The Annals of Applied Probability

Rates of Convergence for the Move-to-Root Markov Chain for Binary Search Trees

Robert P. Dobrow and James Allen Fill

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The move-to-root heuristic is a self-organizing rule that attempts to keep a binary search tree in near-optimal form. It is a tree analogue of the move-to-front scheme (also known as the weighted random-to-top card shuffle or Tsetlin library) for self-organizing lists. We study convergence of the move-to-root Markov chain to its stationary distribution and show that move-to-root converges two to four times faster than move-to-front for many examples. We also discuss asymptotics for expected search cost. For equal weights, $\operatorname{cn}/\ln n$ steps are necessary and sufficient to drive the maximum relative error to 0.

Article information

Ann. Appl. Probab., Volume 5, Number 1 (1995), 20-36.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68P10: Searching and sorting 68P05: Data structures

Markov chains convergence to stationarity self-organizing search binary search trees move-to-root rule move-to-front rule


Dobrow, Robert P.; Fill, James Allen. Rates of Convergence for the Move-to-Root Markov Chain for Binary Search Trees. Ann. Appl. Probab. 5 (1995), no. 1, 20--36. doi:10.1214/aoap/1177004825.

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