Abstract
In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [EmamjomehZadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of EmamjomehZadeh et al. maintains a candidates’ set for the target, while each query asks an appropriately chosen vertex– the "median"–which minimises a potential \Phi among the vertices of the candidates' set. In this paper we address three open questions posed by EmamjomehZadeh et al., namely (a) detecting a target when the query response is a direction to an approximately shortest path to the target, (b) detecting a target when querying a vertex that is an approximate median of the current candidates' set (instead of an exact one), and (c) detecting multiple targets, for which to the best of our knowledge no progress has been made so far. We resolve questions (a) and (b) by providing appropriate upper and lower bounds, as well as a new potential Γ that guarantees efficient target detection even by querying an approximate median each time. With respect to (c), we initiate a systematic study for detecting two targets in graphs and we identify sufficient conditions on the queries that allow for strong (linear) lower bounds and strong (polylogarithmic) upper bounds for the number of queries. All of our positive results can be derived using our new potential \Gamma that allows querying approximate medians.
BibTeX  Entry
@InProceedings{deligkas_et_al:LIPIcs:2017:8058,
author = {Argyrios Deligkas and George B. Mertzios and Paul G. Spirakis},
title = {{Binary Search in Graphs Revisited}},
booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)},
pages = {20:120:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770460},
ISSN = {18688969},
year = {2017},
volume = {83},
editor = {Kim G. Larsen and Hans L. Bodlaender and JeanFrancois Raskin},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8058},
URN = {urn:nbn:de:0030drops80589},
doi = {10.4230/LIPIcs.MFCS.2017.20},
annote = {Keywords: binary search, graph, approximate query, probabilistic algorithm, lower bound.}
}
Keywords: 

binary search, graph, approximate query, probabilistic algorithm, lower bound. 
Seminar: 

42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017) 
Issue Date: 

2017 
Date of publication: 

22.11.2017 