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Greedy Search on the Binary Tree with Random Edge-Weights

Published online by Cambridge University Press:  12 September 2008

David Aldous
David Aldous
Affiliation:
Department of Statistics, University of California, Berkeley CA 94720
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Abstract

There is a simple greedy algorithm for seeking large values of a function f defined on the vertices of the binary tree. Modeling f as a random function whose increments along edges are i.i.d., we show that (under a natural assumption) the values found by the greedy algorithm grow linearly in time, with rate specified in terms of a fixed-point identity for distributions.

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Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 4 , December 1992 , pp. 281 - 293
Copyright
Copyright © Cambridge University Press 1992

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References

[1]Aldous, D. J. (1992) Random Walk on Galton-Watson trees, unpublished.Google Scholar
[2]Asmussen, S. (1987) Applied Probability and Queues, Wiley.Google Scholar
[3]Athreya, K. B. (1985) Discounted branching random walks. Adv. in Appl. Probab. 17 53–66.CrossRefGoogle Scholar
[4]Benjamini, I. and Peres, Y. (to appear) Correlation inequality for tree-indexed random walks. In: Seminar on Stochastic Processes 1991, Birkhauser.Google Scholar
[5]Benjamini, I. and Peres, Y. (1992) Random walk on trees and capacity in the interval, technical report, Hebrew University, Jerusalem.Google Scholar
[6]Benjamini, I. and Peres, Y. (1992) Tree-indexed random walk and first passage percolation, technical report, Hebrew University, Jerusalem.Google Scholar
[7]Biggins, J. D. (1977) Chernoff's theorem in the branching random walk. J. Appl. Probab. 14 630–636.CrossRefGoogle Scholar
[8]Biggins, J. D., Lubachevsky, B. D., Shwartz, A. and Weiss, A. (1991) A branching random walk with a barrier. Ann. Appl. Probab. 1 573–581.CrossRefGoogle Scholar
[9]Durrett, R. (1991) Probability: Theory and Examples, Wadsworth, Pacific Grove CA.Google Scholar
[10]Durrett, R. and Liggett, T. M. (1983) Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete 64 275–301.CrossRefGoogle Scholar
[11]Feller, W. (1966) An introduction to Probability Theory and its Applictations, volume II, Wiley.Google Scholar
[12]Karlin, S. and Taylor, H. M. (1981) A Second Course in Stochastic Processes, Academic Press.Google Scholar
[13]Karp, R. M. and Pearl, J. (1983) Searching for an optimal path in a tree with random costs. Artificial Intelligence 21 99–116.CrossRefGoogle Scholar
[14]Karpelevitch, F. I., Kelbert, M. Ya. and Suhov, Yu. M. (1991) A queueing network model leading to a second-order Lindley equation, technical report, Cambridge University.Google Scholar
[15]Knuth, D. E. (1968) The Art of Computer Programing, volume 1, Addison Wesley.Google Scholar
[16]Lagarias, J. C. and Weiss, A. (1992) The 3x + 1 problem: Two stochastic models. Ann. Appl. Probab. 2 229–261.CrossRefGoogle Scholar
[17]Lyons, R. and Pemantle, R. (1992) Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 125–136.Google Scholar
[18]Mahmoud, H. M. (1992) Evolution of Random Search Trees, Wiley.Google Scholar
[19]McDiarmid, C. J. H. (1990) Probabilistic Analysis of Tree Search. In: Disorder in Physical Systems, Oxford University Press.Google Scholar
[20]Millar, P. W. (1978) A path decomposition for Markov processes. Ann. Probab. 6 345–348.CrossRefGoogle Scholar
[21]Rachev, S. T. and Ruschendorf, L. (1992) Probability metrics and recursive algorithms, technical report, U.C. Santa Barbara.Google Scholar
[22]Rosler, U. (1992) A fixed point theorem for distributions. Stochastic Proc. Appl. 42 195–214.CrossRefGoogle Scholar