A Minimum Cost Active and Backup Path Algorithm with SRLG Constraints

Jianhui Zhang , Bin Wang, Binqiang Wang ,Jinqiu Ren National Digital Switching System Engineering & Technology R&D Center (NDSC), Zhengzhou, 450002

Henan Province, China e-mail:zhangjh.365@gmail.com

Abstract—As more and more mission critical service are now transported over high-speed networks, It becomes an important issue to make use of the relativity of multiple link/node failures and improve transport performance for high-speed networks. In this paper, we propose a novel algorithm called Low cost an S disjoint paths algorithm (LCSD for short) to Establish a SRLG disjoint active path and backup path pair for network protection . Based on the TF algorithm, LCSD algorithm introduces the number of links which relate to the SRLG as a new constraint .Through link weight adjustment before the second time SPF calculate, the backup path can avoid the risk sharing with active path. Theoretical analysis and experiment results have shown that LCSD algorithm resolved the trap problem and path pair total cost non-optimal problem under SRLG disjoint constraint. Compared with RF and TF algorithms, LCSD algorithm is significantly superiority in the successful ratio to find feasible solution and can guarantee minimum total cost of active and backup path.

Keywords- SRLG constraint; active and backup path; S disjoint;Low cost S disjoint paths algorithm; minimal total cost

I. INTRODUCTION As more and more mission critical service are now

transported over high-speed networks, survivability is a critical problem when concurrent multiple node and link failures happen. It becomes an important issue to make use of the relativity of multiple failures and improve transport performance for high-speed networks.

Shared Risk Link Group (SRLG) has been widely recognized as an important concept in survivable high-speed networks. An SRLG is associated with an entity at risk, typically a fiber span [1], though more general risks might be modeled. Based the SRLG concept, Nodes and Links in network can be grouped into different S set. In this way, pair of active and backup path finding problem in multiple network failures environment can be modeled as an S disjoint low cost paths computation problem.

To date, only a few heuristic algorithms have been proposed to find SRLG disjoint path[7-13].A method has been proposed[12] to increasing virtual node and link in logic graph and transforming topology graph for finding S disjoint path. KSP (K-shortest path) algorithm has also been used to calculate S-disjoint path pair. In [13], there is an innovative failure-dependent scheme called PROtection using MultIple

SEgments (PROMISE) to tolerate link/node failures. The basic idea of PROMISE is to divide an AP into a set of possible overlapping active segments (AS), and provide protection for some or all of the links along each AS using a backup segment (BS), which is link/node-disjoint with the AS. These heuristic algorithms are easy to use, but have a drawback that can’t find the active backup path pair even they are exist in some condition.

The algorithms used to compute SRLG disjoint low cost path pair in network can be divided into two categories. One is “remove-find algorithm”， the other is “transform- find algorithm”.

The first category is also named RF algorithm, which first calculate shortest path as active path and then delete links which belong to active path from network topology graph. Based on the deleted network graph, backup path can be found. RF algorithm is easy to extend for calculating S disjoint path if risk separated set is introduced into link remove process. But RF algorithm has drawback to deal with SRLG disjoint constraint. The major is trap problem which means algorithm may fail to find the second best path pair even they are exit in fact. In particular trap problem become more serious in the condition that the first calculate active path have too many links or in the remove process too many links have been deleted. In typical network topology, trap problem probability of RF algorithm is 10% to 30%.The second major problem is non-optimal problem. RF algorithm can guarantee the minimum cost of active path, but the sum cost of active path and backup path is high. RF algorithm is non-optimal and lead to occupying too much network resource to perform 1+1 protection strategy.

The second category is widely known as TF algorithm, which is proposed by Surrballe [8]. Base idea of TF algorithm is used the shortest path algorithm twice to calculate path pair. And between the two calls transform link weights in network topology. This method can find the minimum total cost active and backup path pair. Node disjoint can be easy to meet if we make a slight adjustment when carrying out transform. However， the algorithm does not apply to SRLG separation condition. If the minimum total cost path pair have links belong to same SRLG set, the algorithm will failed to return the ideal result.

This paper introduced a novel algorithm which is called Low cost an S disjoint paths (LCSD for short) algorithm. Based on the TF algorithm, LCSD algorithm introduce the

2009 Pacific-Asia Conference on Circuits,Communications and System

978-0-7695-3614-9/09 $25.00 © 2009 IEEE

DOI 10.1109/PACCS.2009.146

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number of link which relate to the S-set k as a new constraint .Then through link weight adjustment, the backup path can avoid the risk sharing with the active path. This algorithm can provide the S disjoint between active path and backup path and the total cost of path pair is optimal.

II. PROBLEM FORMULATION

A. S Collection Definition The concept of a Shared Risk Link Group (SRLG) is used

to represent a group of links that are subject to a common risk, such as a conduit cut [11]. SRLG represent a generic set of risks, including individual link, wavelength, conduit, node or port failures. Or even, in order to guarantee network transmission service in earthquakes, floods and other extreme cases, you can group physical links which cross through the same disaster area into the same SRLG.

In order to fully reflect risk-sharing feature in network, SRLG definition can be divided into two levels. The first level is Link group which share the same physical resource such as power facilities. The first level SRLG concept can be used to describe multi-failure relation in common network environment. The second level is artificial designated link group or link group which may failure at the same time for multiple causes in the actual operation network. The second level SRLG concept can describe extreme network circumstances of multi-failure correlation.

SRLG has the following attributes: (1) One link may belong to more than one SRLG. (2) Two link belongs to the same SRLG may belongs to

other different SRLG in some conditions. (3) SRLG of a path is composed of all the SRLG which

each link in the path belongs to. Let G (V, E) be an undirected graph representing the

network. We denote the set of network (source and destination) nodes by V, while E is the set of duplex communications links that connect the network nodes., associated with each node n∈V, each link e∈E, denote the adjacent links set of node n by A(n).And we also record node number of G as ( ) | |G Vν = and link number of G as

( ) | |G Eε = . Given a first lelvel SRLG set which is recorded as

{ ( )}SRLG i , we can map nodes and links in G to the collection of S , { , }L NS S S= . Mapping criteria are as follows:

(1) Links belonged to SRLG are mapped to the set NS :

( )( ) { }N e SRLG i

S i e∈

=∪ .

(2) Links which are not belonged to SRLG are mapped to the set LS : ( ) { }LS i e= , e SRLG∉ .

(3) All the adjacent links of node n are mapped to the set NS :

( )( ) { }N e A n

S i e∈

=∪ .

Set S has the following attributes:

(1) S contains all of the links in G and risk-sharing relationship between the links, node malfunction in network equals to all the adjacent links of node n failure.

(2) Various elements of S are independent of each other. The corresponding physical meaning is the network equipment failure for the incident independent of each other.

(3) The element number of S Meets the following formula: | | | | | | | | | | | |SRLG V S SRLG V E+ ≤ ≤ + + (1) If SRLG= ∅ ,then |S|=|V|+|E|. Correspondingly, if given a second level SRLG, we can

map graph G to corresponding collection S .

B. Problem Definition A graph ( , , , )G V E C S represent to network topology. The

meaning of V，E，S is the same as the definition in section 2.1. C is link cost matrix, the symbol ( , )u v denotes a bidirectional link connecting u and v, the cost associated with link ( , )u v is ( )C u v→ .We will use ( , )s t denote t he path between source node s and destination node t .

We define the problem to find an SRLG separated path as low cost S disjoint path pair problem. LCSD problem can be converted to QoS routing problem which has an additional constraint and an optimization goal. (1) Constraint: the path pair of ( , )s t 1P , 2P meet S

disjoint requirement, which means

1 2( ) ( )S P S P∩ = ∅ ， ( ) ( , )u v P

S P S u v→ ∈

= ∪ . In the equation ( , )S u v means a collection consist of all element of S which contain link ( , )u v .

(2) Optimization goal: the path pair of ( , )s t should have the 1 2min{ ( ) ( )}C P C P+ , where ( )C P is the total cost of path, ( ) ( )

u v PC P C u v

→ ∈= →∑ .If there is a Bound

ceiling for path cost, then path pair of 1P and 2P should obey 1 2max{ ( ), ( )}C P C P L≤ .

III. LOW COST S DISJOINT PATH PAIR FIND ALGORITHM A novel simple and efficient method will be proposed in

this section to compute active and backup path in the network global protection mode. The method guarantee S disjoint between the two path and total cost of the path pair is minimum. The algorithm named Low cost and S disjoint Paths algorithm and LCSD for short.

The main idea of LCSD algorithm is perform SPF calculation based on a twice Link weights transformation. First introduce number of S set which related to link as a new constraint, adjust link weight and then calculate active path.Secondly,as TF algorithm, transforming weight of links which belongs to active path and call SPF calculation to get backup path.

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A. Parameter definition In this subsection, we define the decision parameters in

our algorithm as follows: ( , , , ')G V E C S represents the network to pology. 'S is the

operation subset of S set, which meet the different level requirements of SRLG Separation.

suvα indicates that link ( , )u v belongs to element s of

'S . 1suvα = ,if and only if ( , )u v s∈ ,and 0s

uvα = ,otherwise. ( )N s indicates the total number of links which belong to

s , where ( ) suvu v

N s α=∑ ∑ .

( , )N u v indicates the total number of links which have S relationship with link ( , )u v , where ( , ) ( ) s

uvsN u v N s α=∑ .

( , )S u v indicates a subset of 'S ,whose every element contains link ( , )u v ,where

( , ) '( , )

u v sS u v s

∈=∪ .

( )S P indicates the s set which each link of path ( , )P s t sequentially related to S , where

( ) ( , ) \{ ( ), ( )}N Nu v PS P S u v S s S t

→ ∈=∪ .

'C is defined as a New link cost matrix, Which link weight has been adjusted in accordance with factor of S related link number. Through adjustment of weight coefficient, the ratio of S disjoint factor in all link cost factors can be flexible.The new cost of link u v→ is defined as:

max max

1'( ) ( ) ( , )C u v C u v N u vC N

α α−→ = → + (2)

where max

,max ( )

u vC C u v= → ， max

,max ( , )

u vN N u v= ，

α indicates weight coefficient。The relation of 'C and C meet the expression 'C C== ,if and only if α is zero.And

( )C u v→ is directly proportional to ( , )N u v ,if and only if 1α = .

B. The Steps of LCSD Algorithm In this subsection, we describe LCSD algorithm that has

only a polynomial time complexity and is feasible for very large networks. The steps of LCSD algorithm are as follows:

Step1.initialize parameters: min 0α α= = ,

max 1α = , P = ∅ , 1P = ∅ , 2P = ∅ ; Step2.calculate 'C in accordance with formula (2); Step3.calculate 1 SP( , , ', )P s t C G= ,if 1P P≠ ，

then 1P P= and go to Step4; if 1P P= , 2P = ∅ ,then minα α= and go to Step7;if

1P P= and 2P ≠ ∅ ,then则 go to Step8;

Step4.calculate ''C in accordance with ( )

''( ) ( )C v u

C u v C u v− →⎧⎪→ = →⎨⎪ ∞⎩

， 1

1( , ) ( )v u Pu v S P

else

→ ∈∉

Step5.calculate shortest path 2 SP( , , '', )P s t C G= ;

Step6.if 2P ≠ ∅ and 0α = ， then go to Step8;else

if 2P ≠ ∅ and 0α ≠ ,then maxα α= ，go to Step7;else

if 2P = ∅ then minα α= and go to Step7；

Step7.update the value ofα ： min max( )/2α α α= + , max mine α α= − ;

if e ε< then go to Step9,else go to Step2； Step8.Chek whether there is link ( , )u v in path 1P or 2P meet

the follow expression: 1 2u v P v u P→ ∈ ∨ → ∈ .

If not find such ( , )u v ，then go to Step9. If such link ( , )u v is exist, then remove link (u, v) and constitute the remaining links into the new shortest path 1 'P , 2 'P . 1 2 1 2 1 2' ' ( ) \ ( )P P P P P P∪ = ∪ ∩− ， where 2P− indicates the reverse link collection of 2P .

If 1 2( ') ( ')S P S P∩ = ∅ , then go to Step9; Otherwise go to Step7.

Step9.end.

IV. THEORETICAL ANALYSIS LCSD algorithm resolved the trap problem and path pair

total cost non-optimal problem of traditional TF algorithm under SRLG disjoint constraint. Through the analysis on the relation between the algorithm solution and shortest path, the optimal feature of LCSD algorithm will be proved.

A. Aanalysis of Relationship between LCSD Algorithm Solution and Shortest Path For a given graph ( , , , )G V E C S , the solution set of LCSD

for (s, t) is 1 2{ , }d dP P .we denote the shortest path for ( , )s t as ( , , , )sP SP s t C G= . The relationship between Ps and

1 2{ , }d dP P can be divided into the following four types: (1) 1dP or 2dP is the same as the shortest path sP .That is,

1s dP P= or 1s dP P= . (2) sP obeys the S disjoint rule with 1dP and 2dP .That is ,

1 2( ) ( ( ) ( ))s d dS P S P S P∩ ∪ = ∅ . (3) sP obey the S disjoint rule with 1dP or 2dP ,and at the

same time sP don’t obey S disjoint rule with other path. That is, Either expression

1( ) ( )s dS P S P∩ ≠ ∅ ∧ 2( ) ( )s dS P S P∩ = ∅ ∧ 1s dP P≠ is

398

true ,or expression 2( ) ( )s dS P S P∩ ≠ ∅ ∧

1( ) ( )s dS P S P∩ ≠ ∅ ∧ 2s dP P≠ is true. (4) sP does not obey the S disjoint rule with 1dP or

2dP .That is, expression 1( ) ( )s dS P S P∩ ≠ ∅ ∧ 1s dP P≠ ∧

2( ) ( )s dS P S P∩ ≠ ∅ ∧ 2s dP P≠ is true.

B. Optimality Proof for LCSD AlgorithmSolution Theorem 1:The solution of LCSD algorithm is optimal. Proof:Giver 1 2' , 'P P as the optimal solution for (s, t).Where

1 2( ' ) ( ' )C P C P≤ . (1) Assume the relationship between 1 2' , 'P P and Ps is the

first type, we can conclude that if set 0α = ,LCSC can return optimal solution as 1 1 'P P= ,and 2 2 'P P= .

(2) Assume the relationship between 1 2' , 'P P and Ps is the second type or the third type, since the relation

1 2( ) ( ' ) ( ' )sC P C P C P≤ ≤ is true, we can conclude

1 2' , 'P P is not the optimal solution for (s, t) .That hold and contradicts our initial assumption.

(3) Assume the relationship between 1 2' , 'P P and Ps is the fourth type. There are three possible relations between them as shown in Figure 1. If and only if 1 2' ' sP P P∪ − can form a new shortest path 'sP and

( ' ) ( ( ))s s sS P P S P− ∩ ∩ = ∅ , We can let α is equal to 0,then LCSD algorithm can return optimal solution 1 2' , 'P P .

1'P

2'P

1'P

2'P

1'P

2'P

SP SPSP

Figure 1. The solution of LCSD S relationship with shortest path

In other conditions of the fourth type,when 0α = ,LCSD algorithm can not directly get optimal

solution.We can adjust the value of α ,make the shortest path P of new network graph diverse with sP .When the relationship P is the same as one path of set 1 2{ , }d dP P ,the first type relation is obeyed ,then we can get optimal solution. And When P meet the condition of

1 2( ) min{ ( )}, ,i i d i dC P C P P P P P= ∩ ≠ ∅ ∩ ≠ ∅ , We can also get optimal solution.

Therefore, we can conclude LCSD algorithm can find optimal solution for S disjoint path.

V. EXPERIMENT AND ANALYSIS

A. Network topology and experiment parameter In the simulation experiments, network topology shown in

Figure 2 and Figure 3 have been used. Each Link is bidirectional. Different network scenarios with diverse protect requirement are used. Cost of each Link and SRLG relation is Figure out, such as symbol “3/5” where “3” indicate link cost and “5” indicate SRLG id. If two link have the same SRLG ID, then they can share the same network risk. RF, TF and LCSD algorithms are implemented on a Windows XP machine (AMD Athlon 64 Processor 3000+ CPU, 1.00GB size memory) using programming the language C.

Firgure 2 NSFNET topology used in simulation

5/13

3/1

2/12

2/1

3/5

1/6

Firgure 3 CERNET topology used in simulation

B. Rresult Analysis Simulation results for network NSFNET are shown in

Table I. From Table I, we can see for source destination pair (1, 10), RF algorithm failed to calculate path pair and TF algorithm can return result but the path pair not obey the SRLG disjoint constraint. Through adjustment of α value, LCSD algorithm can return feasible solution and meet the SRLG disjoint requirement. This shows that LCSD is noticeably better than RF and TF algorithms.

TABLE I. NSFNET SIMULATION RESULT

(S,T) algorithm path cost SRLG disjoint

Network |E|/|N|

(1,10)

RF 1-7-6-4-5-10 9

-- 1.71 -- ∞

TF 1-0-3-4-5-10 10

N 2.64 1-7-9-10 11

LCSD 1-7-9-10 11 Y 2.36

399

(α=0.5) 1-2-5-10 12

Simulation results for network CERNET are shown in Table II. From Table II, we can see the path pair calculated by TF algorithm is not SRLG disjoint. And we can also find that total cost of LCSD algorithm solution is less than TF and RF algorithm solution. In other words, LCSD algorithm has solved path pair total cost Non-optima problem.

TABLE II. CERNET SIMULATION RESULT

(S,T) algorithm path cost SRLG disjoint

Network |E|/|N|

(3,6)

RF 3-4-1-2-6 7

Y 2.2 3-0-2-9-6 13

TF 3-4-1-2-6 7

N 2.8 3-0-5-6 8

LCSD

(α=0)

3-4-5-6 9 Y 2.6

3-0-2-6 9

C. Additional experiment we also use all the three algorithm to calculate all SRLG

disjoint path pair for all source and destination node in accordance with network Net(|N|，|E|，|SRLG|) shown in [10]. The result in Table III shown that LCSD has much lower trap problem probability than RF. LCSD has improve trap problem probability in an order of magnitude.Though KSP has better trap problem probability than LCSD ,but algorithm complexity is much high than LCSD.

TABLE III. TRAP PROBLEM PROBABILITY COMPARISON

Algorithm Net1(119,190,338) Net2(47,97,65) Net3(144,298,198) RF 20.91% 8.05% 30.98%

KSP, K=20 0.44% 0.00% 0.51%

LCSD 2.36% 1.19% 4.56%

VI. CONCLUSION In this paper, we have proposed a novel low cost an S

disjoint paths calculate algorithm .It can compute active and backup path for network against multi network failures. LCSD algorithm guaranteed SRLG disjoint between the two paths and minimum total cost of the two paths. In performance evaluation, we show that LCSD algorithm can always compute optimal solution for S disjoint path calculating problem. Furthermore, we applied our algorithm

to real network scenarios using realistic topologies and SRLG-maps.

Experiment results have shown LCSD algorithm is noticeably better than RF and TF algorithms in the aspect of success to find feasible solution and path pair total cost optimization.

ACKNOWLEDGMENT We would like to acknowledge the assistance of Jinqiu

Ren for the LCSD implementations. This research was supported by China National 863 Plan (Project No. 2007AA01Z2a1). The research was supported in part by China National Key Underpinning Research Development Plan (973 Plan, Project No.2007CB307102）.

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