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DOI: 10.4230/OASIcs.SOSA.2019.4
URN: urn:nbn:de:0030-drops-100305
URL: http://drops.dagstuhl.de/opus/volltexte/2018/10030/
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Dereniowski, Dariusz ; Tiegel, Stefan ; Uznanski, Przemyslaw ; Wolleb-Graf, Daniel

A Framework for Searching in Graphs in the Presence of Errors

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Abstract

We consider a problem of searching for an unknown target vertex t in a (possibly edge-weighted) graph. Each vertex-query points to a vertex v and the response either admits that v is the target or provides any neighbor s of v that lies on a shortest path from v to t. This model has been introduced for trees by Onak and Parys [FOCS 2006] and for general graphs by Emamjomeh-Zadeh et al. [STOC 2016]. In the latter, the authors provide algorithms for the error-less case and for the independent noise model (where each query independently receives an erroneous answer with known probability p<1/2 and a correct one with probability 1-p). We study this problem both with adversarial errors and independent noise models. First, we show an algorithm that needs at most (log_2 n)/(1 - H(r)) queries in case of adversarial errors, where the adversary is bounded with its rate of errors by a known constant r<1/2. Our algorithm is in fact a simplification of previous work, and our refinement lies in invoking an amortization argument. We then show that our algorithm coupled with a Chernoff bound argument leads to a simpler algorithm for the independent noise model and has a query complexity that is both simpler and asymptotically better than the one of Emamjomeh-Zadeh et al. [STOC 2016]. Our approach has a wide range of applications. First, it improves and simplifies the Robust Interactive Learning framework proposed by Emamjomeh-Zadeh and Kempe [NIPS 2017]. Secondly, performing analogous analysis for edge-queries (where a query to an edge e returns its endpoint that is closer to the target) we actually recover (as a special case) a noisy binary search algorithm that is asymptotically optimal, matching the complexity of Feige et al. [SIAM J. Comput. 1994]. Thirdly, we improve and simplify upon an algorithm for searching of unbounded domains due to Aslam and Dhagat [STOC 1991].

BibTeX - Entry

@InProceedings{dereniowski_et_al:OASIcs:2018:10030,
  author =	{Dariusz Dereniowski and Stefan Tiegel and Przemyslaw Uznanski and Daniel Wolleb-Graf},
  title =	{{A Framework for Searching in Graphs in the Presence of Errors}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{4:1--4:17},
  series =	{OpenAccess Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2018},
  volume =	{69},
  editor =	{Jeremy T. Fineman and Michael Mitzenmacher},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10030},
  URN =		{urn:nbn:de:0030-drops-100305},
  doi =		{10.4230/OASIcs.SOSA.2019.4},
  annote =	{Keywords: graph algorithms, noisy binary search, query complexity, reliability}
}

Keywords: graph algorithms, noisy binary search, query complexity, reliability
Seminar: 2nd Symposium on Simplicity in Algorithms (SOSA 2019)
Issue Date: 2018
Date of publication: 18.12.2018


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