Multiway Trees of Maximum and Minimum Probability under the Random Permutation Model

1996 ◽  
Vol 5 (4) ◽  
pp. 351-371 ◽  
Author(s):  
Robert P. Dobrow ◽  
James Allen Fill
Keyword(s):  
Probability Model ◽  
Mass Function ◽  
Random Permutation ◽  
Search Tree ◽  
Probability Mass Function ◽  
Search Trees ◽  
Asymptotic Expressions ◽  
Minimum Probability ◽  
Permutation Model ◽  
Random Permutation Model

Multiway trees, also known as m–ary search trees, are data structures generalising binary search trees. A common probability model for analysing the behaviour of these structures is the random permutation model. The probability mass function Q on the set of m–ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m–ary search tree. We study some basic properties of the functional Q, which serves as a measure of the ‘shape’ of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values.

scholarly journals Almost sure asymptotics for the random binary search tree

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Matthew Roberts
Keyword(s):  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Saturation Level ◽  
Binary Search Tree ◽  
International Audience ◽  
Permutation Model ◽  
Random Permutation Model

International audience We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.

scholarly journals A Random Permutation Model Arising in Chemistry

2008 ◽  
Vol 45 (04) ◽  
pp. 1060-1070
Author(s):  
Mark Brown ◽  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Random Permutation ◽  
Permutation Model ◽  
Single Cluster ◽  
Random Permutation Model

We study a model arising in chemistry where n elements numbered 1, 2, …, n are randomly permuted and if i is immediately to the left of i + 1 then they become stuck together to form a cluster. The resulting clusters are then numbered and considered as elements, and this process keeps repeating until only a single cluster is remaining. In this article we study properties of the distribution of the number of permutations required.

scholarly journals On the Resilience of Even-Mansour to Invariant Permutations

2021 ◽  
Author(s):  
Bart Mennink ◽  
Samuel Neves
Keyword(s):  
Invariant Subspace ◽  
Block Cipher ◽  
Random Permutation ◽  
Cryptographic Primitives ◽  
Distinguishing Attack ◽  
Cryptographic Primitive ◽  
Permutation Model ◽  
Random Permutation Model

AbstractSymmetric cryptographic primitives are often exposed to invariances: deterministic relations between plaintexts and ciphertexts that propagate through the primitive. Recent invariant subspace attacks have shown that these can be a serious issue. One way to mitigate invariant subspace attacks is at the primitive level, namely by proper use of round constants (Beierle et al., CRYPTO 2017). In this work, we investigate how to thwart invariance exploitation at the mode level, namely by assuring that a mode never evaluates its underlying primitive under any invariance. We first formalize the use of invariant cryptographic permutations from a security perspective, and analyze the Even-Mansour block cipher construction. We further demonstrate how the model composes, and apply it to the keyed sponge construction. The security analyses exactly pinpoint how the presence of linear invariances affects the bounds compared with analyses in the random permutation model. As such, they give an exact indication how invariances can be exploited. From a practical side, we apply the derived security bounds to the case where the Even-Mansour construction is instantiated with the 512-bit ChaCha permutation, and derive a distinguishing attack against Even-Mansour-ChaCha in $$2^{128}$$ 2 128 queries, faster than the birthday bound. Comparable results are derived for instantiation using the 200-bit Keccak permutation without round constants (attack in $$2^{50}$$ 2 50 queries), the 1024-bit CubeHash permutation (attack in $$2^{256}$$ 2 256 queries), and the 384-bit Gimli permutation without round constants (attack in $$2^{96}$$ 2 96 queries). The attacks do not invalidate the security of the permutations themselves, but rather they demonstrate the tightness of our bounds and confirm that care should be taken when employing a cryptographic primitive that has nontrivial linear invariances.

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.

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 A Random Permutation Model Arising in Chemistry

2008 ◽  
Vol 45 (4) ◽  
pp. 1060-1070 ◽  
Author(s):  
Mark Brown ◽  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Random Permutation ◽  
Permutation Model ◽  
Single Cluster ◽  
Random Permutation Model

We study a model arising in chemistry where n elements numbered 1, 2, …, n are randomly permuted and if i is immediately to the left of i + 1 then they become stuck together to form a cluster. The resulting clusters are then numbered and considered as elements, and this process keeps repeating until only a single cluster is remaining. In this article we study properties of the distribution of the number of permutations required.

scholarly journals Detecting Different Road Infrastructural Elements Based on the Stochastic Characterization of Speed Patterns

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mario Muñoz-Organero ◽  
Ramona Ruiz-Blázquez
Keyword(s):  
Total Variation ◽  
Traffic Congestion ◽  
Automatic Detection ◽  
Mass Function ◽  
Total Variation Distance ◽  
Statistical Moments ◽  
Probability Mass Function ◽  
Traffic Lights ◽  
Traffic Conditions ◽  
Variation Distance

The automatic detection of road related information using data from sensors while driving has many potential applications such as traffic congestion detection or automatic routable map generation. This paper focuses on the automatic detection of road elements based on GPS data from on-vehicle systems. A new algorithm is developed that uses the total variation distance instead of the statistical moments to improve the classification accuracy. The algorithm is validated for detecting traffic lights, roundabouts, and street-crossings in a real scenario and the obtained accuracy (0.75) improves the best results using previous approaches based on statistical moments based features (0.71). Each road element to be detected is characterized as a vector of speeds measured when a driver goes through it. We first eliminate the speed samples in congested traffic conditions which are not comparable with clear traffic conditions and would contaminate the dataset. Then, we calculate the probability mass function for the speed (in 1 m/s intervals) at each point. The total variation distance is then used to find the similarity among different points of interest (which can contain a similar road element or a different one). Finally, a k-NN approach is used for assigning a class to each unlabelled element.

scholarly journals Modified Recursions for a Class of Compound Distributions

1996 ◽  
Vol 26 (2) ◽  
pp. 213-224 ◽  
Author(s):  
Karl-Heinz Waldmann
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Mass Function ◽  
Cumulative Distribution ◽  
Probability Mass Function ◽  
Initial Value ◽  
Compound Distributions ◽  
Claim Frequency ◽  
Probability Mass ◽  
Stop Loss

AbstractRecursions are derived for a class of compound distributions having a claim frequency distribution of the well known (a,b)-type. The probability mass function on which the recursions are usually based is replaced by the distribution function in order to obtain increasing iterates. A monotone transformation is suggested to avoid an underflow in the initial stages of the iteration. The faster increase of the transformed iterates is diminished by use of a scaling function. Further, an adaptive weighting depending on the initial value and the increase of the iterates is derived. It enables us to manage an arbitrary large portfolio. Some numerical results are displayed demonstrating the efficiency of the different methods. The computation of the stop-loss premiums using these methods are indicated. Finally, related iteration schemes based on the cumulative distribution function are outlined.

HEAVY TRAFFIC APPROXIMATIONS FOR A SYSTEM OF INFINITE SERVERS WITH LOAD BALANCING

1999 ◽  
Vol 13 (3) ◽  
pp. 251-273 ◽  
Author(s):  
Philip J. Fleming ◽  
Burton Simon
Keyword(s):  
State Space ◽  
Heavy Traffic ◽  
Joint Probability ◽  
Mass Function ◽  
Accurate Estimate ◽  
The State ◽  
Probability Mass Function ◽  
Traffic Loads ◽  
Wide Range ◽  
Heavy Traffic Approximations

We consider an exponential queueing system with multiple stations, each of which has an infinite number of servers and a dedicated arrival stream of jobs. In addition, there is an arrival stream of jobs that choose a station based on the state of the system. In this paper we describe two heavy traffic approximations for the stationary joint probability mass function of the number of busy servers at each station. One of the approximations involves state-space collapse and is accurate for large traffic loads. The state-space in the second approximation does not collapse. It provides an accurate estimate of the stationary behavior of the system over a wide range of traffic loads.

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