{
  "id": "math/0701741",
  "version": "v2",
  "published": "2007-01-25T15:12:28.000Z",
  "updated": "2009-09-01T06:53:12.000Z",
  "title": "Search cost for a nearly optimal path in a binary tree",
  "authors": [
    "Robin Pemantle"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AAP585 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2009, Vol. 19, No. 4, 1273-1291",
  "doi": "10.1214/08-AAP585",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean $p\\leq1/2$. How many of these Bernoullis one must look at in order to find a path of length $n$ from the root which maximizes, up to a factor of $1-\\epsilon$, the sum of the Bernoullis along the path? In the case $p=1/2$ (the critical value for nontriviality), it is shown to take $\\Theta(\\epsilon^{-1}n)$ steps. In the case $p<1/2$, the number of steps is shown to be at least $n\\cdot\\exp(\\operatorname {const}\\epsilon^{-1/2})$. This last result matches the known upper bound from [Algorithmica 22 (1998) 388--412] in a certain family of subcases.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-09-01T06:53:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68W40",
      "68Q25",
      "60J80",
      "60C05"
    ],
    "keywords": [
      "binary tree",
      "search cost",
      "optimal path",
      "assigned independent bernoulli random variables",
      "upper bound"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......1741P"
    }
  }
}