Total Path Length for Random Recursive Trees

1999 ◽  
Vol 8 (4) ◽  
pp. 317-333 ◽  
Author(s):  
ROBERT P. DOBROW ◽  
JAMES ALLEN FILL
Keyword(s):  
Path Length ◽  
Recurrence Relations ◽  
Rooted Tree ◽  
Random Variable ◽  
Finite Variance ◽  
Density Estimator ◽  
Search Cost ◽  
Total Path Length ◽  
Recursive Trees ◽  
Mutually Independent

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud [10] showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W. Here we give recurrence relations for the moments of Wn and of W and show that Wn converges to W in Lp for each 0 < p < ∞. We confirm the conjecture that the distribution of W is not normal. We also show that the distribution of W is characterized among all distributions having zero mean and finite variance by the distributional identityformula herewhere [Escr ](x) := − x ln x − (1 minus; x) ln(1 − x) is the binary entropy function, U is a uniform (0, 1) random variable, W* and W have the same distribution, and U, W and W* are mutually independent. Finally, we derive an approximation for the distribution of W using a Pearson curve density estimator.

scholarly journals Inversions in Split Trees and Conditional Galton–Watson Trees

2018 ◽  
Vol 28 (3) ◽  
pp. 335-364 ◽  
Author(s):  
XING SHI CAI ◽  
CECILIA HOLMGREN ◽  
SVANTE JANSON ◽  
TONY JOHANSSON ◽  
FIONA SKERMAN
Keyword(s):  
Unique Solution ◽  
Path Length ◽  
Random Variable ◽  
Sufficient Condition ◽  
Explicit Formulas ◽  
Total Path Length ◽  
Split Trees ◽  
Common Ancestors

We studyI(T), the number of inversions in a treeTwith its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants ofI(T) have explicit formulas involving thek-total common ancestors ofT(an extension of the total path length). Then we considerXn, the normalized version ofI(Tn), for a sequence of treesTn. For fixedTn's, we prove a sufficient condition forXnto converge in distribution. As an application, we identify the limit ofXnfor completeb-ary trees. ForTnbeing split trees [16], we show thatXnconverges to the unique solution of a distributional equation. Finally, whenTn's are conditional Galton–Watson trees, we show thatXnconverges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that significantly strengthen and broaden previous work by Panholzer and Seitz [46].

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (02) ◽  
pp. 363-376
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

Limiting Distributions for Path Lengths in Recursive Trees

1991 ◽  
Vol 5 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Hosam M. Mahmoud
Keyword(s):  
Path Length ◽  
Elementary Proof ◽  
Random Variable ◽  
Limiting Distribution ◽  
Limiting Distributions ◽  
Quadratic Mean ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Path Lengths ◽  
Recursive Trees

The depth of insertion and the internal path length of recursive trees are studied. Luc Devroye has recently shown that the depth of insertion in recursive trees is asymptotically normal. We give a direct alternative elementary proof of this fact. Furthermore, via the theory of martingales, we show that In, the internal path length of a recursive tree of order n, converges to a limiting distribution. In fact, we show that there exists a random variable I such that (In – n In n)/n→I almost surely and in quadratic mean, as n → α. The method admits, in passing, the calculation of the first two moments of In.

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (2) ◽  
pp. 363-376 ◽  
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

scholarly journals Differences between Tempos of Step Error and Postural Sway in the Stipulated Tempo Step Test for Children and Their Relationships

2021 ◽  
Vol 11 (6) ◽  
pp. 31
Author(s):  
Hiroki Aoki ◽  
Shin-ichi Demura ◽  
Kenji Takahashi ◽  
Hiroshi Hirai
Keyword(s):  
Repeated Measures ◽  
Path Length ◽  
Postural Sway ◽  
Dynamic Balance ◽  
Step Test ◽  
Total Path Length ◽  
Fast Tempo ◽  
Balance Ability ◽  
Path Lengths ◽  
Slow Tempo

Recently, to evaluate dynamic balance ability, a stipulated tempo step test has been developed, and a step error between tempo and contact time of feet has been used as an evaluation variable. The step error, postural sway, and their relationships may differ between the slow tempo (40 bpm) and fast tempo (120 bpm). This study aimed to examine the aforementioned problem with 62 participant children (30 boys and 32 girls). The step error and postural sway variables (X-axis path length, Y-axis path length, total path length, peripheral area, and rectangular area) during stepping while matching both tempos were measured. Means of one minute and three intervals (0–20 sec, 20–40 sec, and 40–60 sec) for each variable were calculated in both tempos. The results of the paired t-test showed that means of all variables were larger in the 40 bpm tempo than in the 120 bpm tempo. In the multiple comparison tests after the results of the two-way repeated measures ANOVA, the means of three intervals in all variables were larger in the 40 bpm tempo than in the 120 bpm tempo; the means of the sway variables, excluding that of the X-axis path length, in the 40 bpm tempo were larger in the 0–20 sec interval than in the 20–40 sec interval or the 40–60 sec interval. Correlations between step errors and those between the step error and sway variables of both tempos were insignificant or under moderation. The correlations between the step error and sway variables in both tempos were insignificant or significant but low, and those among sway variables were high, except between the X- and Y-axis path lengths. The relationship between both axis path lengths differed according to the tempo. In conclusion, in the case of the stipulated tempo step test targeting children, the slow tempo has a greater step error and postural sway than the fast tempo, and the sway in the early step stage is greater in the slow tempo. The relationships between step errors and between the step error and sway variables of both tempos are low; hence, the ability related to the test may differ in both tempos. The relationships among sway variables in both tempos are high, except between the X- and Y-axis path lengths.

On the limiting behaviour of a non-homogeneous Markovian manpower model with independent Poisson input

1982 ◽  
Vol 19 (2) ◽  
pp. 433-438 ◽  
Author(s):  
P.-C. G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Random Variable ◽  
Homogeneous Markov Chain ◽  
Chain Model ◽  
Poisson Random Variable ◽  
Poisson Input ◽  
Manpower System ◽  
Limiting Behaviour ◽  
Mutually Independent

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

Control of Stance and Developmental Coordination Disorder: The Role of Visual Information

2004 ◽  
Vol 21 (1) ◽  
pp. 19-33 ◽  
Author(s):  
Eryk P. Przysucha ◽  
M. Jane Taylor
Keyword(s):  
Visual Information ◽  
Path Length ◽  
Developmental Coordination Disorder ◽  
Postural Sway ◽  
Quiet Standing ◽  
Eyes Closed ◽  
Total Path Length ◽  
Eyes Open ◽  
Stability Limits ◽  
Coordination Disorder

The purpose of this study was to compare the postural sway profiles of 20 boys with and without Developmental Coordination Disorder (DCD) on two conditions of a quiet standing task: eyes open and eyes closed. Anterior-posterior (AP) sway, medio-lateral sway (LAT), area of sway, total path length, and Romberg’s quotient were analyzed. When visual information was available, there was no difference between groups in LAT sway or path length. However, boys with DCD demonstrated more AP sway (p < .01) and greater area of sway (p < .03), which resulted in pronounced excursions closer to their stability limits. Analysis of Romberg’s quotient indicated that boys with DCD did not over-rely on visual information.

On the limiting behaviour of a non-homogeneous Markovian manpower model with independent Poisson input

1982 ◽  
Vol 19 (02) ◽  
pp. 433-438 ◽  
Author(s):  
P.-C. G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Random Variable ◽  
Homogeneous Markov Chain ◽  
Chain Model ◽  
Poisson Random Variable ◽  
Poisson Input ◽  
Manpower System ◽  
Limiting Behaviour ◽  
Mutually Independent

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

scholarly journals The Dimensionless Squared Jerk: An Objective Parameter That Improves Assessment of Hand Motion Analysis during Simulated Shoulder Arthroscopy

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Erica Kholinne ◽  
Maulik J. Gandhi ◽  
Arnold Adikrishna ◽  
Hanpyo Hong ◽  
Haewon Kim ◽  
...  
Keyword(s):  
Motion Analysis ◽  
Path Length ◽  
Shoulder Arthroscopy ◽  
Tracking System ◽  
Arthroscopic Surgery ◽  
Optical Tracking ◽  
Hand Movements ◽  
Total Path Length ◽  
Acceleration Threshold ◽  
Objective Parameter

Purpose. Attempts to quantify hand movements of surgeons during arthroscopic surgery faced limited progress beyond motion analysis of hands and/or instruments. Surrogate markers such as procedure time have been used. The dimensionless squared jerk (DSJ) is a measure of deliberate hand movements. This study tests the ability of DSJ to differentiate novice and expert surgeons (construct validity) whilst performing simulated arthroscopic shoulder surgical tasks. Methods. Six residents (novice group) and six consultants (expert group) participated in this study. Participants performed three validated tasks sequentially under the same experimental setup (one performance). Each participant had ten performances assessed. Hand movements were recorded with optical tracking system. The DSJ, time taken, total path length, multiple measures of acceleration, and number of movements were recorded. Results. There were significant differences between novices and experts when assessed using time, number of movements with average and minimal acceleration threshold, and DSJ. No significant differences were observed in maximum acceleration, total path length, and number of movements with 10m/s2 acceleration threshold. Conclusion. DSJ is an objective parameter that can differentiate novice and expert surgeons’ simulated arthroscopic performances. We propose DSJ as an adjunct to more conventional parameters for arthroscopic surgery skills assessment.

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