a new solution to the k-shortest paths problem and its application in wavelength routed optical...

Photon Netw Commun (2006) 12:269–284DOI 10.1007/s11107-006-0027-0

ORIGINAL ARTICLE

A new solution to the K-shortest paths problem and itsapplication in wavelength routed optical networks

Junjie Li · Hanyi Zhang

Received: 4 July 2005 / Revised: 2 August 2005 / Accepted: 6 December 2005 / Published online: 9 September 2006© Springer Science+Business Media, LLC 2006

Abstract In communication networks, traffic carriedover long paths suffers from a higher call blocking prob-ability (CBP) than those carried over short paths. Thisis a well-known fairness problem. Such a problem be-comes more serious in wavelength-routed opticalnetworks (WRONs) due to the wavelength continu-ity constraint. This paper aims to enhance the fairnesscharacteristic in WRONs by the novel classified alter-nate routing (CAR) approach. As a foundation of thisapproach, the first K-shortest paths (KSP) between adesignated source/destination node pair in a given net-work should be obtained simultaneously. This has beenstudied as the KSP problem in the literature. In this pa-per, a new loopless deviation (LD) algorithm for solv-ing such a loopless KSP problem will be proposed. Itoutperforms existing solutions in terms of running timein real-life implementations. In order to measure thefairness characteristic, we first classify all connection re-quests into categories such as 1-hop traffic, 2-hop traffic,etc., according to the number of minimal hop count be-tween the corresponding source/destination node pair.We then quantify the fairness characteristic by the ratioof the average CBP of each traffic category to that of1-hop traffic such that CBP ratios with a value closer to1 are preferred. We will show that such a measure cri-terion is more precise and robust than existing ones inthe literature. Finally, numerical experiments will revealthat the CAR approach outperforms existing fairnessenhancement methods when considering the compre-

J. Li (B) · H. ZhangDepartment of Electronic Engineering,Tsinghua University,Beijing 100084,Chinae-mail: [email protected]

hensive performance in terms of the balance betweenthe fairness characteristic and the overall CBP feature.

Keywords Wavelength-Routed Optical Networks(WRONs) · Fairness · K-Shortest Paths (KSP) ·Loopless Deviation (LD) · Classified AlternateRouting (CAR)

Introduction to the fairness problem in WRONs

The wavelength division multiplexing (WDM) networkhas already been accepted as the preferred method tohandle the ever-increasing bandwidth demands requiredby customers [1, 2]. In a wavelength-routed optical net-work (WRON) consisting of optical cross-connect(OXCs) nodes and WDM transmission links, end userscommunicate with each other via all-optical wavelengthchannels, which are called lightpaths [3]. Because wave-length conversion techniques are still too immature andexpensive to be commercially available, a lightpath mustoccupy the same wavelength on all the fiber links throughwhich it traverses. This is called the wavelength continu-ity constraint.1 Under such a constraint, the networkoperator should not only find a proper route from thesource node to the destination node for a connection re-quest, but also assign a free wavelength along the entireroute. Such a problem is called the routing and wave-length assignment (RWA) problem. Hui Zang et al. have

1 Although many publications concerned with optical networksthat are fully or partly equipped with wavelength converters, in thispaper, without special indication, the acronym WRON means bydefault a wavelength routed optical network without wavelengthconversion.

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presented a good review on the existing literature aboutthis problem in Ref. [4].

With the explosive increase of data service, currenttraffic in optical networks shows its dynamic feature, i.e.,both the arrival of connection requests and the durationtime of the service are stochastic. The RWA algorithmfor such optical networks should set up and tear downlightpaths dynamically according to the stochastic ar-rival of connection requests (calls) and terminations ofon-line services, which is known as the dynamic RWAproblem [4]. To appraise the diverse dynamic RWA algo-rithms, the overall call blocking probability (CBP) is themost common metric. It has been defined as the ra-tio of the amount of blocked calls (no lightpath canbe established when the call request arrives) to thatof the total calls in the entire network [4, 5]. But itdoes not always capture the full effect of a particulardynamic RWA algorithm on other aspects of networkbehavior, in particular, fairness. Here, fairness refers tothe variability in blocking probabilities experienced byconnection requests between various node pairs, suchthat lower variability is associated with a better fairnesscharacteristic [5]. In general, any network has the prop-erty that paths over more links are likely to experiencehigher blocking than those over less links, and such anunfairness is more serious in WRONs due to the wave-length continuity constraint [6]. The fairness problem isalso an important consideration of QoS (quality-of-ser-vice) RWA algorithms [7]. Rouskas presented a reviewon the existing studies in Ref. [5] and achieved threeguidelines for solving it: (1) Adoption of wavelengthconversions significantly enhances the fairness charac-teristic in optical networks because of the eliminationof the wavelength continuity constraint [8, 9]; (2) Adop-tion of an alternate routing strategy; (3) Wavelengthassignment policies play an important role in terms offairness. Two fairness enhancement methods associatedwith wavelength assignment have been proposed in Ref.[10]: the Protecting Threshold (Thr) method assigns anidle wavelength to a single-hop connection only if thenumber of idle wavelengths on the link is larger or equalto a given threshold; the Wavelength Reservation (Rsv)method reserves some specified wavelengths on spec-ified links for connection requests with multiple links.Both the Thr and Rsv methods enhance the fairnesscharacteristic significantly, but unfortunately, they alsodeteriorate the overall CBP because of the concomi-tant waste of wavelength resources [10, 11]. A TrafficClassification and Service (ClaServ) method was pro-posed in Ref. [11] to optimize the fairness character-istic as well as to reduce the overall CBP, which firstclassifies the connection requests by the hop counts(number of the sequential links of the path) they are

going to traverse, and then implements a waveband ac-cess range (WAR) scheme or a waveband reservation(WRsv) scheme or both for assigning lightpaths to thearrived calls according to their classifications. In fact,such a reduction of the overall CBP comes from thewavelength access range arrangement for diverse con-nection requests in a distributed routing algorithm. Thiscan reduce the CBP caused by wavelength collision, i.e.,two or more connection requests try to use the samewavelength in a particular link simultaneously. WARand WRsv methods are modifications of the Thr and Rsvmethods, and will also deteriorate the overall CBP fea-ture. Harai et al. [12] considered the potential perfor-mance enhancement brought by an alternate routingmethod, but its study was based on approximate anal-ysis modeling with a fixed alternate path set and onlyconsidered small and regular network topologies. So itis out of practicability in real-life networks. Bisbal et al.[13] proposed a novel genetic algorithm for solving thedynamic RWA problem considering the fairness amongconnections.

This paper will propose the classified alternate rout-ing (CAR) approach as a novel method to improve thefairness characteristic without deteriorating the overallCBP feature in WRONs. The paper will also investigatethe performance of such an approach comparativelywith some existing fairness enhancement methods interms of both fairness and overall CBP features. Theessential concept of our CAR approach is to deter-mine more alternate paths for traffic to be carried overlonger paths (in terms of hop count). In this paper, theCAR approach will collaborate with the maximal used(MU) wavelength assignment method to form a com-plete RWA solution. Such an MU method is a modi-fication of the most-used (MU) wavelength assignmentmethod in a single routing strategy and also performs thefunction of selecting the most suitable alternate pathin addition to its basic function of wavelength assign-ment [4]. Our CAR routing approach will adopt thefirst K shortest loopless paths2 as the alternate path set.To solve such a loopless KSP problem, a novel loop-less deviation (LD) algorithm will be proposed in ad-vance. We will prove that our LD algorithm can find thefirst K-shortest loopless paths in a directed non-nega-tive weighted network with less computation complexitythan existing algorithms in literature.

The remainder of this paper is organized as follows:In the next section, we will give a brief review on the

2 In this paper, we only consider weighted directed networks, andif without further specialization, shortest path(s) are equivalent tominimal cost path(s), which means path(s) with the minimum pathcost, and not path with the number of sequential links (hops).

Photon Netw Commun (2006) 12:269–284 271

KSP problem, followed by a detailed description of ourLD algorithm as well as its computational complexityanalyses. Our CAR routing approach together with theadopted MU wavelength assignment and path selectionmethod will be proposed in the next following section.In the section ‘Numerical experiments and discussions’,we will show the performance of our solution via numer-ical experiments on several real-life network topologies.The conclusive remarks will be given at the end of thepaper, and the detailed proofs to some theorems in thepaper will also be presented as Appendix.

KSP problem and LD algorithm

In the alternate routing strategy, a number of alternatepaths with minimal total costs in a network betweena given source and destination node pair are requiredto be determined simultaneously. This is an importantimplementation of the widely studied KSP problem [14,15]. The KSP problem was also studied in many publica-tions as the shortest paths ranking problem [16]. In thissection, we give a brief review on the KSP problem andwe will present our new LD algorithm for solving theloopless KSP problem.

Notations

We present a weighted directed network by a graphG = (V, E, C), where V is the set of nodes, E is the set ofedges, and C is the set of link costs. The parameter n =|V| denotes the amount of nodes and e = |E| denotes theamount of links in network G. Within the scope of thispaper, we do not permit any link cost to be negative. Thenode pair [S, D] denotes a connection request betweensource node S and destination node D, where S ∈ Vand D ∈ V. We present a path P from S to D with thesequence of nodes in P as P = {P(0), P(1), P(2), . . . , P(h−1), P(h)}, where P(0) = S, P(h) = D, and h denotesthe hop count of P. The value Cst(P) denotes the pathcost of P which is defined in such a way that Cst(P) =∑h

i=1 C(P(i − 1), P(i)), where the term C(P(i − 1), P(i))denotes the cost of the link connecting node P(i − 1) tonode P(i).

Definition 1 (K-shortest paths (KSP) problem) The KSPproblem means the determination of a set of paths Pth= {Pth(1), Pth(2), . . ., Pth(K)} that root from the givensource node S and terminate at the given destinationnode D in network G when the objective function isconsidered in such a way that Cst(Pth(k)) � Cst(P), forany path P ∈ {Pth(1), Pth(2), . . ., Pth(k − 1)} and Cst(Q)

� Cst(Pth(K)), for any path Q that also roots from Sand terminates at D in network G but Q /∈ Pth.

Generally, the KSP problem can be classified into twocategories. The first one allows loops in the obtainedpaths. It permits a path passing one node twice or more.Hoffman and Pavley have proposed a very classical andeffective solution to this kind of problems in Ref. [17].The second one only allows loopless paths (i.e., elemen-tary paths or simple paths) [16]. Due to the implemen-tation in WRONs where any loop in a path means thewaste of the rare wavelength resources, we only con-sider loopless paths in this paper. Such a problem canbe expressed more precisely as a loopless KSP prob-lem. Yen has proposed the most popular algorithm tosolve such a problem in Ref. [18]. Due to additionalconstraints, the solutions for loopless KSP problem aregenerally coupled with higher computation complexity[14].

KSP problem is a network optimization problem witha large range of real-world applications [19]. A small butrepresentative portion of these publications is listed inthe Refs. [14–21]. Brander and Sinclair [14] surveyedseveral related papers and gave a comparative studyof four selected algorithms including Hoffman’s algo-rithm [17], Yen’s algorithm [18], and Lawler’s algorithm[20] in terms of running time complexity via theoreticanalyses and numerical simulations. The conclusion wasthat Hoffman’s algorithm wins by its advantage of theshortest running time, but it allows loops in the obtainedpaths. De Azevedo et al. [16] also gave a short review onthe proposed algorithms and classified them into threecategories: one is based on the principle of optimal-ity, represented by the Dreyfus’ algorithm [21] and theLawler’s algorithm [20]; an other class comprises thegenerations of labeling shortest path algorithms; andthe last class encompasses the algorithms based on thepath deletion concept proposed by the author, whichworks over a sequence {G1, G2, . . . , GK} of networks,where G1 is equivalent to G, and its jth shortest path istrivially determined from the shortest path in Gj, whichis an enlarged network obtained from Gj−1 by addingas many auxiliary nodes as the number of intermedi-ate nodes in the shortest path in Gj−1. Hoffman pro-posed the most classical algorithm for the KSP problemwithout the loopless constraint in Ref. [17]. Yen’s algo-rithm [18] was a milestone in this area, which developedthe concept of deviation path. Lawler proposed a gen-eral viewpoint of K best solutions problem and adoptedYen’s algorithm as an implementation. Dreyfus studiedthe problem of determining the KSP from all nodes to afixed terminal node in Ref. [21], which was a modifica-tion of Hoffman’s algorithm [17] but required a larger


running time when only concerning the KSP between asingle node pair.

In order to describe our LD algorithm for the loop-less KSP problem, we define the concept of LD path inadvance.

Definition 2 (Loopless deviation path) We define a loop-less deviation path Q from an original path P, having thesame source node and destination node, which then con-tains exactly one link, called a deviation link, which isnot a link of P, but whose terminal node is containedby P. The tail portion of Q behind the deviation linkcoincides with P. As we illustrate in Fig. 1, path P ori-gins from source node S, transverses node A, and thenterminates at destination node D. Loopless deviationpath Q origins from the same source node S and reachesB (one neighbor node of A) through the shortest path,then passes the deviation link B → A and coincideswith the tail portion of P from A to D. In order to assurethe loopless feature of Q, the shortest path from S toB cannot traverse any node contained by P between Aand D (including these two). A deviation node can beany node on path P including the destination node D,but excluding the source node S. We use Dev(P, A, B)to denote such a loopless deviation path Q. We also callP as the Q’s father path and Q as the P’s child pathhereafter.

Pay attention that Dev(P, A, B) is unique because weonly choose the shortest path from S to B. Althoughthere may be several shortest paths with the same costs,we only choose the first obtained one to construct Dev(P,A, B) and do not worry about overlooking others be-cause those paths can be obtained as the deviation pathsof Dev(P, A, B) if necessary.

Theorem 1 Given any loopless path P in network G froma given source node S to a given destination node D, andA is an arbitrary node contained in P excluding S. If thehead part of P from S to A is the shortest path under theconstraint of the absence of any node contained in the tailportion of P between A and D, the cost of any looplessdeviation path Dev(P, X, Y) is larger or equal to that ofP, where X can be any node on P between A (included)

and S (excluded), and Y is an arbitrary neighbor node ofX that is not contained in P.

S A D

BP: S A DQ: Dev(P,A,B): S B A D

Fig. 1 Illustration of a loopless deviation path

Proof Please refer to Appendix A. �

Theorem 2 Any path Q in network G with source nodeS and destination node D, whose cost is not minimal, is adeviation path from a path P also from S to D in networkG., such that Cst(Q) > Cst(P). If Q is with minimal cost,then either Q is unique, or Q is a deviation form anotherminimal cost path. [17]

Proof Please refer to Ref. [17] or Appendix B. �

Theorem 1 tells us that if we search for all the LDpaths of the shortest path, we can obtain the secondshortest path with a cost equal or larger than the short-est one. By searching all the loopless paths of the secondshortest path and the left deviation paths of the short-est one, we can obtain the third shortest path, and soon. This suggests the iterative manipulation of our LDalgorithm. Theorem 2 tells us that any path, obviouslyincluding any loopless path, can be derived from theshortest path via single or multiple deviation manipula-tions. These two theorems form the theoretic foundationof our LD algorithm.

In order to reduce the running time of our algorithm,we try to avoid generating reduplicate paths in our itera-tive manipulation. But in order to guarantee the validityof the algorithm, we should also prove that all possiblepaths can be examined in our algorithm. We introducetwo auxiliary concepts here, i.e., Start Deviation Node(SDN) and Ignorable Neighbor Node (INN), to help toachieve the above purposes.

We first introduce the concept of SDN, which canshorten the running time without any negative effecton the validity of the algorithm. Please refer to Fig. 2.Suppose that path Q is deviated from its father path P atnode A, then node A is assigned to be the SDN of Q, i.e.,only the node between S (excluded) and A (included)will be considered as deviation node when investigatingthe deviation paths of Q, because the deviation path ofQ from any node between its SDN (i.e., A) and its desti-nation node (i.e., D) has already been generated as thedeviation path of its father path or even its father’s fatherpath, and so on. For instance, we randomly select nodeC: (1) if C locates between A and path P’s SDN (i.e.,

Fig. 2 Illustration of Start Deviation Node (SDN)


node E in Fig. 2). Since the tail portion of Q between Cand D coincides with that of P, the LD path of Q fromC is equivalent to that of P; (2) if C locates between Eand D, then the LD path of both P and Q at node C hasalready been generated by the P’s father path or even itsfather’s father path, according to the analogy presentedin (1). As we have shown, the introduction of SDN willreduce the running time without weakening the validityof our algorithm.

Subsequently, we will introduce the concept of INN,which will support our algorithm in the aspects of bothreducing its running time and guaranteeing its validity.As we have mentioned, our algorithm will adopt itera-tive manipulations to obtain the first K-shortest looplesspaths. Thus in order to reduce the computation time, weshould avoid to utilize the deviation method in Defini-tion 2 straightforward. This method would calculate adeviation path for each neighbor node one-by-one froma deviation node. We give a modified deviation methodto obtain the LD path of P at deviation node A withthe minimal cost directly. Please refer to Fig. 3a. We firstisolate all the nodes contained in the tail portion of Pfrom A (excluded) and D (included), and delete the linkbetween A and its previous node (labeled as F in Fig.3a) on P. Then we compute the shortest path from S toA. Such a path plus the tail portion of P from A to Dforms the shortest loopless path of P at deviation node A,which is labeled as Dev(P, A)−1 in Fig. 3. For the purposeof finding the first K-shortest loopless paths, our inves-tigation should examine all possible LD paths. The iter-ative manipulation in our algorithm will help to achievesuch a goal, but it only obtains the shortest LD pathat each deviation node in our modified method. There-fore, the other possible deviation paths of P at A shouldbe deviated from Dev(P, A)−1 or other deviation pathsgenerated later. Please refer to Fig. 3. If we only performthe same pre-treatment on Dev(P, A)−1 (this means iso-lating the node between A (excluded) and D (included)on Dev(P, A)−1 and deleting the link between A andB1), then its father path (P) would be obtained as itsdeviation node. We call such a phenomenon as a Dead-Lock, and the introduction of INN helps us to breaksuch a Dead-Lock. In this illustration, we define thatnode F belongs to the INN of path Dev(P, A)−1 at nodeA. And the contents of INN are inheritable. This meanswhen a child path is deviated from a particular deviationnode from its father path, all the nodes in the INN ofits father path at this deviation node also belong to theINN of this child path at the same node, e.g., the INN ofDev(P, A)−1 at node A contains node F, and if Dev(P,A)−2 is deviated from Dev(P, A)−1 at node A, the INNof Dev(P, A)−2 would contain node F and node B1, inwhich node F is inherited from Dev(P, A)−1. With the

S A D

B1P: S A DDev(P,A)-1: S B1 A D

F

S A D

B1P: S A DDev(P,A)-1: S B1 A D, INN = {F}

F

B2

Dev(P,A)-2: S B2 A D, INN = {F,B1}

(b)

(a)

2nd loopless deviation path of P at A

1st loopless deviation path of P at A

Fig. 3 Illustration of the modified deviation method

introduction of INN, the pre-treatment of our deviationmethod contains three steps: (1) isolating all the nodesfrom the deviation node (excluded) and the destinationnode (included); (2) deleting the link between the devi-ation node and its previous node on the investigatedfather path; (3) deleting the links connecting the devi-ation node and all the nodes contained in the INN ofthe investigated father path at the very deviation node.For example, referring to Fig. 3b, we delete the link con-nection between A and F as well as that between A andB1 in the pre-treatment, Dev(P, A)−2 would obtainedfrom the deviation manipulation at deviation node A ofpath Dev(P, A)−1. We find that A is also the SDN ofpath Dev(P, A)−1, and in fact we need only to considerthe INN of a path at its SDN. The purpose of INN isto prevent the deviation manipulation to a path fromobtaining its father path, and the SDN is also the firstnode of its tail portion coinciding with that of its fatherpath. For any other deviation node between S and itsSDN, INN becomes unnecessary because the portionfrom deviation node to the destination already differsfrom its father path.

Loopless deviation (LD) algorithm:Input: Network graph G = (V, E, C), n = |V|, e = |E|;A connection request [S, D], S ∈ V denoting the sourcenode and D ∈ V denoting the destination node; K de-notes the number of the required shortest paths.Output: The K-shortest loopless paths root from sourcenode S and terminate at destination node D sorted bypath cost.Algorithm description:Initiation: Let Pth to denote the set of the KSP; Pth(i)means the ith shortest path, Cst(i) denotes the cost ofPth(i); SDN(i) denotes the SDN of Pth(i), and as theinitiation, set Cst(i) = INF with (i = 1, 2, . . . , K); Let


INN(i) denote the set of INN of Pth(i)’s SDN, and setINN(i) to be empty with (i = 1, 2, . . . , K).

Step 1: Compute the shortest path from S to D in net-work G by the Dijkstra algorithm [22] and insert it intothe Pth as Pth(1), then set the value of Cst(1) as the costof this shortest path and D → SDN(1).Step 2: Seek the K shortest loopless paths one-by-onewith up to K − 1 iterations.

(1)for k = 1 to K − 1/* If the cost of kth path is equal to that of the Kth,all K shortest paths are obtained.*//*If the kth path does not exist, there is totally onlyk − 1 shortest paths between the designatednode pair.*/

(2) if Cst(k)=Cst(K) or Cst(k)=INF;return; /* End the algorithm*/

(2)end if(2)else

Pth (k) is presented by the sequence of itsconsisting nodes as {S, Pth(k, 1), Pth(k, 2),…,SDN(k),…, D};Save network file G to a temporary file G(k),and isolate all the nodes on the tail portion ofpath Pth(k) between SDN(k) (excluded) andD (included) from the other nodes in G(k);

(3)for node A denote each node from SDN(k)backward to Pth(k, 1) on Pth(k)

/*Construct the loopless deviation pathDev(Pth(k), A)*/Set the cost of the link between A andits previous node on Pth(k) to be INF,and save this modification in G(k);

(4)if A =SDN(k)Set the costs of the links connectingA to all the nodes in INN(k) to be INF,and save these modifications in G(k);

(4)end ifFind the shortest path from S to A in G(k),denoted by P(S → A);(4)If the cost of P(S → A) �= INF

(5)if A = SDN(k)Let B to denote the previous nodeof A on Pth(i), and then push B intoINN(k);

(5)end ifP(S → A) pluses the tail portion of Pth(k)from A to D forms the loopless deviationpath Dev(Pth(k), A) and let CstDev todenote its path cost;/*Insert Dev(Pth(k), A) into the path set Pthaccording to its path cost.*/

(5)for i= k + 1 to K − 1(6)if Cst(i) � CstDev and Cst(i+1) > CstDev,(7)for j = K to min(K, i + 2) (in adescending order)Pth(j − 1) → Pth(j); SDN(j−1) → SDN(j);Cst(j−1) → Cst(j); INN(j−1) → INN(j);(7)end for

Dev(Pth(k), A) → Path(i+1); A → SDN(i+1);CstDev → Cst(i+1);(7)ifA = SDN(k)

INN(k) → INN(j+1);(7)else

Set INN(j+1) to be empty, and thenpush the previous node of A on Pth(k) into INN(j+1);

(7)end if(6)end if

(5)end for(4)end ifIsolate node A from the other nodes in G(k);(3)end for

(2)end if(1)end for

Step 3: Output the set of the K-shortest paths, i.e., Pth.

In the algorithm described above, we stop the iter-ation when the cost of the Kth path in the obtainedpath set is equal to that of the investigated father path.Why we can do this? As we proved in Theorem 1,the cost of any obtained LD path from Pth(k) cannotbe smaller than that of Pth(k), and we start the itera-tion from the shortest path in the entire network. Sowe can conclude that (1) the cost of the obtained LDpath in each iteration is in an non-decreasing sequence;and (2) when the cost of Pth(k) is equal to that of ex-isted Pth(K), the LD treatment to {Pth(k), Pth(k+1), . . .,Pth(K−1)} is unnecessary. Therefore we can stop theiteration immediately.

Computational complexity

In this subsection, we investigate the computational com-plexity in terms of running time, and then give a com-parison of our LD algorithm with existing algorithms.The author of Ref. [14] has presented the running timecomplexity of several algorithms, such as O(Kn2) forthe Hoffman’s algorithm [17] and O(Kn3) for both theYen’s algorithm [18] and the Lawler’s algorithm [20],under the assumption that the shortest path functioncosts O(n2). In order to assure the significance of thecomparison, we first accomplish our investigation underthe same assumption.


Comparing our LD algorithm with the Yen’s algo-rithm [18], there are three main differences: (1) The LDalgorithm will terminate when the cost of the Kth pathis equal to that of the investigated father path in anintermediate iteration; (2) The introduction of SDN canreduce the amount of deviation nodes for investigatingduring the LD algorithm; (3) The introduction of INNcan ensure the validity of our modified deviation methodand also can avoid the generation of duplicate paths inour LD algorithm. In the Yen’s algorithm, when inves-tigating a particular father path, all possible deviationnodes will be considered, and each generated deviationpath should be compared to all the existing paths for thepurpose of omitting the duplicate ones. Since the devi-ation method of the Yen’s algorithm is very similar toour LD algorithm, i.e., only the shortest path from thesource node to the investigated deviation node will begenerated to form the LD path at that deviation node,the same limitation of the Dead-Lock should also behandled in the Yen’s algorithm. But the treatment ofYen’s algorithm is straightforward and relatively time-consuming. It only compares the investigated fatherpath with all existing paths one-by-one, and if the tailportion between the deviation path and destination ofthem coincide, it deletes the link between the deviationnode and its previous node on the existing path as part ofthe pre-treatment before calculating the shortestpath.

Since the main structure of our LD algorithm is simi-lar to that of the Yen’s algorithm, its worst case runningtime complexity can also be expressed as O(Kn3). Butas we have mentioned above, three main differences be-tween our LD algorithm and the Yen’s algorithm canreduce the running time significantly in real-life imple-mentations. We will prove such a claim via numeri-cal experiments in section ‘Numerical experiments anddiscussions’.

The above discussion is under the assumption thatthe computational complexity of shortest path algorithmis O(n2), but in fact this cost can be reduced notably.If the links in the networks is uniformly weighted orwe need to seek the path with minimal hops, the BFS(breadth first search) algorithm [22] can be adopted,which costs O(n + e). Even if implemented under thenon-negative weighted network condition. Then in casethe network is relatively sparse, which is satisfied inmost real-life optical networks, the running time com-plexity of the adopted Dijkstra algorithm can be re-duced to O((n + e) ∗ lg(n)), and even to O(e + n ∗lg(n)) if a Fibonacci Heap is used as priorityqueue [22].

Classified alternate routing (CAR) approachand maximal used (MU) wavelengthassignment method

We have presented our new LD algorithm for solvingthe loopless KSP problem in Sect. ‘KSP Problem andLD algorithm’. In this section, we will propose our novelClassified alternate routing (CAR) approach and Maxi-mal used (MU) wavelength assignment method for thepurpose of optimizing the fairness characteristic amongdiverse connection requests without deterioration of theoverall CBP feature. First of all, we suppose the networkcan be presented by a weighted directed graph and allnotations are the same as those defined in Section ‘KSPproblem and LD algorithm’.

CAR routing method

We will first describe our novel CAR approach, whoseessential concept is determining the number of alter-nate paths for a particular connection request adaptivelyaccording to the minimal hop count between the corre-sponding source/destination node pair. Thus, traffic tobe carried over long paths will be offered more opportu-nities for obtaining an available lightpath because morealternate paths are given than traffic to be carried byshort paths. Such a benefit will counteract the unfair-ness bought by the link cascading effect and the wave-length continuity constraint to a certain extent. Due tothe diversity of real-life networks, the rule of how manyalternate paths should be associated with connectionrequests to be carried over lightpaths with a particu-lar hop count should respectively be determined for therespective investigated network.

Our CAR routing approach will adopt the looplessKSP paths as the alternate path set, and such pathsare generated adaptively according to the real-time re-source utilization state for any connection request byour LD algorithm proposed in Sect. ‘KSP problem andLD algorithm’. The adaptive routing (AR) strategy hasshown its efficiency in WRONs under dynamic trafficconditions [4], which determines the route of an arrivedconnection request depending on the real-time resourceutilization state. The most common AR algorithm is theadaptive minim-cost (weight) routing approach, whoseconcept is adopted in our CAR approach. Under suchan approach, the link cost is changeable according to thewavelength utilization status, and in this paper a linearlink weight function is supposed. We use W and w to de-note the total number of wavelengths and the number


of occupied wavelength on a link, and the weight of sucha link is shown in the following equation:

C(w) ={

1 + a × w, 0 ≤ w ≤ W∞, w = W

}

,

where a is the linear coefficient.

In our numerical experiments in Sect. ‘Numericalexperiments and discussions’, our CAR approach willdetermine the amount of alternate paths for a particularconnection request proportionally according to the min-imal hop count between the corresponding source–des-tination node pair. Due to the relatively highcomputational complexity of a loopless KSP algorithm(typically it is in the order of O(Kn3)), we suggest a net-work topology, which is optimized for minimizing themaximal hop count between all the node pairs. With thedevelopment of the concept and technologies associatedwith multi-granularity optical networks (MG-ONs) [23,24], a two-stage (fiber/wavelength or waveband/wave-length) or a three-stage (fiber/waveband/wavelength)multiplexing scheme will help to build a fiber networktopology based on the physical network topology lim-ited by geographical or historical constraints. Thus thefiber network topology can be optimized for the goal ofreducing the maximal hop count between all the nodepairs. The adoption of MG-ON to optimize the maximalhop count in a network is beyond the scope of this paper,and it will be considered in our future research.

MU wavelength assignment method

We will describe the adopted Maximal Used (MU) wave-length assignment method. We suppose that W denotesthe amount of wavelengths on a link, and those wave-lengths are numbered according to a certain order, suchas their wavelengths. After the K alternate paths havebeen obtained, we first check the wavelength utilizationstatus on each alternate path and record the results in aW × K matrix WA, where WA(w, k) is set to be 1 if thewth wavelength on the kth alternate path is available andotherwise set to be 0. Therefore, we list the wavelengthsaccording to the number of the alternate paths on whichit is available. This means for the wth wavelength, cal-culate

∑Kk=1 WA(w, k). Furthermore, sort all the wave-

lengths in an ascending order according to the obtainedvalues. Finally, select the first wavelength with the min-imal non-zero value, and also the first alternate path onwhich the selected wavelength is free. Though we callsuch a method as a wavelength assignment method, itis also responsible for the alternate path selection. Wename such a method as maximal used (MU) becauseits concept is derived from the most-used (MU) wave-length assignment method in a single routing strategy

[4]. With the combination of the CAR routing approachand the MU method, our CAR-MU scheme will show itssignificant performance enhancement in terms of bothoverall CBP and fairness characteristics in the follow-ing numerical experiments in Sect. ‘Numerical experi-ments and discussions’. Here, the performance of ourCAR-MU scheme will be compared with some existingfairness enhancement methods in the literature, suchas the well-known Rsv and protection threshold (Thr)methods [10].

Numerical experiments and discussions

In this section, we will present the numerical experimen-tal results to show the performance of the LD algorithmdescribed in Sect. ‘KSP problem and LD algorithm’ aswell as the CAR routing approach proposed in the pre-vious section. We will first verify the efficiency of the LDalgorithm, and then show the performance of the CARapproach comparatively with other RWA solutions interms of both fairness characteristic and overall CBPfeature.

Efficiency of LD algorithm

In this subsection, we will examine the efficiency of theLD algorithm proposed in Sect. ‘KSP problem and LDalgorithm’. We will show that our LD algorithm outper-forms the Yen’s algorithm by a much shorter runningtime. From the description of our LD algorithm andthe descriptions of the formerly proposed algorithmsin literature [14–21], we can find out that in most KSPalgorithms, the major part of the running time to calcu-late the KSP paths occurs in the execution calls to theshortest path algorithm. So we can compare the runningtime complexity of the LD and Yen’s algorithms via theamount of calls to shortest path function. Such a metricperforms better than real-life running time because itis fixed if the network topology and source–destinationnode pair is designated, but the real-life running timeis somewhat chromatic and uncertain. The same metrichas already been adopted in Ref. [14].

In order to verify the performance of our LD algo-rithm in diverse network topologies, our numericalexperiments are implemented for four networks, respec-tively: (1) The NSFNET topology with 14 nodes and 21bidirectional fiber links; (2) The ARPANET topologywith 21 nodes and 31 bidirectional fiber links; (3) TheChinaNet topology with 39 nodes and 72 bidirectionalfiber links; (4) A randomly generated network topol-ogy with 50 nodes and 80 bidirectional fiber links. Theirtopologies are revealed in Fig. 4a–d, respectively, and the


experimental results for each network are presented inFig. 5a–d, respectively. From those results, we can seethat the running time complexity of LD algorithm re-vealed by average calls to shortest path function is sig-nificantly better than that of the Yen’s algorithm despitethe diversity of the investigated network topologies. Thevalue of average calls is obtained by calculating the num-ber of calls for each source/destination node pair andtaking the average value.

The LD algorithm shows its advantage over the Yen’salgorithm in terms of computational complexity charac-terized by the average number of execution calls to theshortest path function in this experiment. Furthermore,if considering the real-life running time, such an advan-tage will be more significant because besides reducingthe number of execution calls to deviation manipula-tion, the LD algorithm also omits all the comparison

manipulations of the investigated father path andobtained deviation path with the existing paths adoptedby the Yen’s algorithm. If K is relatively large or the net-work size is large, the costs brought by such comparisonmanipulations are considerably heavy.

Performance of CAR-MU scheme

In this part, we investigate the performance of diverseRWA schemes in terms of both overall CBP and fairnessamong calls to be carried over lightpaths with differenthop counts. First of all, we suppose a Poisson trafficmodel that the connection requests between any givensource–destination node pair arrive stochastically witha Poisson process, whose mean arrival rate is denotedby λ and will hold for a negative exponentially distrib-uted duration time with a mean value of 1/µ. We also

Fig. 4 Illustration of theinvestigated networktopologies

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assume the values of λ and µ to be uniform for anysource/destination node pair. Thus we can measure thetraffic load of the entire network with the traffic load be-tween a single node pair, which is defined as λ/µ with theunit named by Erlang. For each experimental result, weadopt 5×106 connection requests, and only consider therelatively light traffic condition (i.e., low overall CBP)because we believe that the real-life communication net-works always work under such a condition. We imple-ment our experiments on the NSFNET with its topologyshown in Fig. 3a. The number of wavelength on each link,

symbolized as W, is supposed to be 16 in our experimentshereafter.

In order to comparatively study the performance ofdiverse RWA schemes, we consider several combina-tions of diverse routing approaches and wavelengthassignment methods. Such schemes are adaptive routingand random wavelength assignment (AR-RA), adaptiverouting and most-used (MU) wavelength assignment(AR-MU), adaptive routing and first-fit (FF) wavelengthassignment with and without the Rsv or Thr method(AR-FF, AR-FF-Rsv, AR-FF-Thr), and the CAR routing

Fig. 5 Average number ofexecution calls to the shortestpaths functions vs. number ofrequired KSP paths

NSFNET, N=14, L=21 LD Algorithm Yen's Algorithm

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approach and the MU wavelength assignment method(CAR-MU). As the description in the previous sec-tion, our AR strategy (including the AR and CAR ap-proaches) adopts the minimal cost routing method witha linear link weight function and the wavelength weightparameter α is set to be 0.2 in our experiments. We as-sign the number of alternate paths for each connectionrequest equally to the minimal hop count between thecorresponding source/destination node pair. This meansthere will be 1, 2, and 3 alternate paths for 1-hop, 2-hop,and 3-hop traffic, respectively. Since the investigatedNSFNET topology only contains 1-hop, 2-hop, and

3-hop traffic, the considered wavelength reservation(Rsv) approach will reserve the 15th wavelength oneach link for 2-hop and 3-hop traffic and the 16th wave-length for 3-hop traffic. Furthermore, and the inves-tigated protection threshold (Thr) would block aconnection request belonging to 1-hop traffic, if less thantwo idle wavelengths were available on the obtainedpath.

First, we show the experimental results of the overallCBP in Fig. 6, from which, we can see that the CAR-MUscheme wins the best performance with significant supe-riority, especially under a light traffic condition. Because


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call blocking under light traffic primarily comes from thewavelength discontinuity on the obtained path route,the alternate routing approach enhances the chance ofobtaining a path route with an available wavelength.And as indicated in Refs. [10] and [11], both the Rsvand Thr methods deteriorate the overall CBP feature,but our study shows that the influence of the Rsv methodis much severer than the Thr method. The absence of theoverall CBP under the CAR-MU scheme when the traf-fic load is equal to 0.5 Erlang is due to the fact that theexperimental result is too low to be measured. Thoughwe did not investigate the performance of the AR-MU-Rsv and AR-MU-Thr schemes, considering the negligi-ble difference between AR-FF and AR-MU, we believetheir performance are very similar to that of AR-FF-Rsvand AR-FF-Thr.

Second, we investigate the fairness among diverseconnection requests. An unfairness factor was definedas the ratio of the CBP on the longest path to that onthe shortest path in Ref. [5]. But this factor has manydisadvantages, e.g., it will be meaningless if the CBP ofthe shortest path is zero. Furthermore, it only shows therange of CBPs of diverse node pairs but cannot revealthe distribution of them. Since the most significant exhi-bition of unfairness problem in WRONs is the sharpincrease of CBP with the increase of lightpath hops,we classify all traffic requests into categories such as 1-hop, 2-hop, 3-hop traffic, and so on, according to thenumber of minimal hops from the source node to thedestination node. And therefore we quantify the fair-ness characteristic by the ratio of average CBP of eachtraffic category to that of 1-hop traffic. Obviously, weprefer values of CBP ratios closer to 1 because they areassociated with better fairness characteristic. As the pre-treatment of our CAR approach, referring to previoussection, we have already obtained the number of min-

imal hops of all source–destination node pairs, and thenumber of source–destination node pairs correspondingwith 1-hop, 2-hop, and 3-hops traffic in NSFNET is 42,72, and 68, respectively.

The experimental results of the CBP ratios underall the investigated schemes are presented in Fig. 7a–f,respectively. We can observe that (1) without any fair-ness enhancement method, the unfairness amongconnection requests belonging to diverse traffic classi-fications is very serious, e.g., the average CBP of 3-hoptraffic can be larger than that of 1-hop traffic with morethan 100 times; (2) all the investigated fairness enhance-ment methods, including the Rsv and Thr approaches,and our CAR approach, can improve the fairness char-acteristic significantly; (3) Either of the investigated fair-ness enhancement method performs better under lighttraffic condition than under heavy traffic condition. Thisis exactly opposite to the status without any such meth-ods, and because most real-life optical networks nor-mally work under a light traffic condition, such a featurestrengthens their practicability in real-life networks; (4)The increase of overall traffic load can influence thefairness characteristic of both the Rsv and Thr methodsremarkably, but such an effect on the CAR approach isrelatively inconspicuous.

As a summary of the investigations in this subsec-tion, one can say that although the fairness character-istic of our CAR approach seems to be in the shadowof that of the Rsv and Thr methods, it is consideredto have significant superiority in terms of overall CBP.Furthermore, the low and steady values of CBP ratiosdespite the variety of traffic load show its excellent per-formance. Thus, our CAR-MU scheme is preferred forits outstanding comprehensive performance. This is alsothe case because of the especially good performancein terms of both overall CBP and fairness under lighttraffic condition, which is the actual status in real-lifeoptical networks. Therefore, our CAR approach showsits potential practicability in the next generation opticalnetworks.

Conclusion

This paper aimed to enhance the fairness characteris-tic without deterioration to the overall CBP feature inWRONs without wavelength converters by a novel CARapproach. It determines a number of alternate paths fora connection request adaptively according to the mini-mal hop count between the corresponding source/desti-nation node pair. It then selects the most suitable one bya certain rule. As foundation of such a CAR approach,the first K-shortest loopless paths should be determined


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simultaneously. Such a problem is well known as theKSP problem. After a short review on this problem,this paper proposed a new LD algorithm to solve theloopless KSP problem in a non-negative weighted net-work in the Section ‘KSP problem and LD algorithm’.We also showed that our LD algorithm outperforms the

most popular Yen’s algorithm in terms of computationalcomplexity via both theoretic analysis and numericalexperiments on diverse real-life networks. In Section‘Classified alternate approach...’, we described theCAR approach as well as the collaborated MaximalUsed (MU) wavelength assignment and path selection


method. Numerical experimental results in Section‘Numerical experiments and Discussions’ revealed thatthe CAR-MU scheme outperforms existing fairnessenhancement methods (e.g., Rsv and Thr methods) interms of the balance between the fairness characteris-tic and overall CBP feature. Experimental results alsorevealed that our scheme performs much better underlight traffic condition, which is also the normal workingstratus of real-life optical networks. Thus, such a fea-ture suggests the potential practicability of the proposedscheme in real-life networks.

Our study in this paper also suggests that a well-designed optical network topology should be optimizedfor reducing the maximal number of minimal hop countsbetween all the node pairs. With the development ofMG-ONs, two-stage or three-stage multiplexing schemeswill be adopted in next generation optical networks. InMG-ONs, such optimized fiber network topologies canbe realized via fiber switching technology. This idea isbeyond the scope of this paper and will be consideredin our future studies.

Acknowledgements This work was supported by National Natu-ral Science Foundation of China (NSFC 60132020 and 90104003),National 863 Program (2003AA12202X), and Tsinghua-Bell LabsJoint Laboratory on Optical Communication Networking Sys-tems. The authors would also like to thank the reviewers for theirconstructive comments on improving this paper.

Appendix A: Proof to Theorem 1

Theorem 1 Given network G = (V, E, C), a looplesspath P in G starts at S and terminates at D, S ∈ V, andD ∈ V, is given. P = {S = P(0), P(1), P(2), . . . , P(h −1), D = P(h)}. Any node A ∈ P, but A �= S. If the headpart of P from S to A, i.e. {S = P(0), P(1), . . . , A} is theshortest path under the constraint of the absence of anynode contained in P between A and D (including thesetwo), then any X ∈ {P(1), P(2), . . . , A}, any Y neigh-bored to X and Y /∈ P, the cost of the loopless deviationpath Cst(Dev(P, X, Y)) � Cst(P).

Proof Please refer to Fig. 1. Let S → A → D be thegiven loopless path, let {P(1), P(2), . . . , A} be the set ofdeviation nodes. We symbolize the next node to A on Pas F. According to the definition of loopless deviationpath, any X ∈ {P(1), P(2), . . ., A}, the tail portion ofDev(P, X, Y) will coincide with that of P, i.e., F → D,and any loop is forbidden. Refer to the given conditions,the head portion of P, i.e., {S = P(0), P(1),. . ., A}, is theshortest path from S to A under the constraint of theabsence of any node contained by P between A and D(including these two), the cost of sub-path from P to A

in any loopless deviation path Dev(P, X, Y) cannot besmaller than that of {S = P(0), P(1),. . ., A}. As a result,Cst(Dev(P, X, Y)) � Cst(P) is obviously obtained. �

Appendix B: Proof to Theorem 2

Theorem 2 Any path Q in network G with the sourcenode S and destination node D, whose cost is not mini-mal, is a deviation path from a path P also from S to Din network G., such that Cst(Q) > Cst(P). If P is withminimal cost, then either Q is unique, or Q is a deviationfrom another minimal path.

Proof Please refer to Fig. 1 , let S → B → A → Dbe the given non-minimal path Q. Let A and B be twosuccessive nodes in Q, such that S → B → A is notthe minimal path from S to A but S → B is the min-imal one. By the theorem of graphs, these two nodesdo exist only if Q is non-minimal and there must be aminimal path S → A, which does not traverse B. Wedenote S → A → D as P, and obviously Q is a deviationpath of P. If Q is a minimal path but not unique, it mustbe another minimal path P, which has only a portiondifferent from that of Q with the same starting node andending node, and without loos of generality, we assumethe starting node is S and ending node is A, S → A → Band S → B without traversing node B must have thesame cost, and Q is a deviation path of P. [17] �

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Junjie Li was born in Zhej-ing, China in 1977. He receivedhis Ph.D degree of electron-ics science and technology fromthe Department of ElectronicEngineering, Tsinghua Univer-sity, Beijing, China, in 2005. Heis currently working with ChinaTelecom Beijing Research Insti-tute. His research interests fo-cus on WDM transmission andoptical networking technology.

Hanyi Zhang was born inHeilongjiang, China in 1942. Hegraduated from the Departmentof Electronics Engineering,Tsinghua University in 1965.Since then he has been a facultymember at Tsinghua Universityand engaged in research andeducation on lasers, photonicsand optical communication net-works. During 1965 to 1979 hismain work was on the solid-statelasers, gas lasers, laser range find-ers and laser gyroscopes. Since1980 his research activationhas focused on external cavity

semiconductor lasers, semiconductor optical amplifiers, optical fil-ters, optical wavelength converters, optical fiber sensors, laser fre-quency stabilization, coherent detection, photonic switching, opti-cal ATM switching, OXC, OADM, WDM systems, WDM opticalnetworks and microwave optical transportation. For the past tenyears he has been responsible for more than ten national priorityprojects and won eight awards from the State for his scientificcontributions and invention achievements and published morethan 100 papers. Now Prof. Zhang is the Consultant of the Na-ture Science Foundation of China, the Vice Secretary General ofChinese Optical Society, Vice Secretary General of the Optoelec-tronic Society of the Chinese Electronic Institute, the editor of theChinese Journal of Laser (English edition) and the editor of theJournal of Infrared and Lasers.

a new solution to the k-shortest paths problem and its application in wavelength routed optical...

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