Q ) is a composite quorum set as given above. 3

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Then, the Banzhaf index can be computed recursively as follows ( (Q ; x) (Q ; j ) if j 2 U (Q ; j ) = (nc   j U j? 3 ) otherwise x(Q ; j )jA(Q )j + ncx(Q ; j )jA (Q )j)=(2 1

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Furthermore, if Q is a nondominated coterie under U , 2

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  3 2        8 7 6  5  4    

  

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Figure 1: Tree

(Q ; j ) =

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Proof: Note that jU j = jU ? fxgj + jU j = (jU j ? 1) + jU j. So, (jU j ? 1) = (jU j ? 1) + (jU j ? 1) 3

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and

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2 jU3j? = 2 jU1j? 2 jU2j? By dividing both sides of the equations in Lemma 2 by 2 jU3 j? , we obtain the desired result. (

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3.1.3 Example: tree protocol

The tree protocol is a method to construct coteries [2]. The set of N nodes are logically arranged in a complete binary tree. A path in the tree is a sequence of nodes a ; a ;


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; aj such that ai is a child of ai. A quorum is constructed by grouping all nodes on a path from the root node down to a leaf node. If a node on the path is not available, then paths that start at both children and terminate at leaf nodes can be used instead. They state that any k-ary tree, with k  2, can be used. In fact, we have shown that the protocol can be applied to any tree in which each nonleaf node has at least two children and that coteries constructed by using the protocol are always nondominated [20]. We refer to the resulting coteries as tree coteries. Consider the tree shown in Figure 1. The resulting tree coterie is given by 1

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Q = ff1; 2; 4g; f1; 2; 5g; f1; 2; 6g;


; f2; 6; 7; 8g; f4; 5; 6; 7; 8gg Tree coteries can be formally described by using composition. Let U = fa ; a ;


; ang be a set of n  3 nodes. We de ne a tree coterie of depth two under U by Q = ffa ; aj g j 2  j  ng [ ffa ; a ;


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Table 1. Banzhaf Index j Q nc(Q; j ) (Q; j ) 1, a, b Q 2 1=2 2 4,5,6 3,7,8 1 2 3,7,8 4,5,6

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Qa Qa Qb Q Q Q Q

3=4 1=4 1=2 1=2 3=8 1=4 1=8

6 2 2 64 48 32 16

Node a is viewed as the root node and the remaining nodes are viewed as leaf nodes in the tree. Note that tree coteries of depth two can be de ned by using quorum consensus. Let v(a ) = n ? 2 and v(aj ) = 1 for 2  j  n. Then, TOT(v) = 2n ? 3 is odd. If we set q = MAJ(v) = n ? 1, then the resulting coterie is Q, and Q is nondominated by Theorem 3.2 in [13]. Tree coteries are constructed by repeatedly composing tree coteries of depth two together at one of the leaf nodes. Thus, any tree in which each nonleaf node has at least two children can be constructed. Since tree coteries of depth two are nondominated, it follows from Theorem 3.2 in [20] that all tree coteries are nondominated. For example, the above tree coterie can be represented by composing the following three tree coteries of depth two Q = ff1; ag; f1; bg; fa; bgg Qa = ff2; 4g; f2; 5g; f2; 6g; f4; 5; 6gg Qb = ff3; 7g; f3; 8g; f7; 8gg Let Q = Ta(Q ; Qa), and Q = Tb(Q ; Qb). Then, Q under U = f1; 2;


; 8g is the tree coterie corresponding to the tree shown in Figure 1. For example, G = f1; 2; 4; 5g is a critical set of Q for nodes 1 and 2, but it is not a critical set for nodes 4 and 5. The acceptance sets for Q ; Qa, and Qb are A(Q ) = ff1; ag; f1; bg; fa; bg; f1; a; bgg A(Qa) = ff2; 4g; f2; 5g; f2; 6g; f4; 5; 6g; f2; 4; 5g; f2; 4; 6g; f2; 5; 6g; f2; 4; 5; 6gg A(Qb) = ff3; 7g; f3; 8g; f7; 8g; f3; 7; 8gg By applying the de nitions and the above simpli ed method, we obtain the following results shown in Table 1. For example, nc(Qa; 4) = 2 because the sets f2,4g and f4,5,6g are critical sets of Qa for node 4. Thus, (Qa; 4) = 1=4. Similarly, nc(Q ; a) = 2 and (Q ; a) = 1=2. Since node 4 2 Ua , 1

1

1

2

1

2

1

1

1

1

nc(Ta(Q ; Qa ); 4) = nc(Q ; a) nc(Qa; 4) = 4: 1

1

10

Also, since node 4 62 Ub,

nc(Q; 4) = nc(Tb(Ta(Q ; Qa); Qb); 4) = nc(Ta(Q ; Qa); 4) 2jUbj? = 16: 1

Similarly,

1

1

(Ta(Q ; Qa); 4) = (Q ; a) (Qa; 4) = 1=8 1

and

1

(Q; 4) = (Tb(Ta (Q ; Qa); Qb); 4) = (Ta(Q ; Qa); 4) = 1=8: From Table 1, we can observe that node 1 is four times as powerful as nodes 4, 5, and 6 because (Q; 1) = 1=2 and (Q; j ) = 1=8 for j = 4; 5; 6. Thus, in order to improve the overall reliability of the system, we would want node 1 to be more reliable than the other nodes. 1

1

3.1.4 Complexity

Let Q be a nondominated coterie under a nonempty set of nodes U , with N = jU j. The de nition of nc can be applied directly to compute (Q; j ). However, to compute nc by using the de nition requires a lot of work. A total of jA(Q)j = 2N ? sets must be checked to see if they are critical; that is, for each G 2 A(Q), a check must be made to see if (G ? fj g) 2 A(Q). Since there are N nodes, each test for membership in A(Q) requires comparing N binary digits. So, the time complexity is O(N 2 N ? ) to check all 2N ? sets in A(Q). On the other hand, suppose that Q is a composite coterie constructed by composing M nondominated coteries Q ; Q ;


; QM de ned under pairwise disjoint sets U ; U ;


; UM , respectively. In order to compute the number of critical sets in Qi for node j , at most jUij 2 jU j? comparisons are required. Thus, the number of comparisonsPrequired to compute nc(Q; j ) or (Q; j ), by using the above recursive method, is at most Mi (jUij 2 jU j? ). Since we are only interested in starting with relatively small coteries, we assume that all input coteries are de ned under sets of size at most k for some constant k; that is, 2  jUij  k for 1  i  M , and k