Abstract
We consider the model of random trees introduced by Devroye [SIAM J. Comput. 28 (1999) 409–432]. The model encompasses many important randomized algorithms and data structures. The pieces of data (items) are stored in a randomized fashion in the nodes of a tree. The total path length (sum of depths of the items) is a natural measure of the efficiency of the algorithm/data structure. Using renewal theory, we prove convergence in distribution of the total path length toward a distribution characterized uniquely by a fixed point equation. Our result covers, using a unified approach, many data structures such as binary search trees, $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees, and simplex trees.
Citation
Nicolas Broutin. Cecilia Holmgren. "The total path length of split trees." Ann. Appl. Probab. 22 (5) 1745 - 1777, October 2012. https://doi.org/10.1214/11-AAP812
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