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تسجيل مستخدم جديدCompact Distributed Certification of Planar Graphs
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Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a emph{distributed interactive proof} for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM/ protocol), and uses small certificates, on $O(log n)$ bits in $n$-node networks. We show that a single interaction from the prover suffices, and randomization is unecessary, by providing an explicit description of a emph{proof-labeling scheme} for planarity, still using certificates on just $O(log n)$ bits. We also show that there are no proof-labeling schemes -- in fact, even no emph{locally checkable proofs} -- for planarity using certificates on $o(log n)$ bits.
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A distributed proof (also known as local certification, or proof-labeling scheme) is a mechanism to certify that the solution to a graph problem is correct. It takes the form of an assignment of labels to the nodes, that can be checked locally. There
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Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by us
ing this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a $mathsf{dMAM}(O(log n))$ protocol for this class, that is, a distributed interactive protocol with $O(log n)$-bit proof size in $n$-node graphs, and three interactions between the (centralizer) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a $mathsf{dMAM}(O(log n))$ protocol for the class of planar graphs, as well as for the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of $O(log n)$ bits. This result also holds for the class of graphs with bounded demi-genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded demigenus.
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