We consider labeling nodes of a directed graph for reachability queries. A reachability labeling scheme for such a graph assigns a binary string, called a label, to each node. Then, given the labels of nodes u and v and no other information about the underlying graph, it should be possible to determine whether there exists a directed path from u to v. By a simple information theoretical argument and invoking the bound on the number of partial orders, in any scheme some labels need to consist of at least n/4 bits, where n is the number of nodes. On the other hand, it is not hard to design a scheme with labels consisting of n/2+𝒪(log n) bits. In the classical centralised setting, where a single data structure is stored as a whole, Munro and Nicholson designed a structure for reachability queries consisting of n²/4+o(n²) bits (which is optimal, up to the lower order term). We extend their approach to obtain a scheme with labels consisting of n/3+o(n) bits.
@InProceedings{duleba_et_al:LIPIcs.ISAAC.2020.27, author = {Dul\k{e}ba, Maciej and Gawrychowski, Pawe{\l} and Janczewski, Wojciech}, title = {{Efficient Labeling for Reachability in Directed Acyclic Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.27}, URN = {urn:nbn:de:0030-drops-133710}, doi = {10.4230/LIPIcs.ISAAC.2020.27}, annote = {Keywords: informative labeling scheme, reachability, DAG} }
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