Optimal sequential selection of a unimodal subsequence of a random sequence

  • Authors:

  • Alessandro Arlotto;J. michael Steele

  • Affiliations:

  • Department of operations and information management, wharton school, huntsman hall 527.2, university of pennsylvania, philadelphia, pa 19104, usa (e-mail: alear@wharton.upenn.edu);Department of statistics, wharton school, huntsman hall 447, university of pennsylvania, philadelphia, pa 19104, usa (e-mail: steele@wharton.upenn.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2011

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Abstract

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.