Effective algorithms for finding optimum pairs of link-disjoint paths in α+1 path protection

Telecommun Syst (2013) 52:783–797DOI 10.1007/s11235-011-9577-2

Effective algorithms for finding optimum pairs of link-disjointpaths in α + 1 path protection

Ming-Lee Gan · Soung-Yue Liew

Published online: 2 September 2011© Springer Science+Business Media, LLC 2011

Abstract Path protection ensures continuity of network ser-vices against link failure by assigning a link-disjoint sec-ondary path to protect the primary path. Existing path pro-tection schemes are classified into dedicated- and shared-path protections. However, these schemes either require highredundancy or substantial response time to resolve link fail-ure. The end-to-end partial bandwidth protection scheme,denoted as α +1 protection, offers an alternative where onlycritical real-time information of the primary path is dupli-cated over the secondary path. The parameter α is defined asthe ratio of the protection bandwidth (of secondary path) tothe full bandwidth (of primary path). The challenge of α + 1protection is to identify a pair of primary-secondary pathswith lowest total cost comprising the optimal solution. Thispaper derives the properties inherent in the optimal solutionand subsequently presents two optimum paths-finding algo-rithms. The performance of the proposed algorithms overexisting approaches is then compared through simulation.

Keywords Link-disjoint paths · Path protection · QOS ·Reliable routing

1 Introduction

A communications network is commonly represented bya topology comprising of nodes interconnected by links,whereby data can be transferred through a path from a

M.-L. Gan (�) · S.-Y. LiewFaculty of Information and Communication Technology,Universiti Tunku Abdul Rahman, Kampar, Perak, Malaysiae-mail: [email protected]

S.-Y. Liewe-mail: [email protected]

source node to a destination node. A link failure, such asa fiber cut, is a common cause of communications break-down that could be due to natural disasters, accidents, hu-man error or intentional sabotage. Protection schemes aremeasures implemented on a network to ensure the overallnetwork’s survivability in the event of such unforeseen linkfailures. In the network, the main connection between thesource node to the destination node is known as the primarypath, active path or working path. The objective of protec-tion schemes is thus to maintain the connectivity betweenthe source and destination nodes even if a link failure occursalong the primary path. These protection schemes can, ingeneral, be categorized into link protection, path protectionand partial path protection. They each function to providevarying levels of reliability for the connectivity between thesource and destination nodes.

Two important aspects are basically used to evaluatethe feasibility of protection schemes, namely the resourceutilization and recovery performance. Resource utilizationcan be measured by the amount of redundant backup re-source being reserved to protect a particular connection.Conversely, recovery performance is affected by the timeneeded by the protection scheme to detect a fault and re-sume the transmission. In the existing protection schemes,there is a tradeoff between resource utilization and recoveryperformance. Some protection schemes allow several con-nections to share the same pool of backup resource thus re-sulting in higher resource utilization but at the expense ofrecovery performance and vice versa. The architectures ofthe following protection schemes are discussed in this per-spective.

Link protection [1–3] protects individual links that makeup the primary path. If a link on the primary path fails, linkprotection reroutes all traffic over the failed link while therest of the links remain on the path. The architecture of

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784 M.-L. Gan, S.-Y. Liew

link protection enables fast protection-switching time [1],because the fault detection and recovery are managed by theend nodes of the failed link without the involvement of thesource and destination nodes [2]. This results in a desirablerecovery performance. However, since there is no way topredict which link will fail, all the links along the primarypath have to be considered in order to realize this protectionscheme. This becomes impractical to implement as the sizeof the network increases. Consequently, the resource utiliza-tion for link protection is less efficient compared to path pro-tection [1].

Implementing path protection to a network between asource and destination involves a primary path and a sec-ondary path (or a protection path), where the two paths mustbe link-disjointed. Suppose a link in the primary path is bro-ken, the secondary path would ensure the continuity of ser-vice by assuming the responsibility of data transfer that washandled by the primary path.

Existing path protection schemes are generally classifiedunder two major categories: dedicated path protection (e.g.,1 + 1 protection, 1:1 protection) and shared path protection(e.g., 1:N protection, M:N protection) [2–4]. In 1 + 1 pro-tection, the same information signals are transmitted simul-taneously by the source through both primary and secondarypaths to the destination. Since the destination receives a fullset of duplicate data, fast connection recovery can be en-sured in the case of link breakdown on the primary path(recovery time ≤50 ms) [4]. However, it is less efficient inresource utilization as it requires 100 percent redundancy.

The 1:1 protection, on the other hand, is a special kindof dedicated path protection. It initially transmits data onlyon the primary path, and the reservation of bandwidth on thesecondary path is rather a soft one so that low-priority trafficcan utilize the reserved bandwidth while the secondary pathis idle [4]. Only when the primary path fails, the transmis-sion of data is resumed by switching the connection to thesecondary path. However, the 1:1 protection still requiresthe same amount of bandwidth to be reserved for protectinghigh-priority traffic, thus it does not have significant sav-ing in protection bandwidth because of its dedicated nature.Moreover, in the 1:1 protection, only after a failure is de-tected on the primary path, the secondary path can then re-sume the transmission, which leads to a higher recovery timecompared to the 1 + 1 protection [4].

The 1:N protection allows a single secondary/protectionpath to be shared by N primary/active paths, thus resultingin a more efficient resource utilization compared to the 1+1and 1:1 protections owing to such a shared nature. The draw-back however is the limited protection coverage. That is,once the reserved backup resource has been used in the eventof a path failure, all other paths that rely on this backup re-source will be unprotected. To resolve this problem, M sec-ondary/protection paths are used to protect N primary/active

paths in the M:N protection, where 1 ≤ M ≤ N. Neverthe-less, it still suffers from less tolerance for multi link failurescompared to dedicated path protection [2]. In addition, both1:N and M:N protections notice significant increase in theresponse time to detect and recover from link failure due tothe similarities in architecture with the 1:1 protection.

Another protection scheme known as the partial path pro-tection (PPP) [5] assigns an ‘end-to-end’ protection path foreach link of the primary path. Since the protection paths inPPP function to only protect certain link(s) on the primarypath, it makes this protection scheme more flexible to im-plement as the protection path needs not be completely link-disjoint with the primary path. It is shown in [5] that PPPoutperforms path protection in terms of blocking probabil-ity. But, the disadvantage of the PPP is fairly obvious. Apartfrom the detection at a higher layer for the primary path fail-ure, local information at the lower layers for the location ofthe failed link on the primary path must also be identified.Both pieces of information are needed by the network man-ager in order to activate the necessary protection path asso-ciated to the particular failed link, compromising further therecovery time.

In most cases, when the primary path fails, real-time net-work applications could still be functioning at a satisfac-tory level over the secondary backup path that transmitsonly mission-critical data. In other words, the primary pathshould be able to be protected by a secondary link-disjointpath whose protection bandwidth is less than the primarybandwidth. The concept is to provide sufficient protection toensure continuity of service with lower backup bandwidth,and thus reducing the network cost while the recovery timeis not compromised.

Such a concept was proposed and discussed in [6], la-beled as end-to-end partial bandwidth protection. However,for a call setup, the pair of link-disjoint paths obtained by thealgorithm presented in [6] may not necessarily have a mini-mal combined total cost, which will be further discussed inSect. 2.

Similar to the 1 + 1 protection, a solution in the end-to-end partial bandwidth protection comprises a pair of link-disjoint primary and secondary paths between source anddestination nodes. Unlike the 1 + 1 protection, however, thebandwidth requirement by the secondary path to protect theprimary path can be flexibly adjusted. We denote this protec-tion scheme as the α+1 protection, where α is defined as theratio of the bandwidth required on the secondary path overthat required on the primary, and 0 ≤ α ≤ 1. This scheme isin fact applicable in optical networks as well as connectionoriented networks such as IP/MPLS and ATM.

As the value of α varies, it may result in a different op-timal solution with minimum total cost in implementing theα +1 protection. For instance, when α is equal or very closeto zero, it is obvious that the shortest path, serving as the pri-mary, coupled with another link-disjoint shortest secondary

Effective algorithms for finding optimum pairs of link-disjoint paths in α + 1 path protection 785

path would be an optimal solution because no or only verylittle protection bandwidth is required. Such a solution canbe identified by an algorithm called the Remove Find [7].When α is equal or very close to one, on the other hand,the optimal solution can be identified by another algorithmcalled the Link-disjoint Bhandari, which was proposed in[7] for the 1 + 1 protection.

An interesting optimal solution, which can neither beidentified by the Remove Find nor the Link-Disjoint Bhan-dari algorithms, is found to occur when 0 < α < 1 in somecases. The occurrence of such an optimal solution has notbeen discussed in the literature, and it is referred to as amid-optimal (MO) solution in this paper. We analyze theproperties specific to the mid-optimal solutions and proposetwo optimum paths-finding algorithms to effectively iden-tify optimal solutions in the network with respect to all α

values.The remainder of the paper is organized as follows. Sec-

tion 2 analyzes the cost optimization aspect for α+1 protec-tion and introduces the occurrence of mid-optimal solutionswhen 0 < α < 1. Section 3 discusses the conditions inher-ent in the mid-optimal solutions. In Sect. 4, we propose twooptimum paths-finding algorithm for the α + 1 protection.Section 5 presents simulation results that compare the effi-ciency of our proposed algorithm with existing paths-findingalgorithms. Section 6 concludes the paper.

2 Analysis of optimal solution for α + 1

2.1 Total cost computation of link-disjoint paths in α + 1

As mentioned, the proposed α + 1 protection scheme has avarying parameter, α, where 0 ≤ α ≤ 1, depending on theapplication requirement. As the value of α varies from 0to 1, it will affect the total cost of implementation. With-out the loss of generality, we assume that the bandwidth re-quirement for the primary path is defined as 1, and that forthe secondary is α. Given a valid solution that consists of apair of link-disjoint paths for α + 1 protection, let P denotethe cost (per unit bandwidth) of the primary path, and S thatof the secondary. The total cost can thus be defined as

C(α) = P + αS, (1)

where C(α) is the total cost of the solution, which is also afunction of α.

It should be noted that for any valid solution, the cost ofthe primary and secondary path are such that P ≤ S. This istrue when assuming the network has no capacity constraint.

With reference to the network given in Fig. 1, supposethat a reliable communication needs to be set up between thesource node ‘a’ and destination node ‘f’, where the value on

Fig. 1 Network example

each link represents the cost (per unit bandwidth) incurredfor data to traverse the particular link.

To provide path protection, two approaches have beenproposed in the literature to determine the primary pathand secondary link-disjoint path. They are the Remove Find(RF) method [7], and the Link-disjoint Bhandari (LB) algo-rithm [7, 8]. These two paths-finding algorithms are men-tioned because they provide the optimal solutions in theirrespective (extreme) settings of α. That is, assuming validsolutions can be found by the two algorithms respectively,when α = 0, the solution by RF is optimal, whereas whenα = 1, the solution by LB is optimal.

The RF method consists of two steps to find a link-disjoint pair of paths. From Fig. 2a, the first step is to deter-mine the least-cost path from the source to destination. Thiscan be accomplished by applying familiar shortest path al-gorithms such as the Dijkstra’s algorithm. After the shortest-path has been identified, it is assigned to be the primary path.To find a link-disjoint secondary path, the next step of themethod requires the primary path to be removed from thenetwork and then the shortest path algorithm is applied onceagain to find the least-cost path from the ‘trimmed’ network,as shown in Fig. 2b. Hence, the RF solution consists of theshortest path from “a” to “f” as the primary, and the shortestlink-disjoint path as the secondary. Figure 2c shows the pri-mary path (path: a-b-c-d-e-f, cost = 1 + 1 + 1 + 1 + 1 = 5)and secondary link-disjoint path (path: a-f, cost = 12) as ob-tained by the RF algorithm.

Consider the first extreme value when α = 0. This is ahypothetical case where the secondary backup path is notallocated any bandwidth to protect the primary path. In thisscenario, the RF method [7] is ideal as the primary path isthe shortest path.

The LB solution consists of a pair of link-disjoint pathswhere the sum of their costs is the minimum among all otherpossible pairs [7]. The LB algorithm is summarized into foursteps. The first step is to determine the least-cost path, de-note as P1, from the source to destination. P1 is found by ap-plying a shortest path algorithm. Figure 3a indicates path P1

in the example. The second step is to replace P1 with −P1.−P1 is obtained by reversing the direction of P1 links, andchanging the associate cost of the P1 links with equivalentnegative values. Figure 3b illustrates the modified network


(a)

(b)

(c)

Fig. 2 (a) RF Step 1: Determine the least cost path between node ‘a’and node ‘f ’. (b) RF Step 2: Remove the initial least cost path fromthe network and find the subsequent least cost path from the ‘trimmed’network. (c) A pair of link-disjoint path by the RF method

with −P1. The third step is to find the least-cost path in themodified network, denoted by Pa. The Bellman Ford algo-rithm is used to find Pa in this stage as the network has linkswith negative values. Subsequently, the −P1 link shared byPa is ‘virtually’ removed. Figure 3c shows path Pa as well asindicates the link shared by −P1 and Pa. The forth step in-volves restoring the rest of the reversed links together withtheir initial non-negative cost and grouping the remaininglinks into two paths, P′

1 and P′2. Finally the link that was

‘virtually’ removed in the third step is restored to its initialstate. Figure 3d shows such pair of paths (path: a-e-f andpath: a-b-c-d-f) obtained by the LB algorithm, where thesum of costs is 14 (7 + 7), which is the minimum amongall.

Consider another extreme value when α = 1. This is thecase where the secondary link-disjoint path protects the pri-mary path with the full amount of bandwidth (1 + 1 protec-tion). The LB algorithm is preferred as it ensures the opti-mality of its solution for such 1 + 1 protection scheme [7].

Now, let PRF and SRF be the costs of primary and sec-ondary paths, respectively, of the RF solution, and PLB andSLB those of the LB solution. From the above discussions,we have

PRF ≤ PLB, and (2)

PRF + SRF ≥ PLB + SLB (3)

In fact, (1) has a linear characteristic. Assume that a net-work has distinct primary path and secondary path solutionsby the RF and LB algorithms, respectively. Figure 4 illus-trates a typical graph that shows the relationship betweenthe total cost and α, based on the solutions obtained by RFand LB methods, as α varies from 0 to 1.

Fig. 3 (a) LB Step 1:Determine the least costpath, P1, between node ‘a’ andnode ‘f ’. (b) LB Step 2:Replace P1 with −P1.(c) LB Step 3: Find thesubsequent least cost path, Pa,in the modified network andremove the −P1 link that appearin Pa. (d) A pair of link-disjointpath by the LB algorithm (a) (b)

(c) (d)


Fig. 4 Cost functions of distinct RF and LB solutions

Fig. 5 Optimal solution when α = 0.4

Note that it is obvious when PRF = PLB, the two linescoincide with each other and this results in the same solutionfor both RF and LB algorithms. When the equality does nothold for (2) and (3), however, we can define the intersectionpoint of these two lines as (y,C∗), as shown in Fig. 4, whereC∗ = PRF + SRF · y = PLB + SLB · y. It is easy to solve that

y = PLB − PRF

SRF − SLB, and (4)

C∗ = PLBSRF − PRFSLB

SRF − SLB(5)

2.2 Suitability of RF and LB in α + 1 protection

Suppose now, the α +1 protection is to be implemented intothe example network of Fig. 1. Let α = 0.4. Based on the RFsolution from Fig. 2c, the total cost to implement the α + 1,as defined in (1), is CRF(0.4) = 5 + (0.4) × 12 = 9.8. Forthe LB solution given in Fig. 3d, the total cost to implementthe α + 1 is CLB(0.4) = 7 + (0.4) × 7 = 9.8.

In actual fact, however, the optimal solution is neitherthe solution by RF nor LB when α = 0.4. Figure 5 de-picts such an optimal solution (primary path: a-b-e-f, pri-mary path cost = 1 + 4 + 1 = 6, secondary path: a-c-d-f,secondary path cost = 4+1+4 = 9) which has a lower costthan RF and LB solutions, where the total cost, C′, to adoptthis latest solution is given by C′(0.4) = 6+ (0.4)×9 = 9.6.

In some instances, if there exists an optimal solution atsome value of α where it is neither by the RF nor the LB,

Fig. 6 Typical Total Cost function of distinct RF method and LB pathprotection solution with the occurrence of a mid-optimal solution

we name this solution as the mid-optimal (MO) solution. Letthe MO solution be found at α = q , where 0 < q < 1, and letPMO and SMO be the costs of primary and secondary paths,respectively, given by such an MO solution. We have

PMO + qSMO < PRF + qSRF, and (6)

PMO + qSMO < PLB + qSLB (7)

Figure 6 illustrates the occurrence of an MO solutionwhen x < α < z. In the next section, we study the condi-tions for an MO solution if it does exist.

3 Properties of mid-optimal solution

In this section, we derive several properties of MO solution.These properties are vital for constructing a paths-findingalgorithm to identify the MO solutions in a network.

Lemma Consider a pair of source and destination nodes ina network graph. If the RF and LB algorithms result in thesame pair of link disjoint paths, there is no MO solution.

Proof We prove the lemma by contradiction.If RF and LB have the same solution, then we have, for

all α,

PRF + αSRF = PLB + αSLB = P + αS (8)

Suppose an optimal solution, that is not similar with the RFand LB solution, exists at α = q where 0 < q < 1. Let thecost of the primary and secondary paths be denoted by P ′and S′, respectively. Thus,

(P ′ + qS′) − (P + qS) < 0 (9)

Let

u = P ′ − P (10)

v = S′ − S (11)


We define the cost difference function as,

Cost Diff = (P ′ + αS′) − (P + αS) = u + αv (12)

When α = 1, since the LB solution is always the optimal,we have

u + v > 0 ⇒ v > −u (13)

When α = q , from (9) we have

u + qv < 0 ⇒ v <−u

q(14)

Combining the facts from (13) and (14) yields,

−u < v <−u

q(15)

When α = 0, on the other hand, since the RF solution isalways the optimal, we have

u > 0 (16)

And thus for 0 < q < 1, the following inequality is alwaystrue,

−u

q< −u (17)

Since (15) contradicts against (17), there cannot exist an MOsolution if RF and LB algorithms produce the same solu-tion. �

By referring to Fig. 6, the mid-optimal solution that ap-pears in the α + 1 protection scheme is found to exist onlywhen specific conditions are satisfied.

Theorem Given a pair of link-disjoint paths with primaryand secondary costs P ′ and S′, respectively, for a pair ofsource and destination nodes in a network graph, it is anMO solution if and only if the following conditions are allsatisfied:

(i) PRF < P ′ < PLB, (18)

(ii) PRF + SRF > P ′ + S′ > PLB + SLB (19)

(iii)P ′ − PRF

SRF − S′ <PLB − P ′

S′ − SLB(20)

Proof For the “if” part, (19)–(18) yields

SRF > S′ > SLB (21)

From (19) and (20), define a value q ′ where

0 <P ′ − PRF

SRF − S′ < q ′ < PLB − P ′

S′ − SLB< 1 (22)

we have

P ′ − PRF

SRF − S′ < q ′ ⇒ P ′ + q ′S′ < PRF + q ′SRF, and,

(23)

q ′ < PLB − P ′

S′ − SLB⇒ P ′ + q ′S′ < PLB + q ′SLB (24)

Inequalities (23) and (24) show that the given solution is anMO solution at α = q ′.

For the “only if” part, we assume P ′ and S′ constitute anMO solution, thus P ′ = PMO, S′ = SMO, and (6) and (7) aretrue. Note that an MO solution is neither the same as a RFsolution nor a LB solution. Since the primary path of the RFmethod is the least-cost/shortest path in the graph, hence itis always true that

PRF < P ′ (25)

On the other hand, it has been proven in [7] that LB solutionis the optimal for 1 + 1 full bandwidth protection, so it isalways true that

P ′ + S′ > PLB + SLB (26)

To prove that P ′ < PLB, from (26) we have

−qP ′ − qS′ < −qPLB − qSLB (27)

On the other hand, from (7) we have

P ′ + qS′ < PLB + qSLB (28)

Subsequently, from inequalities (27) and (28), we have

(1 − q)P ′ < (1 − q)PLB (29)

Since q < 1, we have P ′ < PLB.To prove that P ′ + S′ < PRF + SRF , from (25) we have

(1 − q)PRF < (1 − q)P ′ (30)

On the other hand, from (6) we have

P ′ + qS′ < PRF + qSRF (31)

Therefore, from inequalities (30) and (31), we have

q(P ′ + S′) < q(PRF + SRF) (32)

And since q > 0, we have P ′ + S′ < PRF + SRF .To prove that (iii) is also true, from (6) and (7) we have

P ′ − PRF

SRF − S′ < q (33)

q <PLB − P ′

S′ − SLB(34)


Thus, (iii) is true as well. �

It is fairly obvious that there is a possibility of more thanone MO solution to exist in a graph. Suppose that a graphcontains n MO solutions, where n ≥ 1. Let the costs of thepairs of primary and secondary paths of these MO solutionsbe labeled according to the following manner:

(PMO(1), SMO(1)), (PMO(2), SMO(2)), (PMO(3), SMO(3)), . . . ,

(PMO(n), SMO(n))

where

PRF < PMO(1) < PMO(2) < PMO(3) < · · · < PMO(n) < PLB

The derived MO properties in (18), (19) and (20) still ap-ply for all these MO solutions. Moreover, there are addi-tional characteristics that are associated between these MOsolutions as shown in the following corollary.

Corollary 1 Consider a network graph which has n MOsolutions between a source node and a destination node.Let the costs of the primary and secondary paths of theseMO solutions be denoted by PMO(i) and SMO(i), respec-tively, such that PMO(i−1) < PMO(i), where i = 1,2, . . . , n

and PMO(0) and SMO(0) be defined as PRF and SRF , respec-tively. The following inequalities are true for all MO solu-tions.

(i) PMO(i−1) + SMO(i−1) > PMO(i) + SMO(i) (35)

(ii)PMO(i) − PMO(i−1)

SMO(i−1) − SMO(i)

<PLB − PMO(i)

SMO(i) − SLB(36)

Proof We prove (i) by contradiction.Let (PMO(i), SMO(i)) be the cost of the optimal solution

at α = y1, where 0 < y1 < 1. Therefore the subsequent in-equalities are true

PMO(i−1) + y1SMO(i−1) > PMO(i) + y1SMO(i) (37)

PLB + y1SLB > PMO(i) + y1SMO(i) (38)

We know the inequality below to be always true by defini-tion

PMO(i−1) < PMO(i) (39)

Now suppose

PMO(i−1) + SMO(i−1) ≤ PMO(i) + SMO(i) (40)

From (39) and (40), the subsequent inequality is true forall 0 < α < 1.

PMO(i−1) + αSMO(i−1) < PMO(i) + αSMO(i) (41)

However, at α = y1(41) contradicts (37). So (40) cannot betrue. Therefore we can conclude that:

PMO(i−1) + SMO(i−1) > PMO(i) + SMO(i)

To prove that (ii) is also true, from (37) and (38) we have

PMO(i) − PMO(i−1)

SMO(i−1) − SMO(i)

< y1 (42)

PLB − PMO(i)

SMO(i) − SLB> y1 (43)

Thus (ii) is true as well. �

With the above corollary, an effective algorithm for find-ing link-disjoint optimum paths can be constructed. Supposethat PLB is the Lth shortest path in the graph, the primarypath of any existing MO solution must be the K th shortestpath of the graph where 1 < K < L.

For the subsequent section, we propose an algorithm toidentify the MO solutions in a sequential manner.

4 Optimum paths-finding algorithm

4.1 α-Optimum (AO) paths-finding algorithm

In the following, we present an optimum paths-finding al-gorithm capable of identifying the optimal solution withrespect to any α for α + 1 protection. We call this algo-rithm as α-optimum (AO) algorithm. Path notations in bold(P1, S1, PRF , SRF , PLB, SLB, PMO(n), SMO(n), PK and SK )denote the set of links that form their respective path. Reg-ular path notations (P1, S1, PRF , SRF , PLB, SLB, PMO(n),SMO(n), PK , and SK) denote their path cost.

Generally, our algorithm incorporates the K th shortestpath algorithm proposed by Yen [9]. The Yen’s algorithmis selected because it has the lowest worst-case complexityamong K th shortest path ranking algorithm [9, 10]. The K thshortest path algorithm is utilized to identify paths betweenthe source and destination node in increasing cost order se-quentially. In each progressing cycle of our algorithm, afterthe K th shortest path is found, we remove all the links on thepath, and find the link-disjoint shortest path against the K thshortest path. These two paths would be the candidate for anMO solution. The candidate solution is then evaluated uponthe MO properties derived in Sect. 3 to determine the accep-tance or rejection as an MO solution. Importantly, accordingto the properties derived, we can safely terminate the algo-rithm upon reaching the LB solution and this represents thestop rule of our algorithm. The derivation of the MO proper-ties in Sect. 3 also enables our algorithm to be more efficientas we can incorporate rules, based on the properties to ex-clude non-optimal solutions during execution.


The possibility of a ‘trap topology’ [7, 11] in the graphis another concern. In a graph with ‘trap topology’, there isno link-disjoint path with respect to the shortest path, andthus the RF method would return no solution for path pro-tection. This issue is also addressed in our algorithm. Whena ‘trap topology’ occurs, the cost of the secondary path, SRF ,is assumed as very large, or SRF ∼ ∞. When α = 0, the RFsolution is still optimal as SRF will be discounted from thetotal cost calculation. The properties of the MO solution de-rived in Sect. 3 still holds under this assumption.

AO paths-finding algorithm// Define

s: source noded: destination nodeRF: Remove Find method (also known as two-stepmethod)LB: Link-disjoint Bhandari algorithmMO: Mid-Optimal

// Initialization (Note that K is the next shortest path to befound, where 1st shortest path can be easily found by theDijkstra’s algorithm)

K = 2n = 1PRF = SRF = PLB = SLB = PMO = SMO = PK = SK�

// Algorithm startsSTEP 1. Identify the LB solution between s and d, where

PLB and SLB are the primary and secondary pathof the LB solution respectively.if no paths exist // No connection exists between sand d

Go to STEP 11else

{ PLB ← cost {PLB}//LB solutionfoundSLB ← cost {SLB}

}end if

STEP 2. Determine the shortest path, P1, between s and dif P1 = PLB // LB solution is the only optimal so-lution

Go to STEP 11end if

STEP 3. Remove all related links of path P1 from the graphand find the shortest path in the ‘trimmed’ graph,S1, between s and d.if S1 is found

{ PRF ← P1 // RF solution foundSRF ← S1

PRF ← cost {P1}SRF ← cost {S1}Restore graph to original state

}else // Graph has trap topology

{ PRF ← P1 // RF Solution in a traptopology graphSRF ← �

PRF ← cost {P1}SRF ← ∞Restore graph to original state

}end if

STEP 4. Determine the K th Shortest path, PK , between sand d

PK ← cost {PK}if PK ≥ PLB // Stop parameter.

Go to STEP 11end if

STEP 5. Remove all related links of path PK from the graphSTEP 6. Find the shortest path in the modified graph, SK ,

from s and d.if no path exists

Go to STEP 10end if

STEP 7. SK ← cost {SK}if SK < PK

Go to STEP 10else

{ if n > 1Go to STEP 9

elseGo to STEP 8

end if}

end ifSTEP 8. if PRF + SRF < PK + SK < PLB + SLB

{ if PK−PRFSRF−SK

< PLB−PK

SK−SLB{ PMO(n) ← PK

PMO(n) ← cost {PK}// First MO solution foundSMO(n) ← SK

SMO(n) ← cost {SK}n ← n + 1}

Go to STEP 10end if

}else

Go to STEP 10end if

STEP 9. if PMO(n−1) + SMO(n−1) < PK + SK < PLB + SLB

{ if PK−PMO(n−1)

SMO(n−1)−SK< PLB−PK

SK−SLB

{ PMO(n) ← PK

PMO(n) ← cost {PK}// Subsequent MO solutions foundSMO(n) ← SK

SMO(n) ← cost {SK }


n ← n + 1}

Go to STEP 10end if}

elseGo to STEP 10

end ifSTEP 10. Restore graph to original state

K ← K + 1Go to STEP 4

STEP 11. End algorithm

It should be noted that the proposed algorithm has a com-plexity of O(Kono(mo + no logno)) where Ko representsthe number of paths between the source and destinationnodes whose cost are larger than PRF but smaller than PLB,no indicates the number of nodes in the graph, and mo isthe number of edges (links). The core complexity of this al-gorithm is fundamentally constructed from the Yen’s algo-rithm [10]. Such complexity is not considered high becausethe algorithm is applied at the call admission level, which ingeneral allows more time in setting up a call.

4.2 Reversed α-optimum (RAO) paths-finding algorithm

Apart from the MO properties derived in Sect. 3, an addi-tional characteristic can also be observed in the MO solu-tions. That is, the primary and secondary paths of the MOsolution, denoted as PMO and SMO respectively, cannot belink-disjoint with the least-cost path, P1. We prove this char-acteristic in the next corollary.

Corollary 2 Given a pair of source and destination nodesin a network graph, if an MO solution exists and the solutionconstitutes a link-disjoint pair of paths PMO and SMO, bothpaths are not link-disjoint with the least-cost path, P1.

Proof We prove Corollary 2 by contradiction.Let the costs of paths PMO, SMO, P1, PLB and SLB be

denoted accordingly as PMO, SMO, P1, PLB and SLB. Recallthat PLB and SLB are the respective primary and secondarypaths of the LB solution, and they form a pair of link-disjointpaths which has the least cost sum when α = 1.

Suppose PMO is link-disjoint with P1

Since this pair of paths are said to be link-disjoint, there-fore P1 and PMO can be a possible solution for the α + 1protection.

Inequality (18) from the Theorem has shown that PRF <

PMO < PLB where PRF is the primary path cost of the RFsolution and hence PRF = P1.

Given that PLB ≤ SLB, we have P1 < PLB and PMO <

SLB. Thus

P1 + PMO < PLB + SLB (44)

However, inequality (44) contradicts the fact in [7] whichproved that the LB solution’s total cost, PLB + SLB, is opti-mal at 1 + 1 full bandwidth protection.

Therefore PMO cannot be link-disjoint with P1

Now suppose SMO is link-disjoint with P1

Since this pair of paths are said to be link-disjoint, there-fore P1 and SMO can be a possible solution for the α + 1protection.

As P1 is the least-cost path, P1 < PMO. Thus for all 0 ≤α ≤ 1, it is always true that

P1 + αSMO < PMO + αSMO (45)

On the other hand, by definition the MO solution must beoptimal among all other solutions at some α = q , where 0 <

q < 1, so that the following inequality is true

PMO + qSMO < P1 + qSMO (46)

Since (46) contradicts (45), SMO cannot be link-disjointwith P1 �

In fact Corollary 2 provides another method to find theMO solutions between a source and destination node in anetwork graph. That is, once the shortest path is identified,we can find all the possible “link-joint” paths against thisshortest path, and test their validity to become an MO solu-tion.

With the derivation of Corollary 2, we construct a supple-mentary optimum paths-finding algorithm for the α + 1 pro-tection. We name this algorithm as the reversed α-optimum(RAO) algorithm. Likewise, notations in bold (P1, S1, PRF ,SRF , PLB, SLB, PMO(n), SMO(n), P′

K , PK and SK ) denote theset of links that form their respective paths while regular no-tations (P1, S1, PRF , SRF , PLB, SLB, PMO(n), SMO(n), P ′

K ,PK , and SK) denote their respective path costs.

Generally the RAO algorithm is an extended form of theLB algorithm. Similar to the LB algorithm, the graph willinitially be modified. This is done by finding the shortestpath in the graph, P1, and replacing it with −P1. −P1 isobtained by reversing the direction of the links associatedto P1 and converting the cost on the links along path P1 toits equivalent negative value. The K th shortest path algo-rithm is then used to identify paths between the source anddestination node in increasing cost order sequentially, in themodified graph. We denote the K th shortest path as P′

K . Forevery P′

K that is found, the links which are shared with −P1

are temporary removed while the remaining links of −P1

are restored to their original direction and values. The linksof P′

K and P1 which are left will be grouped to form a pairof link-disjoint path, PK and SK where PK ≤ SK . This pairwould be the candidate for an MO solution. The candidatesolution is then evaluated upon the MO properties derivedin Sect. 3 to determine the acceptance or rejection as an MOsolution. The algorithm stops when the K th shortest path(P′

K ) found is link-disjoint with −P1. This stop rule is basedon the derivation of Corollary 2. It can be further validated


using derivations for LB algorithm found in [7] that subse-quent solutions beyond this limit are not optimal.

The ‘trap topology’ is not an issue for RAO algorithmas the core structure of the algorithm is based on the LBalgorithm. Due to its characteristic, the LB algorithm is ableto circumvent the ‘trap topology’ [7].

RAO paths-finding algorithm// Define

s: source noded: destination nodeRF: Remove Find method (also known as two-stepmethod)LB: Link-disjoint Bhandari algorithmMO: Mid-Optimal

// Initialization (Note that K is the next shortest path to befound in the modified graph, where the 1st shortest path canbe found using the Bellman-Ford’s algorithm. This is due tothe presence of negative link cost in the modified graph)

K = 2n = 1PRF = SRF = PLB = SLB = PMO = SMO = PK =SK = P′

K = �

// Algorithm startsSTEP 1. Identify the LB solution between s and d, where

PLB and SLB are the primary and secondary pathof the LB solution respectively.if no paths exist // No connection exists between sand d

Go to STEP 13else

{ PLB ← cost {PLB} // LB solutionfoundSLB ← cost {SLB}

}end if

STEP 2. Determine the shortest path, P1, between s and dif P1 = PLB // LB solution is the only optimal so-lution

Go to STEP 13end if

STEP 3. Remove all related links of path P1 from the graphand find the shortest path in the ‘trimmed’ graph,S1, between s and d.if S1 is found

{ PRF ← P1 // RF solution foundSRF ← S1

PRF ← cost {P1}SRF ← cost {S1}Restore graph to original state

}else // Graph has trap topology

{ PRF ← P1 // RF Solution in a traptopology graph

SRF ← �

PRF ← cost {P1}SRF ← ∞Restore graph to original state

}end if

STEP 4. Replace P1 with −P1 in the graph. −P1 is obtainedby reversing the direction of all related links ofpath P1 from the graph and replacing the associ-ated cost of the P1 links with equivalent negativevalues.

STEP 5. Determine the K th Shortest path, P′K , between s

and d in the modified graph constructed in STEP4.

STEP 6. Remove the links along path P′K , that is shared with

−P1

if P′K ∩ −P1 = � // Stop parameter

Go to STEP 13end if

STEP 7. Restore the remaining links of −P1 back to its orig-inal direction together with their respective initialnon-negative cost

STEP 8. Group the remaining links of P′K and P1 into two

paths PK and SK where; cost{PK} ≤ cost{ SK}STEP 9. PK ← cost{PK}

SK ← cost{SK}if n > 1Go to STEP 11

elseGo to STEP 10

end ifSTEP 10. if PRF + SRF < PK + SK < PLB + SLB

{ if PK−PRFSRF−SK

< PLB−PK

SK−SLB

{ PMO(n) ← PK

PMO(n) ← cost {PK }// First MO solution foundSMO(n) ← SK

SMO(n) ← cost {SK}n ← n + 1

}Go to STEP 12end if

}else

Go to STEP 12end if

STEP 11. if PMO(n−1) + SMO(n−1) < PK + SK < PLB + SLB

{ if PK−PMO(n−1)

SMO(n−1)−SK< PLB−PK

SK−SLB

{ PMO(n) ← PK

PMO(n) ← cost {PK}// Subsequent MOsolutions foundSMO(n) ← SK


SMO(n) ← cost {SK}n ← n + 1

}Go to STEP 12end if

}else

Go to STEP 12end if

STEP 12. Restore graph to original stateK ← K + 1Go to STEP 4

STEP 13. End algorithm

The complexity of RAO is given as O(Komon2o), where

Ko represents the number of paths in the modified graphbetween the source and destination node which are “link-joint” with path −P1, no indicates the number of nodes inthe graph, and mo is the number of edges (links). The com-plexity of RAO is larger by a factor of no when comparedto the AO. This is because there are negative link costs inthe modified graph. As a result, the K th shortest path algo-rithm in RAO will utilize the Bellman Ford algorithm (in-stead of the Dijkstra’s algorithm) which leads to a higheroverall complexity.

Although having a higher complexity than the AO, theRAO has a more novel stop parameter. The main iterationcycle in RAO stops when the K th shortest path found inthe modified graph is link disjoint with −P1. Additionallythe RAO is able to circumvent the ‘trap topology’ therebyeliminating unnecessary iterations that do not produce theexpected candidate solution as may happen in the case ofAO. This generally results in the RAO having to run feweriteration cycles compared to the AO.

4.3 Discussions on relevant proposed paths-findingalgorithms

The paths-finding algorithm proposed in [12] for available-based routing shares some similarities with our proposedAO paths-finding algorithm. However, it only utilizes theLB and RF to identify the primary and secondary paths withminimized unavailability. Under the α +1 protection, the al-gorithm in [12] will not be able to account together the MOsolution which, as mentioned, is an optimal solution that isneither the solution by LB nor RF and exist when 0 < α < 1.

Another algorithm denoted as DP2LC in [13] likewiseincorporates the K th shortest path algorithm to enumeratepairs of possible link-disjoint paths in the network withina determined lower and upper bound which represents thealgorithm’s stop rule. Nevertheless, the application of thisalgorithm is optimized for networks with dual cost metrics.Basically, there are two main iterative cycles in the algo-rithm labeled as ‘Procedure A’ and ‘Procedure B’, each

comprising a subroutine. In general, the subroutine will iter-ate through the graph, using the K th shortest path algorithm,according to a specified set of cost metric permutation to findthe best candidate pair of path. By comparing the DP2LCalgorithm with our AO paths-finding algorithm, the DP2LCwould require double the running time to identify the opti-mal solution for the α + 1 protection scheme.

The study in [14] has a similar problem formulation de-scribed as asymmetrically weighted pair of disjoint pathsand focuses on finding and minimizing the cost functioncomprising from a node-disjointed primary and protectionpath between a given source and destination. Two methodswere proposed in the paper. The first method is an opti-mum paths-finding algorithm that is based on the K th short-est path searching. However, the K th shortest path search-ing is an iterative process. Without the integration of a stoprule (such as that found in our optimum paths-finding al-gorithms), an exhaustive search would be performed whichsubstantially increases the running time. It was acknowl-edged by the authors that the efficiency of their first methodremains a concern.

The second method finds the optimal pair of link-disjointpaths by means of integer linear programming (ILP). Thehigh complexity by the ILP model (especially for large net-works) is subsequently reduced through the relaxation of itsconstraints referred to as Linear Programming Relaxation(LPR) and Single Flow Relaxation (SFR) to become a lin-ear program (LP) and minimal network cost flow (MNCF)problem respectively. Nevertheless, these methods do notguarantee that a solution could be found even though it ex-ists. The cause as pointed out in the paper is due to the“split-flow” problem. The occurrence of “split-flow” yieldsresults which are not useful. Although there are additionalapproaches being presented to remedy the issue, the cor-responding solution obtained may not necessarily be opti-mal. Furthermore, these additional approaches dealing withthe “split-flow” problem add to the complexity of the sec-ond method. We have shown that the RF, MO and LB so-lutions are the optimal solutions at their respective α rangeand our optimum paths-finding algorithms are constructedto efficiently identify all these solutions.

In the following, we present simulation results to showthe performance of our proposed AO and RAO paths-findingalgorithms. Since both the AO and RAO paths-finding algo-rithms are designed to identify the optimal solutions (includ-ing the MO solution) for the α +1 protection, therefore bothalgorithms are generally expected to produce similar perfor-mance outcomes in terms of blocking rate and average costof calls.

5 Simulation results

Simulation is carried out on a 28 node 45 bidirectional linkUSAnet topology as shown in Fig. 7. We assume that the ca-


Fig. 7 USA net with preset linkweights

pacity of each link is given by W = 16 units of bandwidth.The links are assumed to be weighted links. As mentioned,the link weights represent the cost (per unit bandwidth) in-curred for data to traverse the particular link. Although thereare numerous possible link-weights permutations, we selectthis particular combination as it explicitly highlight the av-erage cost difference between the solutions provided by theRF, LB and our proposed AO and RAO paths-finding algo-rithms.

Calls arrive in accordance to a Poisson process withrate λ. Source and destination node is randomly selectedbased on a uniform distribution. The call duration is expo-nentially distributed with a mean of 1/μ. Therefore the Er-lang load offered to the entire network is ρ = λ/μ. Eachpoint on the graph shows the average cost after 1 millioncalls.

A successful call request would reserve 1 unit of band-width on the primary path while on the secondary path thebandwidth reservation is proportional to α. If there is notenough capacity for either the primary or the secondarypath’s request, the call is blocked.

5.1 Simulation 1: static link cost

Utilizing the link-cost configuration of Fig. 7, Fig. 8 showsthe average total costs for calls monitored between node 0and node 27 with respect to α at the network Erlang loadof 40. As this load value is considered low, no call blockingwas noticed in the simulation for all α. The results in Fig. 8show that our proposed algorithms ensure minimum averagetotal cost for all values of α. It was also able to identifythe MO solution (noticeable between α = 0.2 to α = 0.9).These were not achieved by the existing RF and LB paths-finding algorithms.

Figures 9 and 10 show the average total costs of AO,RAO, LB, and RF and their respective blocking probabili-ties at the network Erlang load of 100 while Figs. 11 and 12

Fig. 8 Average total cost at Erlang load = 40 (static link cost)


are respectively the average total cost and blocking prob-ability evaluated at the network Erlang load of 150. As thenetwork load transition to a higher setting, there is an overallincrease in the average total cost as well as blocking proba-bility. The reason being the network is handling more calls,


Fig. 10 Blocking probability at Erlang load = 100 (static link cost)


Fig. 12 Blocking probability at Erlang load = 150 (static link cost)

and it is likely that links along the optimal paths are fully uti-lized most of the time. Hence, it is understandable that thecosts of incoming calls, at times, would be higher becausethese calls have to be routed through a more expensive butstill available set of links. Since at this state the network’s

resource is scarce, there is a higher occurrence of call block-ing.

Figure 11 shows that despite the network’s resource limi-tations at high loads, the AO and RAO managed to generallymaintain a minimum average total cost compared to LB andRF. Figure 12 further demonstrates that in terms of block-ing probability, the trade-off by AO and RAO is minimal incontrast with the cost savings attained over RF and LB.

It is observed in Fig. 9 and 11 that the RF is inclined to-wards a higher average total cost compared to the AO, RAOand LB. This is because unlike the solutions by AO, RAOand LB, the solutions by RF have a large cost difference be-tween its primary path and secondary path. This is due tocharacteristic of the RF solution in which its secondary pathhas to be completely link-disjoint from the current shortestpath. Consequently as the value of α increases, the cost ofthe secondary path in the RF solution would cause the dis-parity in average total cost by the RF to be apparent.

5.2 Simulation 2: dynamic link cost

In the second phase of the simulation, the network is as-signed with dynamic link cost. The implementation of dy-namic link cost provides a more realistic network setting toevaluate the performance of the paths-finding algorithms byenabling the load to be more evenly distributed over all thelinks. The cost of each link is influenced by their respectiveresidual bandwidth. Let W denote the link capacity, ci theconstant link cost (as shown in Fig. 7) and Ri the residualbandwidth of link i, respectively, where 0 ≤ Ri ≤ W . Foran incoming call request, prior to computing the pair of pri-mary and link-disjoint secondary paths, the link costs in thenetwork are adjusted according to (47) as shown below,

c′i = ci + β

(1

Ri + ε− 1

W

)(47)

where c′i is the updated cost of link i, β a positive constant,

and ε a positive constant with a very small value to avoidundefined link cost. In the simulation, we set β = W = 16,and ε = 0.0001. It can be seen from (47) that when the resid-ual of a link reduces, the link cost increases. Therefore, suchcongested links will be avoided by the paths-finding algo-rithm in favor for other links with higher residual bandwidth.

Using the network with initial link cost as in Fig. 7,Figs. 13, 14 and 16 show the average total cost of AO, RAO,LB and RF at the network Erlang load of 40, 100 and 150respectively with dynamic link costs corresponding to (47).Figures 15 and 17 depict the respective blocking probabil-ities at the network Erlang Load of 100 and 150. No callblocking was noticed at the Erlang Load of 40 for all valuesof α. The results in Fig. 13 show that at a low load setting,the AO and RAO algorithms function as expected by ensur-ing minimum average total cost for all values of α. From


Fig. 13 Average total cost at Erlang load = 40 (dynamic link cost)


Fig. 15 Blocking probability at Erlang load = 100 (dynamic link cost)


Fig. 17 Blocking probability at Erlang load = 150 (dynamic link cost)

Figs. 14 and 16 the change in the average total costs can beobserved as the load increases. At a high load setting the av-erage total costs are almost the same with the AO and RAOhaving a slightly lower average cost.

It should be noted that in general the average total costsby the network with dynamic link cost are higher than thenetwork with static link cost when compared against theirrespective Erlang loads. This is attributed to the considera-tion of dynamic link cost. The utilization of links increasesthe link costs. This leads to a higher average total cost.However, with dynamic link cost, the load is more evenlydistributed over the network. By comparing the results ofblocking probabilities, it is obvious that the network withdynamic-link cost achieves better performance by having anoverall lower blocking probability.

6 Conclusions

This paper discussed about the partial bandwidth path pro-tection scheme denoted as α + 1 protection. The analysisof the optimal solution for this protection scheme was in-


troduced. An optimization occurrence named as the mid-optimal (MO) solution, which is also part of the set of op-timal solutions was highlighted. The characteristics inher-ent in the MO solutions were derived and a path findingalgorithm to identify the set of optimal solutions for theα + 1 protection, labeled as α-optimum (AO) paths-findingalgorithm, was subsequently presented. An ensuing propertyconcerning the MO solution enables a supplementary opti-mum path-finding algorithm to be constructed, labeled asreversed α-optimum (RAO) paths-finding algorithm. Simu-lation results show that in terms of cost, the proposed AOand RAO paths-finding algorithm outperforms the existingRF and LB paths-finding algorithms for the implementationof α+1 protection. An extension for this research area is thedevelopment towards reducing the complexity of the pro-posed path finding algorithms.

Acknowledgement This research is supported by grant 01-02-11-SF0084 from the Ministry of Science, Technology and Innovation(MOSTI), Malaysia.

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Ming-Lee Gan is a project researchassistant and a Ph.D. student atUniversiti Tunku Abdul Rahman,Malaysia. He received his B.Eng. inElectrical & Electronics from Uni-versiti Tenaga Nasional, Malaysiain 2004 and M.Sc. in Systems En-gineering & Management from theMalaysia University of Science andTechnology in 2006. His current re-search focus is in network path pro-tection, network routing algorithmsand network reliability analysis.

Soung-Yue Liew received his bach-elor’s degree in Electrical Engineer-ing from the National Taiwan Uni-versity in 1993, and Master’s andPh.D. degrees in Information En-gineering from the Chinese Uni-versity of Hong Kong (CUHK) in1996 and 1999, respectively. From1999 to 2003, he was an assistantprofessor and research associate atCUHK and Polytechnic University(New York), respectively. In August2003, he joined Universiti TunkuAbdul Rahman in Malaysia, wherehe is currently an associate profes-

sor. His research interest is in broadband communications, switch de-sign and switching algorithms, network and routing algorithms, opticaland wireless networking, etc.

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Effective algorithms for finding optimum pairs of link-disjoint paths in α+1 path protection