Repeated fringe subtrees in random rooted trees

Ralaivaosaona, Dimbinaina; Wagner, Stephan

doi:10.1137/1.9781611973761.7

Cited by 10 publications

(8 citation statements)

References 17 publications

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“…We provide the proof of Theorem 3 first, since it is simplest and provides us with a template for the other proofs. Basically, it is a refinement of the proof for the corresponding special case of Theorem 3.1 in [26]. In the following sections, we refine the argument further to prove Theorems 1, 2 and 4.…”

Section: Ordered Fringe Subtreesmentioning

confidence: 90%

“…This result of Flajolet et al was extended to unranked labelled trees in [6] (for a different constant c). Moreover, an alternative proof to the result of Flajolet et al was presented in [26] in the context of simply-generated families of trees.…”

Section: Introductionmentioning

confidence: 90%

“…As the main idea of the proof is to split the number of distinct fringe subtrees into the number of distinct fringe subtrees of size at most k 0 plus the number of distinct fringe subtrees of size greater than k 0 for some suitably chosen integer k 0 , this type of argument is called a cut-point argument and the integer k 0 is called the cut-point (see [17]). This basic technique is applied in several previous papers to similar problems (see for instance [10], [17], [26], [27]). Moreover, we remark that the statement of Theorem 3 can be easily generalized to simply generated families of trees.…”

Section: Ordered Fringe Subtreesmentioning

confidence: 99%

“…To prove the above Theorems 1 and 2, we refine techniques from [26]. Our proof technique also applies to the problem of estimating the number of distinct ordered fringe subtrees in uniformly random binary trees or in random binary search trees.…”

Section: Introductionmentioning

confidence: 99%

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On the Collection of Fringe Subtrees in Random Binary Trees

Benkner¹,

Wagner²

2020

Preprint

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A fringe subtree of a rooted tree is a subtree consisting of one of the nodes and all its descendants. In this paper, we are specifically interested in the number of non-isomorphic trees that appear in the collection of all fringe subtrees of a binary tree. This number is analysed under two different random models: uniformly random binary trees and random binary search trees. In the case of uniformly random binary trees, we show that the number of non-isomorphic fringe subtrees lies between c1n/ √ ln n(1 + o(1)) and c2n/ √ ln n(1 + o(1)) for two constants c1 ≈ 1.0591261434 and c2 ≈ 1.0761505454, both in expectation and with high probability, where n denotes the size (number of leaves) of the uniformly random binary tree. A similar result is proven for random binary search trees, but the order of magnitude is n/ ln n in this case. Our proof technique can also be used to strengthen known results on the number of distinct fringe subtrees (distinct in the sense of ordered trees). This quantity is of the same order of magnitude in both cases, but with slightly different constants in the upper and lower bounds.

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Section: Ordered Fringe Subtreesmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 90%

Section: Ordered Fringe Subtreesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Collection of Fringe Subtrees in Random Binary Trees

Benkner¹,

Wagner²

2020

Preprint

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“…For the case of binary trees (and other tree families representing for example arithmetic expressions), the average ratio of compaction has been first computed in the paper by Flajolet et al [9]. The results have been completed later in [2,19]. We also have started a combinatorial study devoted to the compacted binary tree structures in the paper [11].…”

Section: Introductionmentioning

confidence: 99%

Binary Decision Diagrams: from Tree Compaction to Sampling

Clément¹,

Genitrini²

2019

Preprint

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Any Boolean function corresponds with a complete full binary decision tree. This tree can in turn be represented in a maximally compact form as a direct acyclic graph (dag) where common subtrees are factored and shared, keeping only one copy of each unique subtree. This yields the celebrated and widely used structure called reduced ordered binary decision diagram (robdd). We propose to revisit the classical compaction process to give a new way of enumerating robdds of a given size without considering fully expanded trees and the compaction step. Our method also provides an unranking procedure for the set of robdds. As a by-product we get a random uniform and exhaustive sampler for robdds for a given number of variables and size. For efficiency our algorithms rely on a precomputation step. Finally, we give some key ideas to extend the approach to other strategies of compaction, in relation with variants of bdds (namely qbdds and zbdds).

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Compaction for two models of logarithmic‐depth trees: Analysis and experiments

Bodini

Genitrini

Gittenberger

et al. 2021

Random Struct Algorithms

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We are interested in the quantitative analysis of the compaction ratio for two classical families of trees: recursive trees and plane binary increasing trees. These families are typical representatives of tree models with a small depth. Once a tree of size n is compacted by keeping only one occurrence of all fringe subtrees appearing in the tree the resulting graph contains only O(n∕ ln n) nodes. This result must be compared to classical results of compaction in the families of simply generated trees, where the analogous result states that the compacted structure is of size of order n∕ √ ln n. The result about the plane binary increasing trees has already been proved, but we propose a new and generic approach to get the result. Finally, an experimental study is presented, based on a prototype implementation of compacted binary search trees that are modeled by plane binary increasing trees.

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Repeated fringe subtrees in random rooted trees

Cited by 10 publications

References 17 publications

On the Collection of Fringe Subtrees in Random Binary Trees

On the Collection of Fringe Subtrees in Random Binary Trees

Binary Decision Diagrams: from Tree Compaction to Sampling

Compaction for two models of logarithmic‐depth trees: Analysis and experiments

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