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A Framework for Searching in Graphs in the Presence of Errors

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].

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Details

Category:
Archival
Type:
publikacja w wydawnictwie zbiorowym recenzowanym (także w materiałach konferencyjnych)
Title of issue:
2nd Symposium on Simplicity in Algorithms (SOSA 2019) strony 4:1 - 4:17
ISSN:
2190-6807
Language:
English
Publication year:
2019
Bibliographic description:
Dereniowski D., Tiegel S., Uznański P., Wolleb-Graf D.: A Framework for Searching in Graphs in the Presence of Errors// 2nd Symposium on Simplicity in Algorithms (SOSA 2019)/ ed. Jeremy T. Fineman and Michael Mitzenmacher Dagstuhl, Niemcy: Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2019, s.4:1-4:17
DOI:
Digital Object Identifier (open in new tab) 10.4230/oasics.sosa.2019.4
Verification:
Gdańsk University of Technology

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