The Annals of Applied Probability

Search cost for a nearly optimal path in a binary tree

Robin Pemantle

Full-text: Open access

Abstract

Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean p≤1/2. 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−ɛ, the sum of the Bernoullis along the path? In the case p=1/2 (the critical value for nontriviality), it is shown to take Θ(ɛ−1n) steps. In the case p<1/2, the number of steps is shown to be at least n⋅exp(const ɛ−1/2). This last result matches the known upper bound from [Algorithmica 22 (1998) 388–412] in a certain family of subcases.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 4 (2009), 1273-1291.

Dates
First available in Project Euclid: 27 July 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1248700617

Digital Object Identifier
doi:10.1214/08-AAP585

Mathematical Reviews number (MathSciNet)
MR2538070

Zentralblatt MATH identifier
1176.68093

Subjects
Primary: 68W40: Analysis of algorithms [See also 68Q25] 68Q25: Analysis of algorithms and problem complexity [See also 68W40]
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60C05: Combinatorial probability

Keywords
Branching random walk minimal displacement maximal displacement optimal path algorithm computational complexity

Citation

Pemantle, Robin. Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19 (2009), no. 4, 1273--1291. doi:10.1214/08-AAP585. https://projecteuclid.org/euclid.aoap/1248700617


Export citation

References

  • [1] Aldous, D. (1992). Greedy search on the binary tree with random edge-weights. Combin. Probab. Comput. 1 281–293.
  • [2] Aldous, D. (1998). A Metropolis-type optimization algorithm on the infinite tree. Algorithmica 22 388–412.
  • [3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [4] Biggins, J. D. (1977). Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14 630–636.
  • [5] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31 531–581.
  • [6] Bramson, M. D. (1978). Minimal displacement of branching random walk. Z. Wahrsch. Verw. Gebiete 45 89–108.
  • [7] Chauvin, B. and Rouault, A. (1990). Supercritical branching Brownian motion and K–P–P equation in the critical speed-area. Math. Nachr. 149 41–59.
  • [8] Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493–507.
  • [9] Durrett, R. (2004). Probability: Theory and Examples, 3rd ed. Duxbury Press, Belmont, CA.
  • [10] Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Probab. 2 652–680.
  • [11] Harris, J. W. and Harris, S. C. (2007). Survival probabilities for branching Brownian motion with absorption. Electron. Comm. Probab. 12 81–92.
  • [12] Harris, J. W., Harris, S. C. and Kyprianou, A. E. (2006). Further probabilistic analysis of the Fisher–Kolmogorov–Petrovskii–Piscounov equation: One sided travelling-waves. Ann. Inst. H. Poincaré Probab. Statist. 42 125–145.
  • [13] Karp, R. M. and Pearl, J. (1983). Searching for an optimal path in a tree with random costs. Artificial Intelligence 21 99–116.
  • [14] Kesten, H. (1978). Branching Brownian motion with absorption. Stochastic Process. Appl. 7 9–47.
  • [15] Kingman, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab. 1 883–909.
  • [16] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790–801.
  • [17] Pemantle, R. (1993). Critical RWRE on trees of exponential growth. In Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992) (R. F. Bass and K. Burdzy, eds.). Progress in Probability 33 221–239. Birkhäuser, Boston.