, Volume 80, Issue 1, pp 209–233 | Cite as

On the Complexity of Compressing Two Dimensional Routing Tables with Order

  • Frédéric Giroire
  • Frédéric Havet
  • Joanna Moulierac


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 \(|S|=|T|\).


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


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, Inria, I3SSophia Antipolis CedexFrance

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