A Simple Heuristic Algorithm for Generating All Minimal Paths

488 IEEE TRANSACTIONS ON RELIABILITY, VOL. 56, NO. 3, SEPTEMBER 2007

A Simple Heuristic Algorithm for Generating AllMinimal PathsWei-Chang Yeh, Member, IEEE

Abstract—Evaluating network reliability is an important topicin the planning, designing, and control of network systems. In thispaper, an intuitive heuristic algorithm is developed to find all min-imal paths (MP) by adding a path, or an edge into a network re-peatedly until the network is equal to the original network. Theproposed heuristic algorithm is easier to understand & implementthan the existing known heuristic algorithm. Without generatingany duplicate MP, it is also more efficient. The correctness of theproposed algorithm will be analysed, and proven. One bench ex-ample is illustrated to show how to evaluate the network reliabilityusing the proposed heuristic algorithm.

Index Terms—Heuristic algorithm, minimal cut, minimal path,network reliability.

ACRONYM1

MC/MP Minimal Cut/Minimal Path

NOTATION

The original network which is a connected,undirected network with the node set

, and the edge set ,respectively. For example, Fig. 1 is a connectedundirected network., is the specified source node, and sink

node, respectively.The number of elements of , e.g., is thenumber of nodes in .

,if and,if or

,

where for all , , and .The th updated network, where . If

, is called an initial network created byconstructing node-disjoint MP in , i.e.the intersection of any two MP are only nodes, and . For example, Fig. 2 is one of the initial

networks w.r.t. Figs. 1 and 3 is w.r.t. Figs. 1and 2.

the nodes in the nodes in .

Manuscript received Dec. 6, 2005; revised May 1, 2006; accepted May 14,2006. This research was supported in part by the National Science Council ofTaiwan, R.O.C., under Grant NSC 92-2213-E-035-041. Associate Editor: L.Cui.

The author is with the e-Integration & Collaboration Laboratory, Departmentof Industrial Engineering and Management Engineering, National Tsing HuaUniversity, Hsinchu, Taiwan 300, R.O.C. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TR.2007.903290

1The singular and plural of an acronym are always spelled the same.

Fig. 1. An example network.

Fig. 2. An initial network of Fig. 1.

Fig. 3. The network G w.r.t. Figs. 1 and 2.

the edges in the edges in .

the nodes (edges) in if.

The th MP. It is represented either by an edgeset, or by an ordered node list. For example,

is a MP in Fig. 1.a path or an edge from nodes to with

, and ifis the updated graph after inserting into .For example, Fig. 3 is the updated networkafter inserting into Fig. 2.

is the order of the path added intoto generate . If such path is , then

. Note that whenever. For example,

where is an new MP generated in inFig. 3.The th path inserted into the initial network,where . For example, inFig. 3.The set of all MP.

and for ,e.g. is the set of all new MP generatedin . Note that is a special MP subsetgenerated in s.t. any two MP in haveno common nodes, except nodes & , e.g.

in Fig. 2.

0018-9529/$25.00 © 2007 IEEE

YEH: HEURISTIC ALGORITHM FOR GENERATING ALL MINIMAL PATHS 489

, e.g.is the MP set generated in . Note that

.is the updated network after

inserting in .

The subpath from nodes to in the path .

if for allwith ; otherwise, .

for all and .

for all and.

for all and.

for all and.

.

and.

node and node .

NOMENCLATURE

Reliability The probability of connection of thesource node with the sink node .

MP/MC Minimal path/minimal cut, a path/cut setsuch that if any edge is removed fromthis path/cut set, then the remaining setis no longer a path/cut set. For example,

is a MP between nodes , andin Fig. 1. The MP/MC search is tied

to the Graph Theory concepts, andconnectivity measures. Probabilistictechniques usually associate theconnectivity level of a network withthe availability, and/or reliability of thecommunication paths between specificnetwork node-pairs [10]–[19].

Original Network A network which is required tofind the exact reliability between nodes, and .

Initial Network A special network created byconstructing disjoint MP in the originalnetwork s.t. the intersection ofany two MP are only nodes , and .

Updated Network A new network created by inserting apath into the current network.

Duplicate MP is a duplicate MP iff there is anMP with s.t. .

ASSUMPTIONS

The network must satisfy the following assumptions[10]–[19]:

1) Each node is perfectly reliable, and each edge has twostates: working, or failed.

2) The network is connected, and free of self-loops.3) There are no parallel branches.

4) Degrees of all nodes are at least 2, except for the sourcenode , and sink node . If there is a node, say ,with only one degree, the edge adjacent to node can beremoved without losing any MP, i.e. each node is passedthrough by at least a MP.

5) All flows in the network obey the conservation law.

I. INTRODUCTION

THE network reliability is defined here as the probability ofa live connection between the source node, and the sink

node. In recent years, network reliability theory has been ap-plied extensively in many real-world systems such as computerand communication systems, power transmission and distribu-tion systems, transportation systems, oil/gas production systems[1]–[7], etc. Thus, the network reliability plays important rolesin our modern society; and more authors have been paying at-tention to the network reliability problem [1]–[27].

The evaluation of network reliability is an NP-hard problem[8], [9]. Reliability evaluation approaches exploit a variety oftools for system modeling, and reliability index calculation.Among the most popular tools are network-based algorithmsfounded in terms of either MC or MP [10]–[19]. However, boththe problems in locating all MC/MP, and computing the exactreliability in terms of the known MC/MP, are also NP-hard[8], [9].

A variety of methods for generating MP have been proposed,which are all based on direct (explicit, and implicit) enumer-ation by deriving necessary, and sufficient conditions usingBoolean algebra, and/or set theory. To the author’s best knowl-edge, Al-Ghanim first investigated a heuristic programmingalgorithm to generate all MP [10]. He first created a path, thenused an iterative method by tracing back from an explored nodein the current path via unexplored nodes to the source node, andswapped the sub-paths in the corresponding MP to generatenew MP [10]. The above procedure is repeated until all MP arefound. It is very hard to follow Al-Ghanim’s whole algorithm.It is less efficient, and produced duplicate MP, which neededextensive comparison, and verification. It also worked only forthe directed network (graph); i.e. may have failed to search forall MP in the undirected network. In real cases, many networkssuch as computer, and telecommunications systems are allundirected. The worst part is that there is not any proof for thecorrectness of this algorithm. The need for a more intuitiveheuristic algorithm to search for all MP thus arises.

There is generally a need to evaluate only a few of the modi-fications in an existing network for expansion or reinforcementwithout relocating all MP or MC [11]–[14]. Such problems havebeen solved recently for MP [11], [12], and MC [13], [14], sep-arately. Based on the important result from our previous pub-lished paper [11], to search for all MP in a new network usingthe MP obtained in the network before adding a path, a newheuristic algorithm is proposed to search for all new MP byadding repeatedly a new path (arc) to a network until the orig-inal network (see Fig. 1) is formed completely. It adapted somesimple rules to prevent & detect/delete generating duplicate MPefficiently without determining whether a path is a MP.


This paper is organized as follows. Some important prop-erties, and theorems are discussed in Section II. Section IIIdiscusses how to eliminate duplicate MP using both preven-tion, and detection/deletion. Section IV presents the proposedmethod in detail, together with a discussion of the time com-plexity. One bench example is illustrated to show how togenerate all of the MP using the proposed heuristic algorithmin Section V. Concluding remarks are in Section VI.

II. PRELIMINARIES AND FINDING NEW MP

The main concepts behind the existing heuristic algorithm[10], and the proposed algorithm to search for all MP are:

1) Construct the initial network , find the MP set in ,and let .

2) Find the new MP set in the updated network, where , and ;

let , and .3) Repeat the above Procedure 2 until the current network

is equal to the original network .Obviously, the above Procedure 2 is the most important in

the existing heuristic algorithms. Therefore, some propertiesabout the relationship among , , , and MP are listedas follows.

Property 1: The following statements regarding are true.

Property 2: The following statements regarding are true.

for all

for all

Property 3: The following statements are equivalent.

is a new MP generated in

and for all

Property 4: Let , and . The following state-ments are true.

If then

If then

Adding a path or an edge, say , between two nodes canonly increase the number of MP. Hence, any MP is also an MPafter inserting in the current network. Thus, we will focus onhow to find the new MP in the network after inserting a new path,and eliminate duplicate MP. A special property is consideredfirst, which simply follows immediately from the definitions ofthe four sets: , , , and mentioned in Section I;all of the MP passed through node , and/or can be separatedinto four disjoint subsets as in Property 5.

Property 5: The intersection of any two of , , ,and are empty.

From Property 5, we have the following corollaries. Thesecorollaries are trivial but very useful in the discussion to elimi-nate the duplicate MP.

Corollary 1: The intersection of any two of ,, , and are empty.

Corollary 2: The intersection of any two of ,, , and are empty.

Corollary 3:.

Corollary 4: .Corollary 5:

.To compare the proposed algorithm with the best-known

heuristic method presented in [10], the main calculationprocedure, i.e. procedure 2, of the best-known heuristicmethod for finding new MP is listed by transforming intoour notation.

Theorem 1:

is theset of all new MP in .

Corollary 6: The maximal number of new MP obtained in, using Theorem 1, is

.

Corollary 7:

is the set of all new MP in the directed network.

Theorem 1 is a straightforward method. Without using thespecial characteristics discussed in Corollaries 1-5, it is less ef-ficient, and generates duplicate MP. The following theorem isfirst proposed in [11]. Originally, it is proposed only to find allnew MP in after adding a new path into . It isadapted here to find all MP in .

Theorem 2: If , then let node be node .

Otherwise, let be a node in . If

, then let node be node . Otherwise, let node

be a node in . Let

if , and ; otherwise, let. Then,

are all new MP in [11].Theorem 2 is extended in this study to find all MP. Moreover,

it is improved repeatedly to eliminate duplicate MP to increaseits efficiency. Before that, a special characteristic of MP setis discussed as follows.


Theorem 3: (Menger’s Theorem) The maximum number ofedge-disjoint MP in equals the size of a minimum cutbetween nodes , and .

is the set of node-disjoint MP obtained in by the defini-tion. From the above special case of the Max-Flow Min-Cut the-orem [27], can be obtained in polynomial time. Besides, theintersection of any two MP in is nodes , and ; we have thefollowing property shown that there is no need to verify whichone MP in is a duplicate.

Property 6:

is the set of all new MP in without any duplicateMP.

Corollary 8:is the set of all new MP in the directed network

without any duplicate MP.Corollary 9: If , then ,

and .In the assumption, the degrees of all nodes are at least 2,

except for the source node , and sink node . Hence, both, and are not empty if is

an undirected network, or at least one of , and

is not empty if is a directed network. Basedon the above discussion, a special case of Theorem 2 is listednext. This new theorem plays an important, fundamental rolein the proposed heuristic algorithm. It provides the basis of theproposed algorithm, and is simplified furthermore to eliminateduplicate MP in the next section.

Theorem 4:

is the set of all new MP in .Corollary 10: The maximal number of new generated MP

obtained in using Theorem 4 is

.Corollary 11:

is the set of all new MP in the directed network.

Because

, Theorem 4 is more efficient thanTheorem 1 in finding new generated MP infrom Corollaries 6, and 10. However, each new generated MPstill may have duplicates due to the fact that the intersectionof any two of , , , ,

, , , and may not beempty. The theorems, and properties in the next section arepresented to eliminate the duplicate MP.

III. ELIMINATE DUPLICATE MP

The time complexities to generate new MP in each updatednetwork are directly proportional to the number of old MP fromCorollaries 6, and 10. Because the number of MP grows expo-nentially with network size, and many MP may have duplicates,it is necessary to eliminate duplicate MP before calculating thenetwork reliability at the final stage.

Fig. 4. The updated network G w.r.t. Fig. 3.

To verify whether a new generated MP is a duplicate, wecan simply compare this MP with the other old MP. The fol-lowing corollary follows immediately from Property 2e. It statesa different, easy way to verify a new MP only by comparingeach new MP with the other new generated MP. Thus, we cansearch for duplicate MP in some relatively small scope insteadof searching over the whole MP list.

Corollary 12: is not a duplicate MP iff for all, and .

The new generated MP number also grows exponentially withnetwork size. Therefore, some more efficient methods to elimi-nate duplicate MP are proposed next. Directly from Corollaries1–5, and Theorem 4, duplicate MP are generated only in thefollowing three situations if is the path added to the currentnetwork .

Situation 1:

.

Situation 2:

.

Situation 3: ,

, , and/or

.For example, , , ,

and are all MP in Fig. 3, i.e..

If Theorem 4 is employed to find all new MP in Fig. 4, whichis created by adding edge , i.e. with , and

, into Fig. 3; then we have

and

where , ,, , and

. From above, it is trivial thatis a new MP generated from

both , and ,i.e. belong to Situation 1.

is a new MP generated from

, i.e. belong to Situation 3.


To prevent any duplicate MP before it is generated from Situ-ations 1 & 2, a new notation “ ” is proposed to replace “ ”(which is defined in [11] using notation ): iff

, , and .The following two properties regarding to “ ” are true.Property 7: .Corollary 13: if .Corollary 14: If , then , and

.

Property 8: The intersection of any two of ,

, , , , ,

, and are empty.From Corollaries 1-5, and Property 8, we have the following

property which can prevent any duplicate MP generated fromSituations 1 & 2 discussed above.

Property 9: The intersection of any two of

, , , and/or

are empty.From Theorem 4, and Property 9, to prevent any duplicate MP

generated from Situation 3, avoid duplicate elements occurringin , , , , , and as isdiscussed in the next property.

Property 10: No duplicate MP in

, , ,

and , if no duplicate element in ,

, , , , and

.From Property 10, Theorem 4 is revised s.t. no new duplicate

MP is included, as follows.Theorem 5:

is the set of all new MP inwithout any duplicate MP.

Corollary 15:

is the set of all new MP in the directed networkwithout any duplicate MP.

Corollary 16: iff

, where , , and.

Corollary 17: If , then

, where .Corollary 18: Let .

iff , , and.

All new MP generated in are obtained using the “multi-plication” combinations of , , , , and/or

for all generated in for .

IV. THE PROPOSED ALGORITHM

Based on the discussion in Sections II & III, Theorem 5 pro-vides an easy way to eliminate new generated duplicate MP. In

Fig. 5. The corresponding sub-networks. (a) The 1st MP in Fig. 1. (b) The 2ndMP in Fig. 1. (c) The arcs (shown in dashed line) need to added back after step 3.

TABLE ITHE CORRESPONDING b , P , P , P , AND P

this theorem, its time complexity is equal to Theorem 2. How-ever, no duplicate MP is generated. The proposed heuristic al-gorithm is based on the above theorem. It will be proposed toenumerate all MP. Greater detail is stated as follows.

Algorithm: Find all MP in a network .Input: A connected undirected graph with nodeset , edge set , a source node , and a sink node .Output: All MP in .STEP 0. Let , , and .STEP 1. If there is a MP, say , in , thengo to STEP 2. Otherwise, let ,

, and go to STEP 3.STEP 2. Let , , ,and go to STEP 1.STEP 3. If , then halt, and is the MPset in .STEP 4. Choose a new path, say in , s.t.

; and let , and.

STEP 5. Separate all MP in that pass through nodes ,and/or into , , , and .STEP 6. If , implement Property 6 to find the newMP set generated in , let , , andgo to STEP 3.STEP 7. Find , , ,

, and .STEP 8. Implement Theorem 5 to find the new MP set

generated in , let , ,and go to STEP 3.

In the above algorithm, the disjoint MP, and the initialnetwork is constructed in steps 1–3. Step 4 inserts a newpath which is selected arbitrarily from nodes to in theoriginal network, but not in the current network, and updatesthe current network. Step 5 separates all generated MP passedat least one endpoint of into , , , and . Step6 is based on Property 6 to find all new MP generated in .

, , , , ,

and are found in Step 7 for all . The last step


TABLE IITHE CORRESPONDING b , P , � (P ), AND � (P )

finds all new MP by Theorem 5 using the result obtained inStep 7. Steps 3–8 are repeatedly inserting a new path, updatingthe current network until the original graph is completelyformed.

The number of arcs in a MP is not greater than , andnot less than the number of nodes in any shortest path.Hence, , and the time

complexity is to find allMP [10], where is the number of arcs in the shortestpath. The proposed algorithm is based on paths instead ofarcs. Thus, the time complexity of the proposed paper is

, where is the lastvalue of in STEP 3, and is the total number of theset of all endpoints in the paths found in STEP 4. Because

, and , the proposed algorithmis more efficient than the existing known algorithm proposedin [10].

From the above discussions, we have the following statementimmediately.

Theorem 6: The proposed heuristic algorithm is more effi-cient than the existing known algorithm proposed in [10] forlocating all MP in a network without any duplicate in time com-plexity , where is thelast value of in STEP 3, and is the total number of the setof all endpoints in the paths found in STEP 4 of the proposedalgorithm.

V. AN EXAMPLE

Enumerating all of the MP in a network is a NP-hardproblem [8], [9]. It possesses a computational difficulty that, inthe worse case, grows exponentially with network size. Owingto this inherent problem, instead of presenting practically largenetwork systems, a moderate size benchmark network calledthe modified ARPANET shown in Fig. 1, which is the mostfrequently cited example [10]–[19], was selected to demon-strate this methodology. In this illustrative network, each edgerepresents a transmission line, and consists of several physicaltransmission lines, e.g. T3 cable, E1 cable, optical fiber. Eachnode represents a computer center, and consists of severalswitches. Nodes , and are the source node, and sink node,respectively. To make a branch string easy to be recognized, allof its nodes are underlined.

The proposed algorithm is used to find all MP in Fig. 1 asfollows:

STEP 0. Let , and.

TABLE IIITHE CORRESPONDING b , P , � (P ), AND � (P )

STEP 1. Because (see Fig. 5(a)) isa MP in , go to STEP 2.STEP 2. Let , ,

, and go to STEP 1.STEP 1. Because (see Fig. 5(b)) isa MP in , go to STEP 2.STEP 2. Let , ,

, and go to STEP 1.STEP 1. Because there are no MP in ,let , , and go to STEP 3.STEP 3. Because , go to STEP 4.STEP 4. Choose a branch string, say

(see Fig. 5(c)) in s.t. , let, and .

STEP 5. Let , , and.

STEP 6. Because , let ,, and go to

STEP 3, where

.STEP 3. Because , go to STEP 4.STEP 4. Choose a branch string, sayin s.t. , let ,and .STEP 5. Let , ,

, and (see Table I).STEP 6. Because , go to STEP 7.STEP 7. Find , , ,

, , and . Thecorresponding results are shown in Tables II and III.STEP 8. Let ,

, and go to STEP 3,where

(see Table IV).


TABLE IVTHE CORRESPONDING MP p

Finally, All MP are , , ,, , , , ,

, , , , and(see Table IV).

VI. CONCLUSIONS

Up to now, no method can give birth to real MP withoutduplicating during enumeration. Nevertheless, there is only oneheuristic approach for searching for all MP [10]. This approachis implemented by using a very straightforward procedure.Without utilizing the properties of the network, it is inefficient,and tedious. It is also a very cumbersome, time-consumingtask if all duplicate MP need to be eliminated, and/or appliedin calculating the exact network reliability. By exploiting thespecial characteristic and structure among the subpaths in MP,the proposed algorithm not only requires fewer calculations togenerate new MP, but is also more effective in generating MPwithout duplicates, and unfeasible MP.

ACKNOWLEDGMENT

The author would like to thank both the associate editor andthe referees for their constructive comments and recommenda-tions, which have significantly improved the presentation of thispaper.

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Wei-Chang Yeh is currently a professor of the Department of Industrial En-gineering and Management Engineering at National Tsing Hua University inTaiwan. He received his M.S., and Ph.D. from the Department of Industrial En-gineering at the University of Texas at Arlington. His research interests includenetwork reliability theory, graph theory, deadlock problem, Linear Program-ming, and scheduling. He is a member of the IEEE, and INFORMS.

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A Simple Heuristic Algorithm for Generating All Minimal Paths