A probabilistic analysis of some tree algorithms

Mohamed, Hanène; Pierpoint, Robert

doi:10.1214/105051605000000494

Cited by 20 publications

(26 citation statements)

References 37 publications

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“…Asymptotic behavior of algorithms with an underlying tree structure has been extensively investigated, see Flajolet et al [6], Mohamed and Robert [10] and Mahmoud [8] for a general presentation. By using the terminology of Flajolet et al [6], for non-negative sequences (λ n ) and (µ n ), a series like…”

Section: A Convergence Resultsmentioning

confidence: 99%

Analysis of Steiner subtrees of random trees for traceroute algorithms

Guillemin

Pierpoint

2009

Random Struct Algorithms

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Abstract. We consider in this paper the problem of discovering, via a traceroute algorithm, the topology of a network, whose graph is spanned by an infinite branching process. A subset of nodes is selected according to some criterion. As a measure of efficiency of the algorithm, the Steiner distance of the selected nodes, i.e. the size of the spanning sub-tree of these nodes, is investigated. For the selection of nodes, two criteria are considered: A node is randomly selected with a probability, which is either independent of the depth of the node (uniform model) or else in the depth biased model, is exponentially decaying with respect to its depth. The limiting behavior the size of the discovered subtree is investigated for both models.

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Section: A Convergence Resultsmentioning

confidence: 99%

Analysis of Steiner subtrees of random trees for traceroute algorithms

Guillemin

Pierpoint

2009

Random Struct Algorithms

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“…The renewal structure of split trees. Renewal theory has already been used for studying random trees in [26,28,32,42,43]. The present paper is another example of its wide applicability.…”

Section: 2mentioning

confidence: 87%

“…Even in a fixed class R z , not all the nodes have the same cardinality n r . So, in order to estimate the expected value in (42) we need the following lemma that quantifies the discrepancy of E[Ψ(T n )] under small variations of n. Proof. From the iterative construction, we clearly have E[ Ψ(T n+K )] ≥ E[ Ψ(T n )]; so it suffices to bound the increase in path length when adding K extra items to the tree T n .…”

Section: 3mentioning

confidence: 99%

The total path length of split trees

Broutin¹,

Holmgren²

2012

Ann. Appl. Probab.

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We consider the model of random trees introduced by Devroye [SIAM J. Comput. 28 (1999) . 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.1. Introduction. In this paper we investigate the total path length, that is, sum of all depths, of random split trees defined by Devroye The magnitude of the depths in tree data structures naturally influences their efficiency; in the case where the tree represents the branching choices made by an algorithm, the depths are related to the running time of the algorithm. In this sense, the sum of the depths is a natural and important measure of the efficiency of tree data structures or sorting algorithms.The path length of tree data structures has been studied by many authors, but in most cases the analyses and proofs are very much tied to a specific

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“…If another collision occurs in a subgroup, this subgroup will split again and the process continues in the same manner until all contentions are resolved. Tree algorithm described above is proved that it is stable for a sufficiently small number of users [10]. However, it becomes less effective when there are a large number of users competing for channels.…”

Section: Introductionmentioning

confidence: 98%

Performance evaluation of modified tree algorithm using adaptive permission probability control

Srichavengsup

2012

2012 International Conference on ICT Convergence (ICTC)

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A probabilistic analysis of some tree algorithms

Cited by 20 publications

References 37 publications

Analysis of Steiner subtrees of random trees for traceroute algorithms

Analysis of Steiner subtrees of random trees for traceroute algorithms

The total path length of split trees

Performance evaluation of modified tree algorithm using adaptive permission probability control

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