  The problem of detecting loops in a forwarding network is known to be
NP-complete when general rules such as wildcard expressions are used. Yet,
network analyzer tools such as Netplumber (Kazemian et al., NSDI'13) or
Veriflow (Khurshid et al., NSDI'13) efficiently solve this problem in networks
with thousands of forwarding rules. In this paper, we complement such
experimental validation of practical heuristics with the first provably
efficient algorithm in the context of general rules. Our main tool is a
canonical representation of the atoms (i.e. the minimal non-empty sets) of the
field of sets generated by a collection of sets. This tool is particularly
suited when the intersection of two sets can be efficiently computed and
represented. In the case of forwarding networks, each forwarding rule is
associated with the set of packet headers it matches. The atoms then correspond
to classes of headers with same behavior in the network. We propose an
algorithm for atom computation and provide the first polynomial time algorithm
for loop detection in terms of number of classes (which can be exponential in
general). This contrasts with previous methods that can be exponential, even in
simple cases with linear number of classes. Second, we introduce a notion of
network dimension captured by the overlapping degree of forwarding rules. The
values of this measure appear to be very low in practice and constant
overlapping degree ensures polynomial number of header classes. Forwarding loop
detection is thus polynomial in forwarding networks with constant overlapping
degree.
