Document Zbl 1370.68301

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Discrete Algorithms</a> <a href="/?q=in%3A367707" title="Articles in this Issue">44, 1-20 (2017)</a>. </div> <div class="abstract">Summary: Consider a set of $n$ stochastic lines in $\mathbb{R}^2$, where the existence probability of each line is determined by a fixed probability distribution. For a fixed $x$-coordinate $q$, the $n$ lines from top to bottom can be represented by an ordered $n$-element sequence. Consider all the $\binom{n}{k}$-element sub-sequences of that $n$-element sequence. Each $k$-element sub-sequence has an associated likelihood to be the true $k$-topmost lines at $x$-coordinate $q$, and the one with the largest probability is defined as the most likely k-topmost lines at $q$. This paper studies the most likely $k$-topmost lines of the arrangement of $n$ lines taken over all the $x$-coordinates. Let $cnt$ be the total number of distinct sequences of the most likely $k$-topmost lines over all $x$-coordinates. The main result established is that the expected value of $cnt$ is $O(k n)$, which implies that it is possible to store all the distinct most likely $k$-topmost lines in $O(k^2 n)$ expected space. An example is given showing that $cnt$, in the worst case, can be $\operatorname{\Theta}(n^2)$ even when $k = 1$. This highlights the value of the expected bound. An algorithm is also given to compute the most likely $k$-topmost lines of the arrangement. Applications of this result to the stochastic Voronoi Diagram in $\mathbb{R}^1$ and to the stochastic preference top-$k$ query in $\mathbb{R}^2$ are discussed.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6744845">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A68U05" title="MSC2020">68U05</a> </td> <td class="space"> Computer graphics; computational geometry (digital and algorithmic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P05" title="MSC2020">68P05</a> </td> <td class="space"> Data structures </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aalgorithms">algorithms</a>; <a href="/?q=ut%3Acomputational+geometry">computational geometry</a>; <a href="/?q=ut%3Adata+structures">data structures</a>; <a href="/?q=ut%3Aline+arrangements">line arrangements</a>; <a href="/?q=ut%3Astochastic+problems">stochastic problems</a>; <a href="/?q=ut%3AVoronoi+diagrams">Voronoi diagrams</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 1370.68301" data-ciurl="/ci/06744845" data-biburl="/bibtex/06744845.bib" data-amsurl="/amsrefs/06744845.bib" data-xmlurl="/xml/06744845.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06744845.pdf" title="Zbl 1370.68301 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/10.1016/j.jda.2017.04.003" aria-label="DOI for “On the arrangement of stochastic lines in $\mathbb{R}^2$”" title="10.1016/j.jda.2017.04.003">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">Agarwal, P.; Aronov, B.; Chan, T.; Sharir, M., On levels in arrangements of lines, segments, planes, and triangles, Discrete Comput. 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