{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article10651","name":"Efficient Oracles and Routing Schemes for Replacement Paths","abstract":"Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \\in S x T, where S, T \\subseteq V(G), and every e \\in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)).\r\nMore precisely, we provide the following results:\r\n(1) DSO:\r\nFor every S,T \\subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\\sqrt{|S||T|}) and query time of\r\nO(\\sqrt{|S||T|}). At the expense of having quasi-polynomial query time,\r\nthe size of the oracle can be improved to tilde{O}(n|S|+|T|\\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings.\r\n\r\n(2) FTP:\r\nWe show the construction of a path-reporting DSO of size tilde{O}(n^{4\/3}(|S||T|)^{1\/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1\/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors).\r\n\r\n(3) Routing and Labeling Schemes:\r\nFor the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.","keywords":["Fault tolerant","Shortest path","Oracle","Routing"],"author":[{"@type":"Person","name":"Bil\u00f2, Davide","givenName":"Davide","familyName":"Bil\u00f2"},{"@type":"Person","name":"Choudhary, Keerti","givenName":"Keerti","familyName":"Choudhary"},{"@type":"Person","name":"Gual\u00e0, Luciano","givenName":"Luciano","familyName":"Gual\u00e0"},{"@type":"Person","name":"Leucci, Stefano","givenName":"Stefano","familyName":"Leucci"},{"@type":"Person","name":"Parter, Merav","givenName":"Merav","familyName":"Parter"},{"@type":"Person","name":"Proietti, Guido","givenName":"Guido","familyName":"Proietti"}],"position":13,"pageStart":"13:1","pageEnd":"13:15","dateCreated":"2018-02-27","datePublished":"2018-02-27","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Bil\u00f2, Davide","givenName":"Davide","familyName":"Bil\u00f2"},{"@type":"Person","name":"Choudhary, Keerti","givenName":"Keerti","familyName":"Choudhary"},{"@type":"Person","name":"Gual\u00e0, Luciano","givenName":"Luciano","familyName":"Gual\u00e0"},{"@type":"Person","name":"Leucci, Stefano","givenName":"Stefano","familyName":"Leucci"},{"@type":"Person","name":"Parter, Merav","givenName":"Merav","familyName":"Parter"},{"@type":"Person","name":"Proietti, Guido","givenName":"Guido","familyName":"Proietti"}],"copyrightYear":"2018","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2018.13","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/dx.doi.org\/10.1016\/j.tcs.2003.05.002","http:\/\/dl.acm.org\/citation.cfm?id=1070432.1070524","http:\/\/dx.doi.org\/10.4230\/LIPIcs.STACS.2015.513"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6299","volumeNumber":96,"name":"35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)","dateCreated":"2018-02-27","datePublished":"2018-02-27","editor":[{"@type":"Person","name":"Niedermeier, Rolf","givenName":"Rolf","familyName":"Niedermeier"},{"@type":"Person","name":"Vall\u00e9e, Brigitte","givenName":"Brigitte","familyName":"Vall\u00e9e"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article10651","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6299"}}}