Algorithmica

, Volume 80, Issue 1, pp 209–233

# On the Complexity of Compressing Two Dimensional Routing Tables with Order

• Frédéric Giroire
• Frédéric Havet
• Joanna Moulierac
Article

## Abstract

Motivated by routing in telecommunication network using Software Defined Network (SDN) technologies, we consider the following problem of finding short routing lists using aggregation rules. We are given a set of communications $$\mathcal {X}$$, which are distinct pairs $$(s,t)\subseteq S\times T$$, (typically S is the set of sources and T the set of destinations), and a port function $$\pi :\mathcal {X} \rightarrow P$$ where P is the set of ports. A routing list $$\mathcal {R}$$ is an ordered list of triples which are of the form (stp), $$(*,t,p)$$, $$(s,*,p)$$ or $$(*,*,p)$$ with $$s\in S$$, $$t\in T$$ and $$p\in P$$. It routes the communication (st) to the port $$r(s,t) =p$$ which appears on the first triple in the list $$\mathcal {R}$$ that is of the form (stp), $$(*,t,p)$$, $$(s,*,p)$$ or $$(*,*,p)$$. If $$r(s,t)=\pi (s,t)$$, then we say that (st) is properly routed by $$\mathcal {R}$$ and if all communications of $$\mathcal {X}$$ are properly routed, we say that $$\mathcal {R}$$ emulates $$(\mathcal {X}, \pi )$$. The aim is to find a shortest routing list emulating $$(\mathcal {X}, \pi )$$. In this paper, we carry out a study of the complexity of the two dual decision problems associated to it. Given a set of communication $$\mathcal {X}$$, a port function $$\pi$$ and an integer k, the first one called Routing List (resp. the second one, called List Reduction) consists in deciding whether there is a routing list emulating $$(\mathcal {X}, \pi )$$ of size at most k (resp. $$|\mathcal {X}| -k$$). We prove that both problems are NP-complete. We then give a 3-approximation for List Reduction, which can be generalized to higher dimensions. We also give a 4-approximation for Routing List in the fundamental case when there are only two ports (i.e. $$|P|=2$$), $$\mathcal {X}=S\times T$$ and $$|S|=|T|$$.

## Keywords

Routing Routing tables Order Priority Software Defined Networks Complexity Approximation algorithm Compact tables

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