Document Zbl 1440.68321

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The <strong>and -operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell" role="cell">The <strong>or</strong>-operator <strong>|</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3AEur%2A+J%2A+Mat%2A+Soc%2A+cc%3A14">so:Eur* J* Mat* Soc* cc:14</a></span> <span class="search-explanation" role="cell" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%27%27Partial+diff%2A+eq%2A%27%27+%21elliptic">''Partial diff* eq*'' !elliptic</a></span> <span class="search-explanation" role="cell" role="cell">The <strong>not</strong> -operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to <u>b</u>ooks; alternatively: <u>j</u> for <strong><u>j</u>ournal articles</strong>, <u>a</u> for <strong>book <u>a</u>rticles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000+-+2015+cc%3A%2894A+%7C+11T%29">py: 2000 - 2015 cc:(94A | 11T)</a></span> <span class="search-explanation" role="cell" role="cell">Number <strong>ranges</strong> when searching for <strong><u>p</u>ublication <u>y</u>ear</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href='https://en.wikipedia.org/wiki/ISO_639-1'>ISO 639 - 1</a> (opens in new tab) language codes can also be used.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=st%3A+c+r+s">st: c r s</a></span> <span class="search-explanation" role="cell" role="cell">Find documents that are <strong><u>c</u>ited</strong>, have <strong><u>r</u>eferences</strong> and are from a <strong><u>s</u>ingle author</strong>.</span> </div> </div> <p style="margin-bottom:0.5cm;"></p> <h2>Fields</h2> <div id="help-fields" role="table"> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ab</strong> </span> <span class="search-explanation" role="cell" role="cell"> Text from the summary or review (for phrases use “...”) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>an</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID, i.e.: preliminary ID, Zbl number, JFM number, ERAM number </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>any</strong> </span> <span class="search-explanation" role="cell" role="cell"> Includes ab, au, cc, en, rv, so, ti, ut </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>arxiv</strong> </span> <span class="search-explanation" role="cell" role="cell"> arXiv preprint number </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>au</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name(s) of the contributor(s) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>br</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of a person with biographic references (to find documents about the life or work) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>cc</strong> </span> <span class="search-explanation" role="cell" role="cell"> Code from the Mathematics Subject Classification (prefix with <strong>*</strong> to search only primary MSC) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ci</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID of a document cited in summary or review </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>db</strong> </span> <span class="search-explanation" role="cell" role="cell"> Database: documents in Zentralblatt für Mathematik/zbMATH Open (<em>db:Zbl</em>), Jahrbuch über die Fortschritte der Mathematik (<em>db:JFM</em>), Crelle's Journal (<em>db:eram</em>), arXiv (<em>db:arxiv</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>dt</strong> </span> <span class="search-explanation" role="cell" role="cell"> Type of the document: journal article (<em>dt:j</em>), collection article (<em>dt:a</em>), book (<em>dt:b</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>doi</strong> </span> <span class="search-explanation" role="cell" role="cell"> Digital Object Identifier (DOI) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ed</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of the editor of a book or special issue </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>en</strong> </span> <span class="search-explanation" role="cell" role="cell"> External document ID: DOI, arXiv ID, ISBN, and others </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>in</strong> </span> <span class="search-explanation" role="cell" role="cell"> zbMATH ID of the corresponding issue </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>la</strong> </span> <span class="search-explanation" role="cell" role="cell"> Language (use name, e.g., <em>la:French</em>, or <a href='https://en.wikipedia.org/wiki/ISO_639-1'>ISO 639-1</a>, e.g., <em>la:FR)</em> </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>li</strong> </span> <span class="search-explanation" role="cell" role="cell"> External link (URL) </span> </div> <div class="help-item" 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Interval search with “-” </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>pt</strong> </span> <span class="search-explanation" role="cell" role="cell"> Reviewing state: Reviewed (<em>pt:r</em>), Title Only (<em>pt:t</em>), Pending (<em>pt:p</em>), Scanned Review (<em>pt:s</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>pu</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of the publisher </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>py</strong> </span> <span class="search-explanation" role="cell" role="cell"> Year of publication. Interval search with “-” </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rft</strong> </span> <span class="search-explanation" role="cell" role="cell"> Text from the references of a document (for phrases use “...”) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rn</strong> </span> <span class="search-explanation" role="cell" role="cell"> Reviewer ID </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>rv</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name or ID of the reviewer </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>se</strong> </span> <span class="search-explanation" role="cell" role="cell"> Serial ID </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>si</strong> </span> <span class="search-explanation" role="cell" role="cell"> swMATH ID of software referred to in a document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>so</strong> </span> <span class="search-explanation" role="cell" role="cell"> Bibliographical source, e.g., serial title, volume/issue number, page range, year of publication, ISBN, etc. </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>st</strong> </span> <span class="search-explanation" role="cell" role="cell"> State: is cited (<em>st:c</em>), has references (<em>st:r</em>), has single author (<em>st:s</em>) </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>sw</strong> </span> <span class="search-explanation" role="cell" role="cell"> Name of software referred to in a document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ti</strong> </span> <span class="search-explanation" role="cell" role="cell"> Title of the document </span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"> <strong>ut</strong> </span> <span class="search-explanation" role="cell" role="cell"> Keywords </span> </div> </div> <p style="margin-bottom:0.5cm;"></p> <h2>Operators</h2> <div id="operators-fields" role="table"> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>a & b</strong></span> <span class="search-explanation" role="cell" role="cell">logical and</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>a | b</strong></span> <span class="search-explanation" role="cell" role="cell">logical or</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>!ab</strong></span> <span class="search-explanation" role="cell" role="cell">logical not</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>abc*</strong></span> <span class="search-explanation" role="cell" role="cell">right wildcard</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>ab c</strong></span> <span class="search-explanation" role="cell" role="cell">phrase</span> </div> <div class="help-item" role="row"> <span class="search-fields" role="rowheader"><strong>(ab c)</strong></span> <span class="search-explanation" role="cell" role="cell">parentheses</span> </div> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/czyzowicz.jurek" title="Author Profile">Czyzowicz, Jurek</a>; <a href="/authors/kranakis.evangelos" title="Author Profile">Kranakis, Evangelos</a>; <a href="/authors/krizanc.danny" title="Author Profile">Krizanc, Danny</a>; <a href="/authors/narayanan.lata" title="Author Profile">Narayanan, Lata</a>; <a href="/authors/opatrny.jaroslav" title="Author Profile">Opatrny, Jaroslav</a></div> <h2 class="title"> <strong>Search on a line with faulty robots.</strong> <i>(English)</i> <a class="label nowrap" href="/1440.68321">Zbl 1440.68321</a> </h2> <div class="source"> <a href="/serials/1733" title="Journal Profile">Distrib. Comput.</a> <a href="/?q=in%3A442834" title="Articles in this Issue">32, No. 6, 493-504 (2019)</a>. </div> <div class="abstract"><div class="pre">Summary: We consider the problem of searching on a line using $n$ mobile robots, of which at most $f$ are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a <span class="zbmathjax-textit">target</span> placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target. If $n \ge 2f+2$, there is a simple algorithm with a competitive ratio of 1. For $f< n < 2f+2$ we develop a new class of algorithms, called <span class="zbmathjax-textit">proportional schedule algorithms</span>. For any given $(n, f)$, we give a proportional schedule algorithm $A(n, f)$, whose competitive ratio is \[( \frac{4f+4}{n} ) ^{\frac{2f+2}{n}} ( \frac{4f+4}{n} -2 ) ^{1-\frac{2f+2}{n}} + 1. \] Setting $a=n/f$ as a constant, the asymptotic competitive ratio is $\left( 4/a \right) ^{2/a} \left( 4/a -2\right) ^{1-2/a } + 1$. Our search algorithm is easily seen to be optimal for the case $n=f+1$. We also show that as $n$ tends to $\infty$ the competitive ratio of our algorithm for the case $n = 2f+1$ approaches 3 and this is optimal. More precisely, we show that asymptotically (after excluding small order terms), the competitive ratio of our proportional schedule algorithm $A(2f+1,f)$ is at most $3 + \frac{4\ln n}{n} $, while any search algorithm has a lower bound $3 + \frac{2\ln n}{n}$ on its competitive ratio.</div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7137738">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68W10" title="MSC2020">68W10</a> </td> <td class="space"> Parallel algorithms in computer science </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M14" title="MSC2020">68M14</a> </td> <td class="space"> Distributed systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M15" title="MSC2020">68M15</a> </td> <td class="space"> Reliability, testing and fault tolerance of networks and computer systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68M20" title="MSC2020">68M20</a> </td> <td class="space"> Performance evaluation, queueing, and scheduling in the context of computer systems </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asearch+on+a+line">search on a line</a>; <a href="/?q=ut%3Afaulty+robots">faulty robots</a>; <a href="/?q=ut%3Acow-path+problem">cow-path problem</a>; <a href="/?q=ut%3Acompetitive+ratio">competitive ratio</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1440.68321" data-ciurl="/ci/07137738" data-biburl="/bibtex/07137738.bib" data-amsurl="/amsrefs/07137738.bib" data-xmlurl="/xml/07137738.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07137738.pdf" title="Zbl 1440.68321 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00446-017-0296-0" aria-label="DOI for “Search on a line with faulty robots”" title="10.1007/s00446-017-0296-0">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Agmon, N.; Peleg, D., Fault-tolerant gathering algorithms for autonomous mobile robots, SIAM J. Comput., 36, 56-82 (2006) · <a href="/1111.68136" class="nowrap">Zbl 1111.68136</a> · <a href="https://doi.org/10.1137/050645221" class="nowrap">doi:10.1137/050645221</a></td> </tr><tr> <td>[2]</td> <td class="space">Albers, S.; Henzinger, MR, Exploring unknown environments, SIAM J. Comput., 29, 1164-1188 (2000) · <a href="/0947.68165" class="nowrap">Zbl 0947.68165</a> · <a href="https://doi.org/10.1137/S009753979732428X" class="nowrap">doi:10.1137/S009753979732428X</a></td> </tr><tr> <td>[3]</td> <td class="space">Albers, S.; Kursawe, K.; Schuierer, S., Exploring unknown environments with obstacles, Algorithmica, 32, 123-143 (2002) · <a href="/1052.68127" class="nowrap">Zbl 1052.68127</a> · <a href="https://doi.org/10.1007/s00453-001-0067-x" class="nowrap">doi:10.1007/s00453-001-0067-x</a></td> </tr><tr> <td>[4]</td> <td class="space">Alpern, S., Gal, S.: The Theory of Search Games and Rendezvous, vol. 55. Kluwer Academic Publishers, Alphen aan den Rijn (2002) · <a href="/1034.91017" class="nowrap">Zbl 1034.91017</a></td> </tr><tr> <td>[5]</td> <td class="space">Alpern, S., Fokkink, R., Gasieniec, L., Lindelauf, R., Subrahmanian, V.S.: Search theory: a game-theoretic perspective. Springer Science & Business Media, Berlin (2014) · <a href="/1263.91001" class="nowrap">Zbl 1263.91001</a></td> </tr><tr> <td>[6]</td> <td class="space">Baeza Yates, R.; Culberson, J.; Rawlins, G., Searching in the plane, Inf. Comput., 106, 234-252 (1993) · <a href="/0781.68044" class="nowrap">Zbl 0781.68044</a> · <a href="https://doi.org/10.1006/inco.1993.1054" class="nowrap">doi:10.1006/inco.1993.1054</a></td> </tr><tr> <td>[7]</td> <td class="space">Baeza-Yates, R.; Schott, R., Parallel searching in the plane, Comput. Geom., 5, 143-154 (1995) · <a href="/0839.68104" class="nowrap">Zbl 0839.68104</a> · <a href="https://doi.org/10.1016/0925-7721(95)00003-R" class="nowrap">doi:10.1016/0925-7721(95)00003-R</a></td> </tr><tr> <td>[8]</td> <td class="space">Beck, A., On the linear search problem, Isr. J. Math., 2, 221-228 (1964) · <a href="/0168.39502" class="nowrap">Zbl 0168.39502</a> · <a href="https://doi.org/10.1007/BF02759737" class="nowrap">doi:10.1007/BF02759737</a></td> </tr><tr> <td>[9]</td> <td class="space">Beck, A.; Newman, D., Yet more on the linear search problem, Isr. J. 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