The analysis of a fringe heuristic for binary search trees

Poblete, Patricio V.; Munro, J. Ian

doi:10.1016/0196-6774(85)90003-3

Cited by 41 publications

(24 citation statements)

References 2 publications

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“…The trees produced are called median-of-(2k + 1) trees. They can equivalently be viewed as trees produced by a balancing heuristic applied to the fringe of the tree [24], [29]. It is clear that the external nodes are no longer uniformly picked, and as a consequence Proposition 3 does not apply.…”

Section: 3mentioning

confidence: 99%

Large Deviations for the Weighted Height of an Extended Class of Trees

Broutin

Devroye

2006

Algorithmica

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We use large deviations to prove a general theorem on the asymptotic edge-weighted height H n of a large class of random trees for which H n ∼ c log n for some positive constant c. A graphical interpretation is also given for the limit constant c. This unifies what was already known for binary search trees [11], [13], random recursive trees [12] and plane oriented trees [23] for instance. New applications include the heights of some random lopsided trees [19] and of the intersection of random trees.1. Introduction. This paper gives general laws of large numbers for the height of a class of edge-weighted random trees, which includes as special cases random binary search trees [11], random recursive trees, random plane oriented trees [23] and random split trees [16]. However, it also covers random k-ary trees not analyzed until now. The paper extends the earlier theorems of Devroye [11], [12], [15] where the theory of branching processes was used for this purpose. A special kind of branching random walk permitted Biggins and Grey [5] to obtain the asymptotic height of various random trees including random binary search trees and random recursive trees. We propose in this paper a method based on large deviations for sums of independent random variables. The closest approach was the one of Biggins [4] which used multidimensional branching processes. Our method makes intensive use of Cramér's theorem for large deviations [17], [10] and some properties of the rate functions it defines. The height is characterized as the solution of a two-dimensional optimization problem involving Cramér's functions. We apply our method in some cases where these functions can be expressed in a closed form. In particular, we are able to obtain the height for random binary search trees, random recursive trees, random median-of-(2k + 1) trees and random lopsided trees, thus extending the class of trees covered by a single theorem.We first present the main result and its proof, taking for granted some results about large deviations. The proofs for these have been put in the Appendix. We next make the link between trees of random variables and random trees of size n, leaving the most interesting part on applications as a concluding section.

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Section: 3mentioning

confidence: 99%

Large Deviations for the Weighted Height of an Extended Class of Trees

Broutin

Devroye

2006

Algorithmica

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“…The method presented above is simple and didactical. Another method uses the properties of Polya-Eggenberger urn models, which have been suggested for the analysis of search trees by Poblete and Munro [26]. Bagchi and Pal [4] developed a limit law for general urn models and applied it in the analysis of random 2-3 trees.…”

Section: Urn Modelsmentioning

confidence: 99%

Limit laws for local counters in random binary search trees

Devroye¹

1991

Random Struct Algorithms

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Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for m-dependent random variables. Examples include: the number of leaves (I!,,,), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply.

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“…The latter algorithm can be improved by means of a simple heuristic proposed and analyzed by Poblete and Munro [9]. It turns out that the algorithm for 2-3 trees is efficient whereas that of the binary tree is inefficient.…”

Section: Fringe Analysis Of Search Treesmentioning

confidence: 99%

“…Here, using Eqs. As a final example, Yao's second-order analysis of 2-3 trees (i.e., the consideration of the two bottom tiers of nodes) from our viewpoint is simply the fringe problem for which (c,,C2,c 3) C4,C5,c 6) c 7 ) = (4,5,6,6,7,8,9) and Q is the (irreducible) matrix the a.s. result analogous to Theorem 2.12 of Yao [10]. 12 n 2n(n) 18 (10) Our result, Theorem 2, also applies in more complicated settings such as the generalized k -t binary search trees of Cunto and Gasc6n [3], a generalization of P.M. trees where there are up to / keys per node and rotations are made each time there is a linear array of 2k -1 nodes in the fringe.…”

Section: Fringe Analysis Of Search Treesmentioning

confidence: 99%