Choosing a heuristic and root node for edge ordering in BDD-based network reliability analysis

Author's Accepted Manuscript

Choosing a heuristic and root node for edge ordering in BDD-based network reliability analysis Yuchang Mo, Liudong Xing, Farong Zhong, Zhusheng Pan, Zhongyu Chen

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S0951-8320(14)00155-0 http://dx.doi.org/10.1016/j.ress.2014.06.025 RESS5085

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Reliability Engineering and System Safety

Received date: 18 November 2013 Revised date: 18 June 2014 Accepted date: 26 June 2014 Cite this article as: Yuchang Mo, Liudong Xing, Farong Zhong, Zhusheng Pan, Zhongyu Chen, Choosing a heuristic and root node for edge ordering in BDDbased network reliability analysis, Reliability Engineering and System Safety, http: //dx.doi.org/10.1016/j.ress.2014.06.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Choosing a Heuristic and Root node for Edge Ordering in BDD-based Network Reliability Analysis Yuchang Moa, Liudong Xingb, Farong Zhonga, Zhusheng Pana, Zhongyu Chena a

Dept. of Computer Science & Technology, Zhejiang Normal University, China

b

Dept. of Electrical & Computer Engineering, University of Massachusetts, Dartmouth, USA E-mail: [email protected], [email protected], [email protected], [email protected], [email protected]

Abstract: In the binary decision diagram (BDD)-based network reliability analysis, heuristics have been widely used to obtain a reasonably good ordering of edge variables. Orderings generated using different heuristics can lead to dramatically different sizes of BDDs, and thus dramatically different running times and memory usages for the analysis of the same network. Unfortunately, due to the nature of the ordering problem (i.e., being an NP-complete problem) no formal guidelines or rules are available for choosing a good heuristic or for choosing a high-performance root node to perform edge searching using a particular heuristic. In this work, we make novel contributions by proposing heuristic and root node selection methods based on the concept of boundary sets for the BDD-based network reliability analysis. Empirical studies show that the proposed selection methods can help to generate high-performance edge ordering for most of studied cases, enabling the efficient BDD-based reliability analysis of large-scale networks. The proposed methods are demonstrated on different types of networks, including square lattice networks, torus lattice networks and de Bruijn networks.

Key words: network reliability; binary decision diagram; edge ordering heuristic; lattice network, de Bruijn network.

1

Acronyms BDD

Binary Decision Diagrams

BFS

Breadth First Search

TSBS

Total Size of Boundary Set

IFBSH

Index of First Boundary Set Hit

DFS

Depth First Search

NDS

Network Driven Search

MP

Minimal Path

MC

Minimal Cut

UGF

Universal Generating Function

MCS

Monte Carlo Simulation

Notation G

network

V

node set of network G

ni

a node of node set V

s,t

terminal nodes

E

edge set of network G

ei

a edge of edge set G

O

an edge ordering on E

e(i)

the i-th edge in ordering O

Fi

boundary set of first i edges in ordering O

L

the maximum size of the boundary sets

hi

an edge ordering heuristic

H

an ordering heuristic library H ={h1, h2,…, }

Oni

ordering generated using root node ni∈V

Ohi

ordering generated using hi∈H 2

TSBS(O)

total size of boundary sets of an edge ordering O

IFBSH (O)

IFBSH measure of an edge ordering O

SelTSBS

TSBS-based selection method

SelIFBSH

IFBSH-based selection method

3

1. Introduction Networks are a widely-applied model for analyzing many complex systems such as physical, technological, social, biological and economical systems. Because networks usually display a high degree of tolerance to random failures and attacks due to redundant paths, network reliability analysis has been a challenging task for last three decades. Monte Carlo Simulations (MCS) have been applied for approximate network reliability analysis [1-5]. MCS is performed based on random samples of hazard intensities and corresponding network component responses. It has been shown that for highly reliable networks the crude MCS method is impractical, because the probability of network failure is a rare-event probability [1]. The research for efficient MCS-based algorithms for highly reliable networks has resulted in a number of variance reduction methods. Among the most prominent ones are conditional MCS approaches [2], approximate zero-variance importance sampling [3], and combinations of these two [4]. Refer to [5] for a survey of the MCS-based methods. The simulation-based approaches are often used for relatively large networks because their computational efficiency depends largely on the convergence of probability not the size of networks. However, statistical errors in the simulation-based approaches may result in slow convergence for achieving acceptable accuracy in low probability estimations and in parameter sensitivity calculations. On the other hand, the exact evaluation of network reliability is an NP-hard problem [6]. Numerous studies were devoted to analyze the network reliability based on minimal paths/cuts (MPs/MCs), where all the possibilities for which two specified nodes can communicate (or not communicate) are first enumerated (path/cut set search) and then the reliability expression is evaluated, resorting to different techniques, like inclusion-exclusion method [7, 8] or sum of disjoint products [9]. These traditional MPs/MCs based algorithms can have doubly exponential complexity and thus are not efficient for large-scale networks. Among the efforts to improve the computational efficiency of the traditional MPs/MCs based method, the recursive decomposition algorithm was proposed [10, 11], which integrates a shortest path search algorithm and a disjoint

4

path identification process to efficiently calculate the reliability of a node pair connection. In [12, 13] a matrix-based method was also developed, which calculates the probability of general system events with correlated system components based on efficient matrix manipulations and MC identification. In [14], Chang and Song integrated the methods of [10-13] for the reliability analysis of a real-world network, where a smaller number of non-minimal disjoint sets are identified to fully represent the network connectivity. Although the methods in [10-14] made improvement in the computational efficiency, they still need to deal with the exponentially increasing number of MPs/MCs to find the node-pair reliability/unreliability. As another approach, Levitin [15] proposed to evaluate the network reliability using universal generating functions (UGF)-based recursive procedure. This method has recently been improved by Yeh through some simplification techniques [16, 17]. A survey of recent developments of UGF based methods in the field of network reliability estimation can be found in [18]. The UGF based methods can improve the computational efficiency of MP/MC identification or inclusion-exclusion calculation, but they still suffer from computational inefficiency on large-scale networks. In the last few decades Binary Decision Diagrams (BDD), as an extraordinarily efficient data structure to represent and manipulate Boolean functions [19, 20], have been exploited to model and analyze the reliability of many different classes of systems, such as multi-state systems [21], dynamic systems [22, 23], phased-mission systems [24, 25]. Among these efforts, BDD-based approaches for the reliability analysis of large-scale network were proposed in [26-29] and they involve three main steps: 1) Order the network components by applying a good heuristic ordering approach, 2) Generate a BDD model from the probabilistic graph of the given network, and 3) Evaluate the BDD to obtain the network reliability. The algorithms to construct the BDD from the probabilistic graph of a given network in [26-29] can be classified into three classes. The first class of algorithms involves constructing an edge expansion diagram of a given network by recursively expanding edges of the source node for each sub-graph. The success path function and the equivalent BDD are then constructed by traversing the graph with the edge expansion diagram [26, 27]. The second class of algorithms is based on a suitable factorization algorithm [28], consisting in 5

pivoting along a complete sequence of edges and deriving the right sub-graph when the edge functions (i.e., the edge is contracted) and the left sub-graph when the edge fails (i.e., the edge is deleted). The third class of algorithms constructs the BDD recursively until the sink node is reached [29]. The BDDs of nodes along a path from source node to sink node are combined using logic AND operation. If a node possesses more than one outgoing edge, the BDDs of the paths starting from each edge are combined using logic OR operation. It is of great interest to generate BDDs as small as possible; the smaller the BDDs, the more efficient BDD construction and subsequent operations (e.g., enumeration of minimal cutsets, and reliability assessment) are. The size of a BDD modeling network reliability heavily depends on the chosen edge ordering. Empirical studies show that there is often a variation of several orders of magnitude between sizes of BDDs of the same network built using different orderings [30]. However, finding the best ordering for a relatively large network is an intractable task in theory [31]; heuristics have been used to obtained good orderings for BDD generation. The ordering heuristics commonly used to generate BDDs for computing network reliabilities are typically based on search schemes such as breadth-first search (BFS) [26-30], depth-first search (DFS) [30], and network-driven search (NDS) [32]. Some piecemeal performance comparisons between different heuristics have been made. For example, in [30], comparisons between BFS and DFS heuristics have been performed and it was found that the BFS heuristic outperforms the DFS heuristic for most of example networks under study. However, no formal guidelines or rules are available for selecting an adequate heuristic from the existing heuristic library. Different people can make different decisions on the heuristic selection, leading to dramatically different sized BDD models. For a specific example, consider the Boolean function f=x1x2+ x3x4 + … + x2n-1x2n. Using the variable ordering where all the variables with odd indices are before variables with even indices, i.e., x1
6

x1 1

x2 0

x3 1

x4 0

x5 x6 0

(a)

(b)

0

1

1

Fig.1. BDD for the function f=x1x2+ x3x4 + x5x6 using (a) bad ordering (b) good ordering Application of a heuristic to a network graph for generating variable ordering starts with the selection of a root node. As shown in [33], the sizes of BDD built using the same heuristic but different root nodes for the same network can also be dramatically different with a variation of several orders of magnitude. Empirical study was performed to study the relationship between the root node selection and efficiency of BDD analysis in [33]. But no formal guidelines exist for selecting a high-performance root node for a given ordering heuristic. In this work, we make novel contributions by proposing heuristic and root node selection methods for efficient BDD-based reliability analysis of networks with perfectly reliable nodes. Empirical studies of different types of undirected networks are conducted to illustrate the implementation and performance of the proposed selection methods. The rest of this paper is organized as follows: Section 2 introduces the mathematical background of the proposed selection methods. Section 3 reviews three widely-used edge ordering heuristics: BFS, DFS and NDS. Section 4 is devoted to the development of a boundary set-measure based selection method for choosing the ordering heuristic. Experimental data on applying the proposed selection method to lattice networks and de Bruijn networks are also presented. Section 5 presents boundary set measure-based selection methods for choosing high-performance root nodes. Finally, Section 6 gives conclusions and directions of our future work. 7

2. Mathematica M al backgroun nd Inn network reliability r anaalysis, a netw work is typiically modeleed using an undirected probabbilistic graph G= (V, E), with V as its nodde set and E V ×V as its eddge set. For coonvenience, we naame a node paair as a edge ek. For example, thhe 3×3 lattice network in Fig.2 can be repressented by G = ({n0, n1, n2, n3, n4, n5, n6, n7, n8}, {e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12}). 0

1

1

3 3

4

6

7

9 11

7

2 5

4

8 6

2

5 10

12

8

Fig.2. A 3×3 lattice nettwork E Each edge ek of o the probabilistic graph G is subject to o failure withh a known proobability qk, 0≤qk≤1. ≤ pk =1- qk thhus represents the probabilityy that edge ek functions. Wee assume that all the edge failuree events are sttatistically inddependent andd all the nodess are perfectlyy reliable in thhis work. A subgraaph of G= (V, E) is also a graph g G’= (V’,, E’), with V’ V as its nodee set and E’ E∩(V’ E ×V’) as its edge e set. W now introoduce two graph We g operatioons: edge deeletion and eedge contracttion, which respecctively lead to subgraph G-k obtained by deleting edge ek from G and ssubgraph G*k obtained o by contraacting ek from m G (the nodee pair of ek arre merged intto one node). When edge ek fails the netwoork behavior iss equivalent to o G-k; when ek functions the network behaavior is equivaalent to G*k. Basedd on this deccomposition, the t reliability of network represented bby G can bee calculated recurssively using R(G)= R pi *R((G*k) + qi * R(G R -k)

(1)

S Suppose the contraction c o deletion opperation is appplied to edgges in the orrder of O= or e(1)
by the operations specified by the path from the root to the node. Further, two reduction rules (“deletion of useless nodes” and “merging isomorphic nodes”) can be applied to modify the binary spanning tree as a rooted acyclic graph, that is a BDD, which contains no node whose outgoing edges point to the same child node (referred to as a useless node) and which does not contain two distinct nodes e(i) and e(j) such that the subgraphs rooted at e(i) and e(j) are isomorphic. Fig.3. gives the binary spanning tree representing the whole contraction (left solid arrows) and deletion (right dashed arrows) process for all-terminal reliability of a complete graph K3 and its BDD

2

1

representation.

3

2

2 3

3

3

3

3

1

(a)

0

1

0

1

0

(b)

Fig.3. All-terminal reliability of complete graph K3 (a) Spanning tree (b) BDD model

The size of a BDD is defined to be the number of non-sink nodes, i.e., the number of created non-isomorphic subgraphs in it. The width of the BDD is defined to be the maximum among the numbers of non-sink nodes at each level. The depth of the BDD is |E|, i.e., the number of edges. Suppose Ek ={e(1), e(2),…, e(k-1)} and ¬Ek ={e(k), e(k+2),…, e(|E|)}. Then the subgraphs of G in the kth-level have the edge set ¬Ek. For example, the subgraphs of G in the top level (level 1) have the edge set ¬E1, and the subgraphs of G in the second level (level 2) have the edge set ¬E2. For k=1, 2,…, |E|, define Fk as the set of nodes incident to some edges both in Ek and ¬Ek. Fk is referred to as

9

a boundary set. For example, assume that the considered edge ordering of the 3×3 lattice network in Fig.2 is e1
the Bell number and calculated as BL = ∑ AL, j where AL,, called the Stirling number, is the number j=1

of partitions of j blocks that can be made with L elements. Theorem 2 [28]: For K-terminal network reliability, let L be the maximum size of the L

min(|K |, j )

j=1

l =0

boundary set. Then, the width of a BDD of a computation process is bounded by ∑ ( AL , j i ∑

⎛ j ⎞⎟ ⎜⎜ ⎟⎟) ⎜⎝ l ⎟⎠

.

As discussed above, the concept of boundary sets has been widely explored in the BDD construction process. However their application in edge ordering has not been fully studied. In this work, we make novel contributions by proposing heuristic and root node selection methods based on the concept of boundary sets and performing empirical studies on different types of undirected networks to illustrate the performance of the proposed selection methods.

3. Edge Ordering Heuristics The size of a BDD encoding the binary spanning tree heavily depends on the chosen edge ordering. However, the problem of finding the best ordering for a network has been proved to be a 10

NP-complete problem [31]. Thus, heuristics have been used to obtained good orderings for BDD generation. As stated in Section 1, the heuristic library used in this paper includes three widely-used edge ordering heuristics: DFS, BFS and NDS. 2.1 Heuristic #1 - DFS The DFS begins at a root node and explores as far as possible along each branch before backtracking. In the implementation, edge ordering with DFS uses a last-in-first-out stack data structure to store intermediate results as it traverses a network graph. Consider a 3×3 lattice network where all the network nodes are numbered in a left-right, top-down manner, starting with 0. Fig.4 illustrates the DFS ordering results on all the edges using node 0 as the root node. Note that the successors of the currently processed node are chosen according to the increasing numbers of the nodes during the DFS search process. For example, node 0 has two successor nodes 1 and 3, node 1 is processed before node 3.

Fig.4. Ordering results with DFS and node 0 being root node 2.2 Heuristic #2 - BFS A key difference between BFS and DFS is the order in which the discovered (adjacent) nodes are explored. The BFS begins at a root node and inspects all its successor nodes. Then for each of these successor nodes in turn, it inspects their successor nodes that remain unvisited, and so on. In the implementation, edge ordering with BFS uses a first-in-first-out queue data structure to store intermediate results as it traverses a network graph. Fig. 5 illustrates the BFS ordering results for a 3×3 lattice network using node 0 as the root node. The successors of the currently processed node are processed in the order of their increasing numbers.

11

Fig.5. Ordering results with BFS and node 0 being root node 2.3 Heuristic #3 - NDS The NDS heuristic was proposed and used by Dutuit, Rauzy and Signoret [32]. It follows the common-sense rule that “two variables that are dependent of each other should be close in ordering". The root node is selected first. The node selected next is the one that has the greatest number of adjacent nodes already indexed or the one that minimizes the sum of indices of its adjacent nodes in case of a tie. Each edge is ordered as soon as its two adjacent nodes are chosen. Fig. 6 illustrates the NDS ordering results for a 3×3 lattice network using node 0 as the root node.

Fig.6. Ordering results with NDS and node 0 being root node

4. Proposed Heuristic Selection Method A boundary set measure based selection method is developed in this section to choose the best heuristic from a given library of edge ordering heuristics for a network reliability problem. Applications of the proposed selection method to different types of network reliability problems are illustrated. 4.1 TSBS-based Selection Method We present a boundary set measure called TSBS for heuristic selection, which can be formally defined as follows:

12

Measure 1 (TSBS): The total size of boundary sets (TSBS) of an edge ordering O=e(1)
TSBS (O) = ∑ | Fi |

(2)

i =1

For example, assume that the edge ordering of the 3×3 lattice network in Fig.2 is O=e1
Fi

|Fi |

e1

{}

0

e2

{0,1}

2

e3

{0,1,2}

3

e4

{1,2,3}

3

e5

{2,3,4}

3

e6

{3,4,5}

3

e7

{3,4,5}

3

e8

{3,4,5}

3

e9

{4,5,6}

3

e 10

{5,6,7}

3

e 11

{6,7,8}

3

e 12

{7,8}

2

Sum

31

TSBS-based Heuristic Selection Method: Given an ordering heuristic library H={h1, h2,…, hL } and a network reliability problem. The TSBS based heuristic selection method can be expressed as follows:

SelTSBS ( H ) = hi ⇔ TSBS (Ohi ) = min {TSBS (Ohj ),1 ≤ j ≤| H |}

(3)

where the ordering generated by hi∈H is denoted as Ohi. In words, the selected ordering heuristic from H is the one that generates ordering leading to the minimum TSBS.

13

4.2 Applications and Performance Evaluation In this section we illustrate the performance of the proposed TSBS-based selection method on the reliability analysis of de Bruijn networks and lattice networks. The heuristic library considered in this work is H={DFS, BFS, NDS}. 4.2.1 De Bruijn Networks De Bruijn networks have applications in a variety of different areas [36, 37]. They allow to construct large networks of fixed degree and small diameter, and they can also be used to design dynamic networks. For example, nodes of a complex dynamic network such as Internet can organize themselves into a constant-degree logarithmic-diameter de Bruijn network. A de Bruijn graph or network of order n has 2n nodes labeled with a binary bit representation of length n of the numbers 0, 1, …, 2n-1. Two nodes are connected iff the label of one is the left- or right-shifted label of the other, or it is the left- or right-shifted label of the other and differs only in the first or last bit. As a result, there are 4 edges at each node, and there are totally 2n+1 edges in the network. Fig.7 illustrates a de Bruijn network of order 4.

Fig.7. An example de Bruijn networks of order 4

Empirical study is performed on the de Bruijn network of order 4 in Fig. 7 by following steps listed below: 1) Randomly select four root nodes: {1, 7, 10, 15}; 2) For each heuristic in the library {DFS, BFS, NDS} and root node in {1, 7, 10, 15}, generate an edge ordering Oi, 1≤i≤ 3*4=12, and then calculate the TSBS measure of this ordering TSBS(Oi) 14

(Table 2); 4) Randomly select eight terminal pairs: {(0, 5); (1, 13); (3, 6); (5, 7); (7, 9); (10, 13); (12, 14); (14, 15)} from the example network; 3) For each terminal-pair reliability problem, generate an equivalent BDD using each edge ordering Oi. Table 3 collects the BDD size data for all the combinations. Based on the collected experimental data in Table 2 and Table 3, it can be observed that for the de Bruijn network the smaller the TSBS value of an ordering heuristic, the smaller the BDD model generated using the corresponding ordering. The smallest BDD is generated by using the NDS heuristic and root node 1, which corresponds to the smallest TSBS value in Table 2. It is also observed that the NDS heuristic performs better than the BFS and DFS heuristics for all the cases.

Table 2. TSBS results for the de Bruijn network of order 4 Root node #

DFS

BFS

NDS

1

159

136

112

7

152

167

134

10

160

151

143

15

177

139

123

Table 3. BDD size data for the de Bruijn network of order 4 Root node#

1

7

10

15

heuristic

(0,5)

(1,13)

(3,6)

(5,7)

(7,9)

(10,13)

(12,14)

(14,15)

DFS

4507

8299

4155

7033

5896

5277

4944

3702

BFS

1626

1681

819

1044

1090

822

895

893

NDS

874

922

409

554

871

429

385

439

DFS

3599

6043

3435

4979

4063

3650

3660

3409

BFS

3267

2972

3717

4130

4581

2621

3349

4022

NDS

1022

776

1520

1741

1660

608

1547

1527

DFS

5505

7976

3490

4409

4051

3607

3574

2788

BFS

4537

2856

2321

4473

2956

3431

2018

2123

NDS

4481

2124

1746

4002

2649

2485

2085

2049

DFS

6816

11481

6521

9897

8087

6872

7052

3408

BFS

1306

1517

2221

2249

2341

1212

2224

2527

NDS

921

675

1456

1624

1546

523

1434

1458

15

4.2.2 Square Lattice Networks Square lattice networks have many applications in distributed parallel computation, distributed control and wired circuits such as CMOS circuits and CCD-based devices [38, 39]. Lattice networks can also be considered as an approximation for networks whose nodes are randomly located [40]. In the graph theory, a square lattice network is a graph whose nodes correspond to the points in the plane with integer coordinates, both x and y coordinates being in the same range 1, ..., n. Two nodes in a lattice network are connected by an edge whenever the corresponding points are at distance 1. In other words, it is a unit distance graph for the described point set. Fig. 8 shows a 5*5 two dimensional square lattice network. 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Fig.8. The 5*5 two dimensional square lattice network

Similar empirical study for de Bruijn network is performed for the example square lattice network in Fig. 8, except the following parameters are different due to the random selection: 1) the selected four root nodes are {2, 7, 11, 18}; 2) the selected eight terminal pairs are {(0,1), (0,8), (0,12), (0,24), (10,17), (12,16), (14,19),(18,20)}. From the collected experimental data listed in Table 4 and Table 5, similar observation to that for the de Bruijn network can be made, that is, the smaller the TSBS value of an ordering heuristic, the smaller the BDD model generated using the corresponding ordering. The smallest BDD for the example 5*5 square lattice network is generated by using the BFS heuristic and root node 2, which corresponds to the smallest TSBS value in Table 4. Different from the de Bruijn network where 16

NDS is the best heuristic among the three, BFS performs the best for square lattice networks while the performance of NDS is comparable to that of BFS. While the combination of the BFS heuristic and root node 2 gives the best choice for the example network, we also observe from Table 4 and Table 5 that for root nodes 7 and 18, the TSBS generated for NDS is slightly smaller than that using BFS, but the BDD obtained by NDS is smaller than that by BFS for just 3 out of the 8 terminal pairs (root node 7) and for only one terminal pair (root node 18). This can be explained by the fact that the problem of finding the best ordering for BDD generation is an NP-complete problem. Our proposed measure-based selection method can be used to find a reasonable good heuristic for BDD generation, but we cannot guarantee or prove that the heuristic selected performs the best for all the case. Based on the data in Tables 4 and 5, the BDDs obtained by NDS and BFS actually have similar sizes when their TSBS values are very close, so choosing either NDS or BFS will not make much difference. Nevertheless, the TSBS generated using DFS is much higher than TSBS of NDS and BFS, so the BDD size generated using DFS is much larger than those generated using either NDS or BFS. The TSBS generated using BFS and root node 2 is much smaller than TSBS of NDS and BFS for root node 7, 11 and 18, so the BDD size generated using BFS and root node 2 is much smaller than those generated using either NDS or BFS with other root nodes. In summary, the proposed TSBS-based selection method can provide useful guidelines on choosing a good heuristic for BDD generation. Table 4. TSBS results for a 5*5 two dimensional square network Root node #

DFS

BFS

NDS

2

180

178

180

7

222

201

197

11

276

215

215

18

259

211

209

17

Table 5. BDD size data for a 5*5 two dimensional square network Root node #

2

7

11

18

Heuristic

(0,1)

(0,8)

(0,12)

(0,24)

(10,17)

(12,16)

(14,19)

(18,20)

DFS

1332

2327

1989

1965

1212

1093

811

1029

BFS

987

1571

1549

1425

850

1009

681

829

NDS

1129

1739

1681

1564

921

1041

741

879

DFS

5002

5827

7245

6220

4425

3728

2722

3195

BFS

1585

3396

3071

2710

1668

2159

1515

1663

NDS

1921

2911

2681

2759

1734

1854

1554

1692

DFS

15023

24669

18287

21165

14703

9484

9221

10644

BFS

2243

3770

5449

4338

3873

4127

2149

2890

NDS

3766

4393

6340

4959

4539

4096

2580

3125

DFS

7226

7298

8396

9702

8668

5505

4334

5684

BFS

1899

2993

2864

4028

3550

2448

2348

5145

NDS

1929

3120

3630

4313

3468

2941

3123

5839

4.2.3 Torus Lattice Networks Torus Lattice network is widely used as a network topology for connecting processing nodes in a parallel computer system. It can be visualized as a mesh interconnected with nodes arranged in a rectilinear array of D = 2, 3, or more dimensions, with processors connected to their nearest neighbors, and corresponding processors on opposite edges of the array connected. The lattice has the topology of a D dimensional torus and each node has 2D connections. A number of supercomputers on the TOP500 list use three dimensional torus networks, e.g. IBM's Blue Gene/L and Blue Gene/P, and the Cray XT3 [41]. IBM's Blue Gene/Q uses a five dimensional torus network. Fujitsu's K computer and the PRIMEHPC FX10 use a proprietary six dimensional torus interconnect called Tofu [42]. In a two dimensional torus network, the nodes are laid out in a two dimensional rectangular lattice of rows and columns, with each node connected to its 4 nearest neighbors, and corresponding nodes on opposite edges connected. The connection of opposite edges can be visualized by rolling 18

the rectangular array into a "tube" to connect two opposite edges and then bending the "tube" into a torus to connect the other two. In a three dimensional torus interconnect the nodes are imagined in a three dimensional lattice in the shape of a rectangular prism, with each node connected with its 6 neighbors, with corresponding nodes on opposing faces of the array connected. Higher dimensional arrays can not be directly visualized, but each higher dimension adds another pair of nearest neighbor connections to each node. Fig. 9 gives a 4*4 two dimensional torus network.

Fig.9. The 4*4 two dimensional torus network The empirical study for this 4*4 two dimensional torus network has the following parameters: 1) the randomly selected root nodes are {3, 5, 11, 13}; 2) the randomly selected terminal pairs are {(0, 1), (0, 6), (0, 11), (0, 15), (5, 6), (7, 8), (10, 12), (13, 15)}. Tables 6 and 7 present the results. Similar to de Bruijn and square lattice networks, the smaller the TSBS value of an ordering heuristic, the smaller the BDD model generated using the corresponding ordering for torus networks. The smallest BDD for the example torus network is generated by using the BFS heuristic for all the cases but with different root nodes.

Table 6. TSBS results for a 4*4 two dimensional torus network Root node #

DFS

BFS

NDS

3

195

173

190

5

196

173

190

11

227

173

190

13

197

173

190

19

Table 7. BDD size data for a 4*4 two dimensional torus network Root node #

heuristic

(0,1)

(0,6)

(0,11)

(0,15)

(5,6)

(7,8)

(10,12)

(13,15)

3

DFS

13691

23146

15422

18036

31273

27209

21582

17565

BFS

9397

9036

8956

11353

4709

8670

5976

7771

NDS

27302

23357

25857

24428

10661

17204

15019

15618

DFS

13634

26887

21284

18016

35024

22293

17841

17439

BFS

9575

9969

7737

7307

12742

4143

4444

6162

NDS

27302

22783

21973

21211

22301

12016

14010

15081

DFS

31244

42692

31077

36023

60249

37203

51852

36461

BFS

4828

5358

10792

7443

5385

12389

7911

8930

NDS

13267

14323

26280

15618

11314

32968

15867

16590

DFS

19739

22012

15785

18174

29895

26167

20717

17714

BFS

9541

6654

7122

5688

5363

5176

7835

9521

NDS

19239

14323

15341

14800

15531

14475

15867

25412

5

11

13

5. Proposed Root Node Selection A root node must be chosen before applying a selected heuristic to generate a good edge ordering. As shown in the examples of Section 4, the same heuristic with different root nodes produces different edge orderings, leading to dramatically different sizes of BDD model for the same network. Our recent work [33] shows that 1) though the terminal nodes are normally selected as the root node, they often have poor performance; 2) for square lattice networks, the high-performance root nodes are not randomly distributed; nodes located at the four corners of the lattice are better choices for being the root node than nodes located in the center area. To the best of our knowledge, however no formal method has been developed for guiding the root node selection. In this section selection methods are proposed for choosing a high-performance root node contributing to more efficient terminal-pair network reliability analysis.

5.1 TSBS-based Selection Method For most types of networks, such as De Bruijn networks and square lattice networks, different 20

TSBS values will be obtained for the same heuristic but with different root nodes, then the TSBS-based selection method can be applied to select the best performance root node. TSBS-based Root Node Selection Method: Given an ordering heuristic h and a terminal-pair reliability problem of network G= (V, E). The TSBS based root node selection method can be expressed as follows:

SelTSBS (V ) = ni ⇔ TSBS (Oni ) = min {TSBS (Onj ),1 ≤ j ≤| V |}

(4)

where the ordering generated by root node ni∈V using ordering heuristic h is denoted as Oni. In words, the selected root node from V is the one that generates ordering leading to the minimum value of TSBS. Now, we test the performance of the proposed TSBS-based root node selection methods on De Bruijn networks and square lattice networks. z

De Bruijn Networks

For the experimental data shown in Tables 2 and 3, the favorite heuristic of De bruijn networks is NDS. Since this heuristic can result in different TSBS measure values using different root nodes, we can simply use the TSBS based selection method for root node selection. Based on two arrays BestPerformance and WorstPerformance recording the results of performance comparison, our experiment protocol is designed as follows: 1) Generate a de Bruijn Network of order 4; 2) For each node ni of the total 16 nodes, a) Generate edge ordering Oi using the NDS heuristic and root node ni, b) Calculate TSBS measure of this ordering TSBS(Oi); 3) Sort all 16 TSBS(Oi) values in a non-decreasing order and use orderi to denote the position of Oi in the ordered list; 2

4) For each different terminal-pair reliability of total C16 =120 problems in the network with 16 nodes, a) Generate an equivalent BDD using each edge ordering Oi; 21

b) Collect the BDD size data; c) Find the smallest BDD and its ordering Oj, and increment BestPerformance [j] by one d) Find the largest BDD and its ordering Ok, and increment WorstPerformance [k] by one Part of the results in BestPerformance array is listed in Table 8. For each root node ni given in Row ‘#root node’, Row ‘Performance’ lists the number of smallest BDDs obtained by edge ordering Oi, which is generated using the NDS heuristic and root node ni. Row ‘TSBS(Oi)’ and ‘orderi’ lists the TSBS measure calculated using this ordering Oi and its index in the increased ordering list among all 16 TSBS(Oi). From the collected experimental data, it can be observed that: nodes with smaller orderi, i.e., smaller TSBS measure, can generate more number of smallest BDDs. For example, the best root node 0 can generate smallest BDD for 48 terminal-pair reliability problems out of total 120 problems as shown in the first column, and the value of its TSBS measure, TSBS(O0)=111, is also the smallest; the second largest number of smallest BDD generated by root node 8 is 34 as given in column “2nd_best” and its TSBS value, TSBS(O8)=112, is also the second smallest. Part of the results of WorstPerformance array is listed in Table 9. From the collected experimental data, it can be observed that: the first three worst nodes, 10, 3, and 12, can generate 65+36+18=119 largest BDDs out of total 120 cases, and their TSBS are also among the first three largest ones. Though the node with the largest TSBS is not guaranteed to always generate the largest BDDs, as the size data show, large TSBS values still provide a good guideline on identifying bad choices of root nodes.

Table 8. Part of the Results in BestPerformance array Best

2nd_Best

3rd_Best

#root node

0

8

13

Performance

48

34

15

orderi

1/16

2/16

7/16

TSBS(Oi)

111

112

122

22

Table 9. Part of the Results in WorstPerformance array

z

worst

2nd_worst

3rd_ worst

#root node

10

3

12

Performance

65

36

18

orderi

14/16

16/16

15/16

TSBS(Oi)

143

152

149

Square Lattice Networks

In previous experiments, we have shown that for de Bruijn networks the selection method using the TSBS measure can select an adequate root node for a given heuristic. To find if the same conclusion still holds for square lattice networks, the similar experiment protocol as for de Bruijn networks is implemented for the 5*5 two dimensional square lattice network, where the BFS heuristic is used. Part of the results in arrays BestPerformance and WorstPerformance is listed in Tables 10 and 11. The four BFS orderings calculated from root nodes 10, 24, 0, and 4 have the same smallest values of TSBS measure, thus their orders, order10, order24, order0, and order4 are all equal to 1. They together generate 83+74+69+62=228 smallest BDDs out of total 300 cases. On the other hand, the two BFS orderings calculated from root nodes 12, and 17 have the first two largest values of TSBS measure, and their orders orderi are order12=25 and order17=24. They together generate 242+57=299 largest BDDs out of total 300 cases. From the collected experimental data, it can be concluded that similar to de Bruijn networks, nodes with smaller TSBS measures are better choices for being the root node than nodes with large TSBS measure.

23

Table 10. Part of the Results in BestPerformance array Best

2nd_Best

3rd_Best

4th_Best

5th_Best

5th_Best

#root node

20

24

0

4

3

1

Performance

83

74

69

62

4

3

orderi

1/25

1/25

1/25

1/25

8/25

7/25

TSBS(Oi)

155

155

155

155

168

165

Table 11. Part of the Results in WorstPerformance array worst

2nd_worst

3rd_ worst

4th_ worst

5th_worst

5th_worst

#root node

12

17

13

22

11

18

Performance

242

57

0

0

1

0

orderi

25/25

24/25

23/25

21/25

21/25

20/25

TSBS(Oi)

274

249

223

215

215

211

5.2 IFBSH-based Selection Method There are some classes of networks, such as two dimensional torus networks, it can happen that the same TSBS value is obtained for the same heuristic but with different root nodes, as shown in Table 6. At this time, the TSBS-based selection method cannot work well for the root node selection. Another new measure based on the concept of boundary sets called IFBSH is proposed instead. Measure 2 (IFBSH): Given an ordering O and the terminal pair {s, t} of a network, the IFBSH (Index of First Boundary Set Hit) measure of this ordering, IFBSH(O), is the index of boundary set Fi, which includes at least one of the two terminal nodes:

IFBSH (O) = min{i | Fi ∩ {s, t} ≠ ∅}

(5)

According to the definition, the IFBSH measure of an ordering generated using a given root node will have different values for different terminal pairs of the same network. For example, consider

the

the

network

in

Fig.

2.

The

24

boundary

sets

of

a

given

ordering

O=e1
SelIFBSH (V ) = ni ⇔ IFBSH (Oni ) = max { IFBSH (Onj ),1 ≤ j ≤| V |}

(6)

where the ordering generated by root node ni∈V using ordering heuristic h is denoted as Oni. In words, the selected root node from V is the one that generates ordering leading to the maximum value of IFBSH. Now, we show the performance of IFBSH-based selection method on two dimensional torus networks. From the experimental data shown in Table 6 and 7, the favorite heuristic of two dimensional torus networks is BFS, and this heuristic will result in same TSBS measure values for different root nodes. Thus, the IFBSH based selection method instead of the TSBS based selection method should be used. Different from protocols used in the previous two subsections, the experiment protocol here is based on one matrix Performance recording the results of performance comparison. The following gives the details of the experiment protocol. 1) Generate a 4*4 two dimensional torus network; 2) For each different node ni of total 16 nodes, generate edge ordering Oi using the BFS heuristic and root node ni, 2

4) For each different terminal-pair reliability {s, t} of total C16 =120 problems, a) Calculate IFBSH measure of each edge ordering Oi, IFBSH(Oi), using terminal pair {s, t}; 25

b) Sort all 16 IFBSH(Oi) values in a non-increasing order and use orderi to denote the position of Oi in the ordered list; c) Generate an equivalent BDD using each edge ordering Oi, and collect the BDD size data; d) Sort all 16 BDD size values in a non-decreasing order and use BDDSizei to denote the position

of

the

BDD

generated

using

in

Oi

the

ordered

list;

increment

Performance[orderi][BDDSizei] by one.

Part of the results in Performance matrix for the best performance and worst performance are listed in Tables 12 and 13. For example, the first column of Table 12 shows that the orderings with the largest IFBSH values, orderx=1, can generate 58 smallest BDDs and 18 second smallest BDDs out of the total 120 cases. In Table 13, the last column shows that the ordering with smallest IFBSH value, orderx=16, can generate 106 largest BDDs out of the 120 cases, and the second last column shows the ordering with second smallest IFBSH values can generate the most second largest BDDs. From the collected experimental data, it can be concluded that the orderings generated using root nodes with larger values of the IFBSH measure are better choices for being the root node than nodes with smaller values of the IFBSH measure.

Table 12. Part of the Best Results in Performance matrix orderi

1

2

3

4

5

6

No. of Smallest BDDs

58

12

29

12

3

2

No. of 2nd Smallest BDDs

18

60

6

23

10

1

No. of 3rd Smallest BDDs

14

18

60

6

17

0

Table 13. Part of the Worst Results in Performance matrix orderi

11

12

13

14

15

16

No. of Largest BDDs

0

2

0

12

0

106

No. of 2nd Largest BDDs

4

0

12

0

104

0

No. of 3rd Largest BDDs

12

22

18

58

2

4

26

In this work, we give the priority to TSBS over IFBSH for selecting root node because the size of boundary sets has been theoretically shown to be closely related to BDD size by Theorems 1-2. The IFBSH is used as a secondary method only for networks where the selected heuristic results in the same TSBS measure values for different root nodes. For networks where different TSBS values are obtained with different root nodes, we actually found that the performance of the IFBSH method is not as favorable as what is achieved in two dimensional torus networks. Specifically, consider the de Bruijn Network of order 4. As shown in Table 8, the best root node 0 can generate smallest BDDs for 48 terminal-pair reliability problems out of total 120 problems; among these 48 problems,

only

16

largest/second-largest/third-largest

IFBSH

values

but

12

smallest/second-smallest/third-smallest IFBSH values out of total 48 cases are generated. Similarly, consider the 5*5 two dimensional square lattice network. As shown in Table 10, the best root node 20 using the BFS heuristic can generate smallest BDDs for 83 terminal-pair reliability problems out of total 300 problems. Among these 83 problems, only 31 largest/second largest/third largest IFBSH values but 23 smallest/second smallest/third smallest IFBSH values out of the total 83 cases are generated.

6. Conclusions and Future work In the BDD-based terminal-pair network reliability analysis, a network is encoded by a BDD. This encoding requires selecting a total order over all the edges of the network. The size of the BDD heavily depends on the chosen ordering. There is often a variation of several orders of magnitude between the sizes of BDDs built using different orderings. Several ordering heuristics, such as DFS, BFS and NDS, have been proposed in the literature for the network reliability analysis. They perform differently for different types of networks. In addition, the same heuristic can perform differently when different root nodes are used. Therefore, there is a need for the development of selection methods that can facilitate the selection of an adequate heuristic from a heuristic library and the selection of a high-performance root node to perform edge searching. The 27

main contribution of this paper is to construct such selection methods based on the concept of boundary sets, in particular, two new measures TSBS and IFBSH, and then test the performance of these methods using experiments on three types of widely-used networks. Experimental data revealed that the high-performance orderings and values of TSBS measure are correlated: heuristics resulting in smaller TSBS values typically achieve better performance in BDD generation. For the case where different TSBS measure values are calculated using the same heuristic but different root nodes, the TSBS based selection method can also be used to select a high-performance root node. Similar to the heuristic selection, root nodes resulting in smaller TSBS values are better choices. For the case where the same TSBS measure values are calculated using the same heuristic but different root nodes, the IFBSH based selection method can be used to select a high-performance root node and root nodes resulting in larger IFBSH values are better choices. One direction of our future work is to conduct performance studies of TSBS and IFBSH based selection methods using more general network structures, specifically, randomly generated networks and directed network graphs. We also plan to design different variants of BFS and NDS heuristics based on weight assignment of nodes and edges for better performance. As in most of network reliability studies, we have assumed perfectly reliable nodes in this work. In the real-world network systems, however, both nodes and edges can fail [27]. Thus, another direction of our future work is to extend the TSBS and IFBSH based selection methods to take into account the effect of imperfect nodes.

Acknowledgement This work was supported in part by the NSF of China under Grant No. 61272130, 61272007, U.S. NSF under Grant No. 1112947, the NSF of Zhejiang Province under Grant No. Y1100689.

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Unveils

Post-K

Supercomputer.

fujitsu_unveils_post-k_supercomputer.html

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http://www.hpcwire.com/hpcwire/2011-11-07/