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Two Heuristics for Calculating a Shared Risk Link Group Disjoint Set of Paths of Min-Sum Cost

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Abstract

A shared risk link group (SRLG) is a set of links which share a common risk of failure. Routing protocols in Generalized MultiProtocol Label Switching, using distributed SRLG information, can calculate paths avoiding certain SRLGs. For single SRLG failure an end-to-end SRLG-disjoint path pair can be calculated, but to ensure connection in the event of multiple SRLG failures a set with more than two end-to-end SRLG-disjoint paths should be used. Two heuristic, the Conflicting SRLG-Exclusion Min Sum (CoSE-MS) and the Iterative Modified Suurballes’s Heuristic (IMSH), for calculating node and SRLG-disjoint path pairs, which use the Modified Suurballes’s Heuristic, are reviewed and new versions (CoSE-MScd and IMSHd) are proposed, which may improve the number of obtained optimal solutions. Moreover two new heuristics are proposed: kCoSE-MScd and kIMSHd, to calculate a set of \(k\) node and SRLG-disjoint paths, seeking to minimize its total cost. To the best of our knowledge these heuristics are a first proposal for seeking a set of \(k\, (k>2)\) node and SRLG-disjoint paths of minimal additive cost. The performance of the proposed heuristics is evaluated using a real network structure, where SRLGs were randomly defined. The number of solutions found, the percentage of optimal solutions and the relative error of the sub-optimal solutions are presented. Also the CPU time for solving the problem in a path computation element is reported.

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Notes

  1. As defined in [10] “A TE link is a “logical” link that has TE properties. The link is logical in a sense that it represents a way to group/map the information about certain physical resources (and their properties) into the information that is used by Constrained SPF for the purpose of path computation, and by GMPLS signaling.”

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Acknowledgments

The author acknowledges financial support through project QREN 23301 PANORAMA II, co-financed by European Union’s FEDER through “Programa Operacional Factores de Competitividade” (POFC) of QREN (FCOMP-01-0202-FEDER-023301) by the Portuguese Foundation for Science and Technology under project grant PEst-OE/EEI/UI308/2014 and by Project PERGS of Portugal Telecom Inovação. We thank the anonymous referees for their comments and suggestions, which contributed to improve the paper.

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Correspondence to Teresa Gomes.

Appendices

Appendix 1: Example Illustrating MSH

In the network in Fig. 13 there are three SRLGs, marked as ellipses: \(g_1= \{(1,3),(3,1),(1,4),(4,1)\}\), \(g_2=\{(3,7),(7,3),(4,7),(7,4)\}\) and \(g_3=\{(5,8),(8,5),(6,8)\}\). The label of each arc corresponds to its cost.

The original network graph \(G'\), and the shortest path from node 1 to node 8 is shown in Fig. 13. The directed network \(G'\), after the network transformation of the MSH (as described in Sect. 4.1) is shown in Fig. 14. In Fig. 15 the interlacing arc (3, 6) is removed and the solution is shown in Fig. 16.

Fig. 13
figure 13

Directed network \(G\), where \(s=1\) and \(t=8\). The seed path is \(p_1=\langle 1,3,6,8\rangle\), and three different SRLG are marked

Fig. 14
figure 14

Directed network \(G'\), after dividing the network using the MSH transformation

Fig. 15
figure 15

Shortest path \(q'_1 = \langle 1,2,6',3'',7,8\rangle\) in \(G'\)

Fig. 16
figure 16

Solution \((p,q) = (\langle 1,2,6,8\rangle , \langle 1,3,7,8\rangle )\)

Appendix 2: Auxiliary Heuristic AllPairs

The heuristic kCoSE-MScd requires a version of CoSE-MS that stores all node and SRLG-disjoint pairs discovered, during the \(i_{\max }\) iterations or until the stack of problems is empty. This task is performed by the heuristic AllPairs.

figure d

Given a seed path of problem \(P_c\), calculated in the network where the arcs affected by the SRLGs \(E_c \cap H_c\), have been removed, the MBH or MSH seek to obtain an SRLG path pair of min-sum cost. If no such pair is found, the conflicting SRLG set must be found. The set \(T_c\) is calculated as described in Sect. 4.1. The function SRLG_Exclusion \((I_c,\, p_c)\) in line 28 of AllPairs, corresponds to algorithm “Algorithm. Finding a conflicting SRLG set for a given AP \(p\) from node \(s\) to node \(t\)” in [26] and is now used only when no node-disjoint path pair can be found.

In line 31 of AllPairs is the new procedure for obtaining the conflicting SRLG set \(T_c\), used when a node disjoint path pair, which is not SRLG-disjoint, exists (as described in Sect. 4.1).

Appendix 3: Illustrating the Failure of the IMSH Optimal Condition

In [29] the following proof is presented of the optimality of the best current path pair \((p,q)\), which we reproduce here using our notation.

Let \((p,q)\) be the current optimal SRLG diverse path pair found and its cost be \(c_{(p,q)}\). In the \(i\)-th iteration, let \(c_{p_i}\) be the cost of the shortest path computed using Yen’s algorithm [38]. Let \((p'_i,p_i'')\) be the SRLG diverse path pair computed using modified Suurballe’s heuristic, if such a path pair exists. Let \((p'_i,p_i'')\) be more optimal than the current optimal \((p,q)\), i.e,

$$\begin{aligned} c_{p'_i} + c_{p''_i} < c_{(p,q)} \end{aligned}$$
(14)

Now \(c_{p'_i}, c_{p''_i} \ge c_{p_i}\). Since, without loss of generality, if \(c_{p'_i} < c_{p_i}\) the optimal SRLG path pair must have already been computed using \(p'_i\) as the seed path. Therefore,

$$\begin{aligned} 2 c_{p_i} \le c_{p'_i} + c_{p''_i} \end{aligned}$$
(15)

From Eqs. 14 and 15 we get,

$$\begin{aligned} 2 c_{p_i}&< c_{(p,q)}\\ c_{p_i}&< c_{(p,q)}/2 \end{aligned}$$

Therefore if the cost of the current seed path in the \(i\)-th iteration is greater than or equal to \(c_{(p,q)}/2\) then the optimal SRLG diverse path pair is \((p,q)\).

The problem with this proof, is in the statement “Since, without loss of generality, if \(c_{p'_i} < c_{p_i}\) the optimal SRLG path pair must have already been computed using \(p'_i\) as the seed path.” which does not hold for generic randomly generated SRLG.

Let \(p'_i=p_j, j<i\), be the shortest of the current pair obtained in the \(i\)-th iteration \((c_{p'_i} \le c_{p''_i})\). When \(p_j\) was used as seed path it may have resulted in a path pair which is not SRLG-disjoint, due to the interlacing removal. Hence \(p'_i=p_j, j<i\) may appear later, in an SRLG-disjoint path pair resulting from using a seed path \(p_i\), and this contradicts the previous statement. We next will illustrate, using an example that, in networks with randomly generated SRLGs, that this is the reason why the proof fails.

Fig. 17
figure 17

Network for illustrating the failure of the optimal condition in [29]

In the Fig. 17 an undirected network is represented. The SRLGs are: \(g_1 = \{(1,7), (8,11)\},\, g_2= \{(1,2),(1,9)\}\) and \(g_3= \{(1,7),(1,9)\}\). Initially the best solution is \((p,q) = (\emptyset ,\emptyset )\) of cost \(\infty\). Algorithm IMSH would have the following iterations:

  1. Iteration 1:

    \(p_1 = \langle 1,2,3,4,11\rangle\) of cost 4, \(R_{p_1} =\{ g2\}\) . The shortest path in the modified network is \(q'_1=\langle 1,7,3,2,8,11\rangle\) of cost 20 (in \(G'\)). These paths are SRLG-disjoint, but an interlacing exists, and after removing that interlacing the resulting path pair is:

    • \(p'_1=\langle 1,2,8,11\rangle\) of cost 11, \(R_{p'_1}=\{g_1,g_2\}\);

    • \(p''_1= \langle 1,7,3,4,11\rangle\) of cost 12, \(R_{p''_1}=\{g_1,g_3 \}\).

    Because \(R_{p'_1} \cap R_{p''_1}= \{g_1\}\) the path pair is not SRLG-disjoint and hence is not admissible.

  2. Iteration 2:

    \(p_2 = \langle 1,2,8,11\rangle\) of cost 11, \(R_{p_2} =\{ g_1,g2\}\) . The shortest path in the modified network is \(q'_2=\langle 1,5,6,11\rangle\) of cost 160 (in \(G'\)), \(R_{q'_2}= \emptyset\). There is no interlacing, and the resulting path pair is:

    • \(p'_2=\langle 1,2,8,11\rangle\) of cost 11, \(R_{p'_2} =\{ g_1,g2\}\);

    • \(p''_2= \langle 1,5,6,11\rangle\) of cost 160, \(R_{p''_2} =\emptyset\);

    which is SRLG-disjoint. The best solution is updated: \((p,q) = (\langle 1,2,8,11\rangle ,\langle 1,5,6,11\rangle )\), and \(c_{(p,q)} = 171\).

  3. Iteration 3:

    \(p_3 = \langle 1,7,3,4,11\rangle\) of cost 12, \(R_{p_3} =\{ g_1,g3,\}\) . The shortest path in the modified network is \(q'_3=\langle 1,5,6,11\rangle\) of cost 160, \(R_{q'_3}= \emptyset\). There is no interlacing, and the resulting path pair is:

    • \(p'_3=\langle 1,7,3,4,11\rangle\) of cost 12, \(R_{p_2} =\{ g_1,g2\}\);

    • \(p''_3= \langle 1,5,6,11\rangle\) of cost 160, \(R_{p_2} =\emptyset\);

    which is SRLG-disjoint. The best solution is not updated because \(c_{(p'_3,p''_3)} = 172\) is greater than \(c_{(p,q)} = 171\).

  4. Iteration 4:

    \(p_4 = \langle 1,7,3,2,8,11\rangle\) of cost 21, \(R_{p_4} =\{ g_1,g3\}\) . The shortest path in the modified network is \(q'_4=\langle 1,2,3,4,11\rangle\) of cost 3 (in \(G'\)), \(R_{q'_4}= \{g_2 \}\). These paths are SRLG-disjoint, but an interlacing exists, and after removing that interlacing the resulting path pair is:

    • \(p'_4=\langle 1,7,3,4,11\rangle\) of cost 12, \(R_{p'_4} =\{ g_1,g_3\}\);

    • \(p''_4= \langle 1,2,8,11\rangle\) of cost 160, \(R_{p''_4} =\{ g_1,g_2\}\);

    Because \(R_{p'_4} \cap R_{p''_4}= \{g_1\}\) the path pair is not SRLG-disjoint and hence is not admissible.

  5. Iteration 5

    \(p_5 = \langle 1,9,10,11\rangle\) of cost 156. Because \(c_{(p,q)} \le 2 c_{p_5}\) (\(171 \le 2\times 156\)) the algorithm would end considering that the best solution found so far, of cost 171, is the optimal solution.

It can be easily seen that at the 6-th iteration, with \(p_6 = \langle 1,5,6,11\rangle\) of cost 160, \(R_{p_6} = \emptyset\), in the modified graph \(q'_6=\langle 1,2,3,4,11\rangle\) of cost 4 (which coincides with \(p_1\)), \(R_{p_1}=R_{q'_6} =\{ g2\}\), would result in the path pair:

  • \(p'_6= \langle 1,2,3,4,11\rangle\) of cost 4, \(R_{p'_6} =\{ g2\}\)

  • \(p''_6=\langle 1,5,6,11\rangle\) of cost 160, \(R_{p''_6} =\emptyset\) ;

which is SRLG-disjoint an has cost 164. In this case \(c_{p'_6} < c_{p_6}\) and the optimal solution was not found when the seed path was \(p_1=p'_6\).

However, in a network where the SRLGs are strictly local (that is all the edges in each SRLG have the same node in common), if \(p_i\) and \(q'_i\) are SRLG-disjoint, the interlacing removal will never result in a non SRLG-disjoint solution (as in iteration 1 of this example).

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Gomes, T., Soares, M., Craveirinha, J. et al. Two Heuristics for Calculating a Shared Risk Link Group Disjoint Set of Paths of Min-Sum Cost. J Netw Syst Manage 23, 1067–1103 (2015). https://doi.org/10.1007/s10922-014-9332-6

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