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An improved Lower Bound for Local Failover in Directed Networks via Binary Covering Arrays

Erik van den Akker, Klaus-Tycho Foerster

TL;DR

A network is constructed, in which successful routing is linked to the Covering Array Problem on a binary alphabet, leading to a lower bound of $\Omega(k + \lceil\log\log(\lceil\frac{n}{4}\rceil-k)\rceil)$ for failures in an node network.

Abstract

Communication networks often rely on some form of local failover rules for fast forwarding decisions upon link failures. While on undirected networks, up to two failures can be tolerated, when just matching packet origin and destination, on directed networks tolerance to even a single failure cannot be guaranteed. Previous results have shown a lower bound of at least $\lceil\log(k+1)\rceil$ rewritable bits to tolerate $k$ failures. We improve on this lower bound for cases of $k\geq 2$, by constructing a network, in which successful routing is linked to the \textit{Covering Array Problem} on a binary alphabet, leading to a lower bound of $Ω(k + \lceil\log\log(\lceil\frac{n}{4}\rceil-k)\rceil)$ for $k$ failures in an $n$ node network.

An improved Lower Bound for Local Failover in Directed Networks via Binary Covering Arrays

TL;DR

A network is constructed, in which successful routing is linked to the Covering Array Problem on a binary alphabet, leading to a lower bound of for failures in an node network.

Abstract

Communication networks often rely on some form of local failover rules for fast forwarding decisions upon link failures. While on undirected networks, up to two failures can be tolerated, when just matching packet origin and destination, on directed networks tolerance to even a single failure cannot be guaranteed. Previous results have shown a lower bound of at least rewritable bits to tolerate failures. We improve on this lower bound for cases of , by constructing a network, in which successful routing is linked to the \textit{Covering Array Problem} on a binary alphabet, leading to a lower bound of for failures in an node network.
Paper Structure (20 sections, 1 theorem, 2 equations, 1 figure)

This paper contains 20 sections, 1 theorem, 2 equations, 1 figure.

Table of Contents

  1. Motivation and Background
  2. Motivation.
  3. Related Work on Undirected Graphs.
  4. Related Work on Directed Graphs.
  5. Overview.
  6. Model
  7. Network and Forwarding Model.
  8. Deadend Avoidance.
  9. Goal.
  10. Remark:
  11. The Covering Array Problem
  12. Overview.
  13. Definition.
  14. Example.
  15. Lower Bound.
  16. ...and 5 more sections

Key Result

theorem 1

For each number of failures $k$ there are networks of size at least $4k+5$, such that $\Omega(k + \lceil\log\log(\lceil\frac{n}{4}\rceil-k)\rceil)$ header bits are needed to forward packets from $s$ to $t$.

Figures (1)

  • Figure 1: Network for the lower bound construction. All $a_i$ and $b_i$ have an arc back to $s$. Any of the arcs $a_i \rightarrow x_{i+1}$ can fail (but not both at the same time), there needs to be at least one failover path that does not run into any failed link.

Theorems & Definitions (1)

  • theorem 1