scholarly journals Gaussian approximation for rooted edges in a random minimal directed spanning tree

2021 ◽  
Author(s):  
Chinmoy Bhattacharjee
Keyword(s):  
Spanning Tree ◽  
Gaussian Approximation ◽  
Directed Spanning Tree

scholarly journals Random minimal directed spanning trees and Dickman-type distributions

2004 ◽  
Vol 36 (03) ◽  
pp. 691-714 ◽  
Author(s):  
Mathew D. Penrose ◽  
Andrew R. Wade
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Dirichlet Distribution ◽  
Maximal Length ◽  
Directed Path ◽  
Limiting Distributions ◽  
Limit Distributions ◽  
Tree Construction ◽  
Directed Spanning Tree ◽  
Westerly Direction

In Bhatt and Roy's minimal directed spanning tree construction fornrandom points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for largen) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

scholarly journals On a random directed spanning tree

2004 ◽  
Vol 36 (1) ◽  
pp. 19-42 ◽  
Author(s):  
Abhay G. Bhatt ◽  
Rahul Roy
Keyword(s):  
Spanning Tree ◽  
Poisson Point Process ◽  
Spanning Trees ◽  
Asymptotic Properties ◽  
Record Values ◽  
Extreme Value Statistics ◽  
Minimal Spanning Tree ◽  
Unit Square ◽  
Directed Spanning Tree ◽  
Number Of Branches

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

scholarly journals On a random directed spanning tree

2004 ◽  
Vol 36 (01) ◽  
pp. 19-42 ◽  
Author(s):  
Abhay G. Bhatt ◽  
Rahul Roy
Keyword(s):  
Spanning Tree ◽  
Poisson Point Process ◽  
Spanning Trees ◽  
Asymptotic Properties ◽  
Record Values ◽  
Extreme Value Statistics ◽  
Minimal Spanning Tree ◽  
Unit Square ◽  
Directed Spanning Tree ◽  
Number Of Branches

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

scholarly journals Random minimal directed spanning trees and Dickman-type distributions

2004 ◽  
Vol 36 (3) ◽  
pp. 691-714 ◽  
Author(s):  
Mathew D. Penrose ◽  
Andrew R. Wade
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Dirichlet Distribution ◽  
Maximal Length ◽  
Directed Path ◽  
Limiting Distributions ◽  
Limit Distributions ◽  
Tree Construction ◽  
Directed Spanning Tree ◽  
Westerly Direction

In Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

scholarly journals On the total length of the random minimal directed spanning tree

2006 ◽  
Vol 38 (2) ◽  
pp. 336-372 ◽  
Author(s):  
Mathew D. Penrose ◽  
Andrew R. Wade
Keyword(s):  
Spanning Tree ◽  
Minimal Element ◽  
Normal Component ◽  
Boundary Effects ◽  
Total Length ◽  
Tree Construction ◽  
Directed Spanning Tree ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Set Of Points

In Bhatt and Roy's minimal directed spanning tree construction for a random, partially ordered set of points in the unit square, all edges must respect the ‘coordinatewise’ partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power-weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary plus a contribution introduced by the boundary effects, which can be characterized by a fixed-point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law.

scholarly journals Control Strategies for Multi-UAV Simultaneous Arrival under Communication Topology Changing

2018 ◽  
Vol 36 (3) ◽  
pp. 565-570
Author(s):  
Yanying Tan ◽  
Liang Xue ◽  
Yanning Zhang ◽  
Xiaoping Zhu
Keyword(s):  
Survival Rate ◽  
Spanning Tree ◽  
Control Strategies ◽  
Consensus Algorithm ◽  
Time Of Arrival ◽  
Node Failure ◽  
Directed Spanning Tree ◽  
Communication Topology ◽  
Simulation Results ◽  
Multi Uav

Aimed at changing in communication topology due to partial link/node failure in the battlefield, and for satisfying the weakest connectivity criteria of consensus algorithm, the strategy of judging in real time whether the communication topology among UAVs has a directed spanning tree or sub-tree and the strategy of acquiring the expected minimum partial link change suggestion for the topology containing a directed spanning tree again are designed. Based on the above strategies and selecting ETA (Estimated Time of Arrival) as coordinated variable, the consensus algorithm ensures that after some time all the members or part members of the UAVs arrive simultaneously. The strategy is put forword which the UAVS are controlled arriving simultaneously on the planed virtual sub-goal waypoints one by one along the flight path. Through the strategy, the UAVs fly together soon and arrive simultaneously at last. It is helpful to improve the survival rate of UAVs and preserve the communication connectivity of UAVs which is usually distance-dependent. The simulation results show that the above strategies are effective.

Sign in / Sign up

Export Citation Format

Share Document