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The Maximum of a Random Walk and Its Application to Rectangle Packing

Published online by Cambridge University Press:  27 July 2009

E. G. Coffman Jr ,
Philippe Flajolet ,
Leopold Flatto  and
Micha Hofri
E. G. Coffman Jr
Affiliation:
Bell Labs, Lucent Technologies, 700 Mountain Avenue, Murray Hill, New Jersey 07974
Philippe Flajolet
Affiliation:
INRIA-Rocquencourt F-78153 La Chesney, France
Leopold Flatto
Affiliation:
Bell Labs, Lucent Technologies, 700 Mountain Avenue, Murray Hill, New Jersey 07974
Micha Hofri
Affiliation:
Department of Computer Science, Rice University, Houston, Texas 77005
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Abstract

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate

,

where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 12 , Issue 3 , July 1998 , pp. 373 - 386
Copyright
Copyright © Cambridge University Press 1998

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