Reference Work Entry

Encyclopedia of Algorithms

pp 1-99

# Tail Bounds for Occupancy Problems

1995; Kamath, Motwani, Palem, Spirakis
• Paul SpirakisAffiliated withComputer Engineering and Informatics, Research and Academic Computer Technology Institute, Patras University
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The properties of the resulting distribution of balls among bins have been the subject of intensive study in the probability and statistics literature [<span class="CitationRef"><a href="#CR3_419">3</a></span>,<span class="CitationRef"><a href="#CR4_419">4</a></span>]. In computer science, this process arises naturally in randomized algorithms and probabilistic analysis. Of particular interest is the <em class="EmphasisTypeItalic ">occupancy problem</em> where the random variable under consideration is the number of empty bins.</p><p class="Para">In this entry a series of bounds are presented (reminiscent of the Chernoff bound for binomial distributions) on the tail of the distribution of the number of empty bins; the tail bounds are successively tighter, but each new bound has a more complex closed form. Such strong bounds do not seem to have appeared in the earlier literature.</p></section></body></html><!--4.0--> <!DOCTYPE html SYSTEM "about:legacy-compat"> <html xmlns="http://www.w3.org/1999/xhtml" lang=""><head><title/><meta charset="utf-8"/><script type="text/x-mathjax-config"> MathJax.Hub.Config({ /* load MathML extension */ extensions: ["mml2jax.js"], /* configure input and output */ jax: ["input/TeX", "input/MathML", "output/HTML-CSS"], /* align display equations to the left */ displayAlign: "left", /* set zoom trigger */ menuSettings: { zoom: "Click" }, /* configuration of the tex2jax preprocessor: - set delimiters for inline and display equations - preview while equations are processed is "TeX" - allow \ to represent a literal dollar sign */ tex2jax: { inlineMath: [ ["\\(","\"] ], displayMath: [ ['$$','$$'], ["\$","\$"] ], preview: ["TeX"], processEscapes: true }, /* configuration of the mml2jax preprocessor: - preview while equations are processed is "MathML" */ mml2jax: { 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xmlns:meta="http://www.springer.com/app/meta" xmlns:epub="http://www.idpf.org/2007/ops" class="FulltextWrapper"><aside class="Bibliography" id="Bib1_419"><h3 class="Heading">Recommended Reading</h3><ol class="BibliographyWrapper"><li class="Citation"><div class="CitationNumber">1.</div><div class="CitationContent" id="CR1_419">Kamath, A., Motwani, R., Spirakis, P., Palem, K.: Tail bounds for occupancy and the satisfiability threshold conjecture. J. Random Struct. Algorithms <strong class="EmphasisTypeBold ">7</strong>(1), 59–80 (1995)<span class="Occurrences"><span class="Occurrence OccurrenceAMSID"><a href="http://www.ams.org/mathscinet-getitem?mr=1346284"><span><span>MathSciNet</span></span></a></span><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0834.68051"><span><span>MATH</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">2.</div><div class="CitationContent" id="CR2_419">Janson, S.: Large Deviation Inequalities for Sums of Indicator Variables. Technical Report No. 34, Department of Mathematics, Uppsala University (1994)</div></li><li class="Citation"><div class="CitationNumber">3.</div><div class="CitationContent" id="CR3_419">Johnson, N.L., Kotz, S.: Urn Models and Their Applications. Wiley, New York (1977)</div></li><li class="Citation"><div class="CitationNumber">4.</div><div class="CitationContent" id="CR4_419">Kolchin, V.F., Sevastyanov, B.A., Chistyakov, V.P.: Random Allocations. Wiley, New York (1978)</div></li><li class="Citation"><div class="CitationNumber">5.</div><div class="CitationContent" id="CR5_419">Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, New York (1995)<span class="Occurrences"><span class="Occurrence OccurrenceZLBID"><a href="http://www.emis.de/MATH-item?$0849.68039"><span><span>MATH</span></span></a></span></span></div></li><li class="Citation"><div class="CitationNumber">6.</div><div class="CitationContent" id="CR6_419">Shwartz, A., Weiss, A.: Large Deviations for Performance Analysis. Chapman-Hall, Boca Raton (1994)</div></li><li class="Citation"><div class="CitationNumber">7.</div><div class="CitationContent" id="CR7_419">Weiss, A.: Personal Communication (1993)</div></li></ol></aside></div> </div> </div> </div> </div> <div id="abstract-about" class="expander expander-open"> <div class="expander-title"> <div class="heading"> <button> <h2>About this Reference Work Entry</h2> </button> </div> </div> <div class="expander-content"> <div class="expander-content-inner"> <div class="summary"> <dl> <dt>Title</dt> <dd id="abstract-about-title">Tail Bounds for Occupancy Problems</dd> <dt>Reference Work Title</dt> <dd id="abstract-about-publication"> <a href="/book/10.1007/978-0-387-30162-4">Encyclopedia of Algorithms</a> </dd> <dt id="dt-abstract-about-book-chapter-page-ranges">Pages</dt> <dd id="abstract-about-book-chapter-page-ranges">pp 1-99</dd> <dt id="dt-abstract-about-book-chapter-copyright-year">Copyright</dt> <dd id="abstract-about-book-chapter-copyright-year">2008</dd> <dt>DOI</dt> <dd id="abstract-about-book-chapter-doi" class="doi">10.1007/978-0-387-30162-4_419</dd> <dt id="dt-abstract-about-book-print-isbn">Print ISBN</dt> <dd id="abstract-about-book-print-isbn">978-0-387-30770-1</dd> <dt id="dt-abstract-about-book-online-isbn">Online ISBN</dt> <dd id="abstract-about-book-online-isbn">978-0-387-30162-4</dd> <dt id="dt-abstract-about-publisher">Publisher</dt> <dd id="abstract-about-publisher">Springer US</dd> <dt id="dt-abstract-about-book-copyright-holder">Copyright Holder</dt> <dd id="abstract-about-book-copyright-holder">Springer-Verlag</dd> <dt>Additional Links</dt> <dd id="abstract-about-additional-links"> <ul> <li> <a class="external" href="http://www.springer.com/978-0-387-30770-1?wt_mc=ThirdParty.SpringerLink.3.EPR653.About_eBook" target="_blank" title="It opens in new window">About this Reference Work</a> </li> </ul> </dd> </dl> <dl> <dt>Topics</dt> <dd itemprop="genre"> <ul class="abstract-about-subject"> <li> <a href="/search?facet-subject=%22Theory+of+Computation%22">Theory of Computation</a> </li> <li> <a href="/search?facet-subject=%22Electronic+and+Computer+Engineering%22">Electronic and Computer Engineering</a> </li> <li> <a href="/search?facet-subject=%22Applications+of+Mathematics%22">Applications of Mathematics</a> </li> <li> <a href="/search?facet-subject=%22Computing+Methodologies%22">Computing Methodologies</a> </li> <li> <a href="/search?facet-subject=%22Computational+Mathematics+and+Numerical+Analysis%22">Computational Mathematics and Numerical Analysis</a> </li> </ul> </dd> <dt>Industry Sectors</dt> <dd itemprop="genre"> <ul class="abstract-about-industrysectors"> <li> <a title="/industry/electronics" href="/industry/electronics">Electronics</a> </li> <li> <a title="/industry/telecom" href="/industry/telecom">Telecommunications</a> </li> <li> <a title="/industry/it" href="/industry/it">IT & Software</a> </li> </ul> </dd> <dt>eBook 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