Near-Optimal Compression for the Planar Graph Metric

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using Opnq bits, or to explicitly store the distance matrix with Opk log nq bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA’01], who rule out compressions into a polynomially smaller number of bits, for weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k “ ? n, the upper bound is Opnq and their constructions imply an Ωpn3{4q lower bound. This gap is directly related to other major open questions in labelling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric into Õpminpk, ? k ̈ nqq bits, which is optimal up to log factors. Our data structure breaks an Ωpkq lower bound of Krauthgamer, Nguyen, and Zondiner [SICOMP’14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new Subset Distance Oracle for planar graphs with Õp ? k ̈ nq space, and Õpn3{4q query time. Our work carries strong messages to related fields. In particular, the famous Opn1{2q vs. Ωpn1{3q gap for distance labelling schemes in planar graphs cannot be resolved with the current lower bound techniques. ∗Supported in part by Israel Science Foundation grant 794/13. †Stanford University, Department of Computer Science, [email protected]. ‡University of Haifa, Department of Computer Science, [email protected]. Interdisciplinary Center Herzliya, Efi Arazi School of Computer Science, [email protected]. ¶University of Haifa, Department of Computer Science, [email protected]. 1 ar X iv :1 70 3. 04 81 4v 1 [ cs .D S] 1 4 M ar 2 01 7