Refined quicksort asymptotics. Issue 2 (12th April 2013)

• Record Type: Journal Article
• Authors: Neininger, Ralph
• Is Part Of: Random structures & algorithms. Volume 46:Issue 2(2015)
• Page Start: 346
• Page End: 361
• Publication Date: 2013-04-12
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20497
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of <italic>n</italic> data, permuted uniformly at random, the appropriately normalized complexity <italic>Y</italic><sub><italic>n</italic></sub> is known to converge almost surely to a non‐degenerate random limit <italic>Y</italic>. This assumes a natural embedding of all <italic>Y</italic><sub><italic>n</italic></sub> on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown: <disp-formula content-type="mathematics" id="rsa20497-disp-0001"><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734fzt1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10429832:media:rsa20497:rsa20497-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mrow><mml:mfrac><mml:mi>n</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>log</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:msqrt><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>Y</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mover><mml:mo>→</mml:mo><mml:mi>d</mml:mi></mml:mover><mml:mi mathvariant="script">N</mml:mi><mml:mo> </mml:mo><mml:mo> </mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>∞</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>, </mml:mo></mml:mrow></mml:math></alternatives></disp-formula> where <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3734g01b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20497:rsa20497-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">N</mml:mi></mml:math></alternatives></inline-formula> denotes a standard normal random variable. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 346–361, 2015</p> </abstract>