Compressing Virtual Forwarding Information Bases Using the Trie-folding Algorithm

Abstract

In the latest years network virtualization and virtual routers have gained considerable attention, because with their aid it is possible to realize multiple virtual topologies on the same physical network. This plays a key role in cloud-based systems and VPN (Virtual Private Network), amongst others. In order to be able to forward IP packets according to multiple topologies, the router needs to store numerous Forwarding Information Bases (FIBs), which can lead to memory scalability issues. We address this issue by applying the well-known “trie-folding” FIB compression method to the case of multiple virtual FIBs (VFIBs). We propose two novel approaches to perform the compression, based on different virtual router architectures. We introduce a further opportunity for optimization, based on the distribution of next-hop labels. We formulate a minimization problem using entropy measure, and we provide a heuristic approach of solving the problem. We present numerical evaluations including lookup speed, memory size of compressed VFIBs, and their corresponding entropies. Based on these results and the underlying theoretical reasoning, we can safely say that the presented techniques are not only able to resolve the memory scalability issues of modern virtual routers, but they also improve lookup speed considerably, one of the most important performance measures in a core router, be it virtual or not.

Appendix

Proof of Theorem 1:

$$ H_{0}^{sep} = \frac{{\left( {\sum\limits_{i} {\sum\limits_{j} {n_{j}^{i} log\left( {\frac{{n^{i} }}{{n_{j}^{i} }}} \right)} } } \right)}}{n} = \mathop \sum \limits_{j} \left( {\frac{{n_{j} }}{n}} \right)\mathop \sum \limits_{i} \left( {\frac{{n_{j}^{i} }}{{n_{j} }}log\left( {\frac{{n^{i} }}{{n_{j}^{i} }}} \right)} \right) \le \mathop \sum \limits_{j} \left( {\frac{{n_{j} }}{n}log\mathop \sum \limits_{i} \left( {\frac{{n^{i} }}{{n_{j} }}} \right)} \right) = H_{0}^{com} $$

This is true, because the log function is concave: $\sum {a_{i} = 1} \Rightarrow \sum { a_{i} \,log\,x_{i} } \le log \sum {a_{i} \,x_{i} }$, which proves that $H_{0}^{sep} \le H_{0}^{com}$.

Proof of Theorem 2:

If we allow the permutation of next-hop labels in each VFIB then we can reduce the entropy of $H_{0}^{com}$:

$$ H_{0}^{rea} = \mathop \sum \limits_{j} \left( {\frac{{\sum\limits_{i} {m_{j}^{i} } }}{{\sum\limits_{i} {n^{i} } }}log\left( {\frac{{\sum\limits_{i} {n^{i} } }}{{\sum\limits_{i} {m_{j}^{i} } }}} \right)} \right) = \mathop \sum \limits_{j} \left( {\frac{{m_{j} }}{n}log\frac{n}{{m_{j} }}} \right) $$

We achieve this by rearranging the characters in each string according to their incidence, where m ⁱ _j denotes the incidence of the j ^th next-hop label in the i ^th VFIB. Since this transformation has no effect on $H_{0}^{sep}$, it follows that $H_{0}^{sep} \le H_{0}^{rea}$.

Proof of Theorem 3:

For this we only need to show that if not all next-hop labels of all VFIBs are ordered according to incidence, then we can decrease the entropy by changing the order of them. In other words, i ₁ , i ₂ , j ₁ , j ₂ exist such that n _j1 i ₁  <= n _j2 i ₂ and n _j1 i ₂  > n _j2 i ₂ while n_j1 <= n_j2. Swapping j ₁ and j ₂ in i ₂ the entropy is decreasing. For the sake of simplicity let p denote n _j1 i ₂ − n _j2 i ₂ :

$$ \frac{{n_{j1} }}{n}log\frac{n}{{n_{j1} }} + \frac{{n_{j2} }}{n}log\frac{n}{{n_{j2} }} > \frac{{\left( {n_{j1} - p} \right)}}{n}log\frac{n}{{\left( {n_{j1} - p} \right)}}log\frac{n}{{\left( {n_{j1} - p} \right)}} + \frac{{\left( {n_{j2} + p} \right)}}{n}log\frac{n}{{\left( {n_{j2} + p} \right)}} $$

$$ \left( {n_{j1} - p} \right)log\left( {n_{j1} - p} \right) + \left( {n_{j2} + p} \right)log\left( {n_{j2} + p} \right) > n_{j1} logn_{j1} + n_{j2} logn_{j2} $$

This is satisfied because x log(x) is convex. This proves that $H_{0}^{rea} \le H_{0}^{com}$.