Optimal $K$-survivable backward-recursive path computation (BRPC) in multi-domain PCE-based networks

Abstract

The path computation element (PCE) enables optimal path computation in single-domain (G)MPLS networks. To overcome the lack of traffic engineering (TE) information in multi-domain networks and to preserve both computation optimality and domain confidentiality, the backward-recursive PCE-based computation (BRPC) procedure has been standardized. BRPC procedure is based on PCE protocol (PCEP) and enables synchronized computation of TE label-switched paths with the requested level of reliability and quality of service requirements (i.e., guaranteed bandwidth). In this paper, the problem of computing $K$-survivable optimal multi-domain paths by resorting to BRPC procedure is analyzed. Extensions to PCEP protocol are discussed to achieve path optimality when domain information is kept confidential. The optimality is theoretically proved, and the computational complexity is shown to be more efficient than existing approaches. The discussed extensions are experimentally validated in an MPLS network test bed based on commercial equipments and are shown to have performance comparable to standard PCEP approach.

Appendix

Lemma

When $K\!=\!0$, if the element $\mathbf{w}_{j+1}(i)$ indicates the minimum weight of the path from node $\mathbf{r}_{j+1}(i)$ to $T$ and if $\mathbf{r}_{j+1}$ contains all the nodes in $B_{j+1}^{in}$, then PCE of domain $j$ is able to optimally compute the minimum weight of a path from a node in its own domain, to $T$, in polynomial time. Thus, source PCE is able to compute the minimum weight of the path from $S$ to $T$.

Proof

It is a well-known property of the minimum weight path (with additive weights) that, for a given minimum weight path such as $p = \{S, 1, 2, \ldots m, {m+1},\ldots ,\,q, \ldots , T\}$, the subpath of $p$ from a node $m \in p$ to another node $q \in p$ is also a minimum weight path from $m$ to $q$. In other words, if we know that nodes $m$ and $q$ belong to the minimum weight path $p$, then it is possible to compute the path from $S$ to $T$ by joining together the minimum weight path from $S$ to $m$, with the minimum weight path from $m$ to $q$, and with the minimum weight path from $q$ to $T$. Note that this principle can be extended for any arbitrary number of subpaths of $p$.

Based on this property, the lemma is proven for the following two distinct cases.

• Case 1: A single inter-domain link exists between any pair of adjacent domains, i.e., $|E_{j,j+1}|=|B_{j+1}^{in}| = |B_{j}^{out}| =1\,\forall ~1 \le j < D$. Then, the minimum weight path from $S$ to $T$ in $G$ is forced to pass on each link in the sets $E_{j,j+1}\,\forall ~1 \le j < D$. By applying the aforementioned principle, the path from $S$ to $T$ can be optimally found by connecting the minimum weight path from $S$ to the single node in $B_{2}^{in}$, with the minimum weight path from the single node in $B_{2}^{in}$ to the node in $B_{3}^{in}$ and so on, with finally the minimum weight path from $B_{D}^{in}$ to $T$. Therefore, the computation of the minimum weight path from $S$ to $T$ could be even carried out by computing in parallel the minimum weight paths in each domain and joining them. Vectors $\mathbf{r}_j$ and $\mathbf{w}_j$ would not even be required, i.e., need not be included in PCRep message. In this case, computation by each PCE (even carried out in parallel) leads to optimal solution, and thus, the lemma holds.

• Case 2: The number of inter-domain links between any two domains is greater than one, i.e., $|E_{j,j+1}| \ge 1$ for any $1 \le j < D$. Then the minimum weight path from $S$ to $T$ should pass through one of the links of $E_{j,j+1}$ as they form a cut, but it is not possible to know a priori through which one (i.e., through which node in $B_{j+1}^{in}$).

First, the lemma is proved for the case of $D=2$, i.e., $S$ and $T$ are in two adjacent domains, and then for the case $D > 2$. When $D=2$, the optimal path is the one at minimum weight among the set of minimum weight paths from $S$ to $T$ and has as intermediate node one of the nodes in $B_{2}^{in}$, i.e., $m\in B_{2}^{in}$. Thus, in order to compute the optimal path from $S$ to $T$ across two domains, it is necessary and sufficient to compute the minimum weight paths $(\mathbf{p}_{mT})$ from any node $m\in B_{2}^{in}$ to $T$ and minimum weight paths $(\mathbf{p}_{Sm})$ from $S$ to any node $m\in B_{2}^{in}$. It is a sufficient condition, because, since the links arriving at $B_{2}^{in}$ (i.e., $E_{1,2}$) form a cut, the minimum weight path must pass through one of them, i.e., one of $m \in B_{2}^{in}$. It is a necessary condition, since if node $m\prime \in B_{2}^{in}$ is not considered and thus neither are the corresponding subpaths $\mathbf{p^\prime }_{Sm\prime }=(S,\ldots ,m\prime )$ and $\mathbf{p^\prime }_{m\prime ,T}=(m\prime ,\ldots , T)$, the optimal minimum weight path from $S$ to $T$ may pass through $m\prime$ and thus may not be found.

When $D >2$, the same arguments can be repeated recursively for each domain pair. For example, let $D=3$. Then, the previous demonstration can be applied to the minimum weight path from any node $m \in B_{2}^{in}$ to $T$, i.e., the source node is a BN in $B_{2}^{in}$. PCE of domain $2$ can compute the minimum weight of paths from $m$ to $T$ on an auxiliary graph $G^\prime _2$. $G^\prime _2$ is the graph obtained by adding to $G_2$ the nodes in $B_{3}^{in}$, a dummy node $\delta$, the inter-domain links $E_{2,3}$, and a link connecting each node $m \in B_{3}^{in}$ to $\delta$. Weights of links $(m,\,\delta )$ are given by elements in $\mathbf{w}_{3}$. With the computed $\mathbf{r}_2$ and $\mathbf{w}_2$, the same arguments apply also at the source domain, for the minimum weight path from $S$ to $T$ passing through any $m \in B_{1}^{in}$. $\square$

Theorem

If element $\mathbf{w}_{j+1}(i)$ indicates the minimum weight of the link-disjoint paths from nodes in $\mathbf{r}_{j+1}(i)$ to $T$ and if $\mathbf{r}_{j+1}$ contains all the subsets of $K\!+\!1$ nodes (with repetition) of $B_{j+1}^{in}$, then PCE of domain $j$ is able to compute the minimum weight link-disjoint paths (if they exist), which are $K$-survivable to link faults, from any node in its own domain to $T$, using Suurballe and Tarjan algorithm [9]. Thus, source PCE is able to compute the minimum weight link-disjoint paths from $S$ to $T$, which are $K$-survivable to link faults.

Proof

From the link-disjointness property, the link-disjointness is preserved when interconnecting link-disjoint paths. Let us prove the optimality of the link-disjoint paths by contradiction.

By contradiction, a set of inter-domain $K$-survivable paths from BNs of domain $j$ to $T$ exist but cannot be computed optimally by using Suurballe and Tarjan algorithm on the domain, using the received weights of the downstream domains. This means that either at least one of the $K\!+\!1$ link-disjoint paths should traverse another BN or other path(s) should have been selected within domain $j$. The former case is not possible as all the combinations (with repetition) of BNs leading to $K+1$ link-disjoint paths are considered in set $\mathbf{r}_{j+1}$, by assumption. The latter case means that either 1) the Suurballe and Tarjan algorithm [9] computing the minimum weight link-disjoint paths does not find the optimal solution on the auxiliary graph representing the domain or 2) the choice of considering only $K\!+\!1$ link-disjoint paths that have minimum weight sum is not optimal. The first option is not correct as the Suurballe and Tarjan algorithm is optimal and is run on the domain graph in which additional links at null weight are added. The second option means that other $K\!+\!1$ paths (i.e., another subset of BNs) to $T$ on the downstream domains should have been selected, although the summation of their weights is not minimum. This would imply that weight sum of the link-disjoint paths from BNs of domain $j\!+\!1$ to $T$ is increased, i.e., the solution is not optimal, but it contradicts the hypothesis. $\square$