Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path $\textbf{P}$ between any two vertices $u,v\in V$ is said to be $t$-approximate shortest path if its length is at most $t$ times the length of the shortest path between $u$ and $v$. We consider the problem of building a compact data structure for a given graph $G$ which is capable of answering the following query for any $u,v,z\in V$ and $t>1$. \centerline{\em report $t$-approximate shortest path between $u$ and $v$ when vertex $z$ fails} We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size {\em nearly} match the size of their best static counterparts.
@InProceedings{khanna_et_al:LIPIcs.STACS.2010.2481, author = {Khanna, Neelesh and Baswana, Surender}, title = {{Approximate Shortest Paths Avoiding a Failed Vertex: Optimal Size Data Structures for Unweighted Graphs}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {513--524}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2481}, URN = {urn:nbn:de:0030-drops-24812}, doi = {10.4230/LIPIcs.STACS.2010.2481}, annote = {Keywords: Shortest path, distance, distance queries, oracle} }
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