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.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (4) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (04) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

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 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 The central limit theorem for Euclidean minimal spanning trees II

1999 ◽  
Vol 31 (04) ◽  
pp. 969-984 ◽  
Author(s):  
Sungchul Lee
Keyword(s):  
Central Limit ◽  
Spanning Tree ◽  
Limit Theorems ◽  
Spanning Trees ◽  
Regularity Conditions ◽  
Minimal Spanning Tree ◽  
The Central Limit Theorem ◽  
Common Distribution ◽  
The Common ◽  
Minimal Spanning Trees

Let X i : i ≥ 1 be i.i.d. points in ℝ d , d ≥ 2, and let T n be a minimal spanning tree on X 1,…,X n . Let L(X 1,…,X n ) be the length of T n and for each strictly positive integer α let N(X 1,…,X n ;α) be the number of vertices of degree α in T n . If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X 1,…,X n ) and N(X 1,…,X n ;α). We also study the rate of convergence for EL(X 1,…,X n ).

scholarly journals The central limit theorem for Euclidean minimal spanning trees II

1999 ◽  
Vol 31 (4) ◽  
pp. 969-984 ◽  
Author(s):  
Sungchul Lee
Keyword(s):  
Central Limit ◽  
Spanning Tree ◽  
Limit Theorems ◽  
Spanning Trees ◽  
Regularity Conditions ◽  
Minimal Spanning Tree ◽  
The Central Limit Theorem ◽  
Common Distribution ◽  
The Common ◽  
Minimal Spanning Trees

Let Xi : i ≥ 1 be i.i.d. points in ℝd, d ≥ 2, and let Tn be a minimal spanning tree on X1,…,Xn. Let L(X1,…,Xn) be the length of Tn and for each strictly positive integer α let N(X1,…,Xn;α) be the number of vertices of degree α in Tn. If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X1,…,Xn) and N(X1,…,Xn;α). We also study the rate of convergence for EL(X1,…,Xn).

scholarly journals Converting MST to TSP Path by Branch Elimination

2020 ◽  
Vol 11 (1) ◽  
pp. 177
Author(s):  
Pasi Fränti ◽  
Teemu Nenonen ◽  
Mingchuan Yuan
Keyword(s):  
Spanning Tree ◽  
Closed Loop ◽  
Minimum Spanning Tree ◽  
Travelling Salesman Problem ◽  
Open Loop ◽  
Travelling Salesman ◽  
Elimination Algorithm ◽  
Number Of Iterations ◽  
Number Of Branches

Travelling salesman problem (TSP) has been widely studied for the classical closed loop variant but less attention has been paid to the open loop variant. Open loop solution has property of being also a spanning tree, although not necessarily the minimum spanning tree (MST). In this paper, we present a simple branch elimination algorithm that removes the branches from MST by cutting one link and then reconnecting the resulting subtrees via selected leaf nodes. The number of iterations equals to the number of branches (b) in the MST. Typically, b << n where n is the number of nodes. With O-Mopsi and Dots datasets, the algorithm reaches gap of 1.69% and 0.61 %, respectively. The algorithm is suitable especially for educational purposes by showing the connection between MST and TSP, but it can also serve as a quick approximation for more complex metaheuristics whose efficiency relies on quality of the initial solution.

DEVELOPING PROXIMITY GRAPHS BY PHYSARUM POLYCEPHALUM: DOES THE PLASMODIUM FOLLOW THE TOUSSAINT HIERARCHY?

2009 ◽  
Vol 19 (01) ◽  
pp. 105-127 ◽  
Author(s):  
ANDREW ADAMATZKY
Keyword(s):  
Foraging Behavior ◽  
Delaunay Triangulation ◽  
Spanning Tree ◽  
Physarum Polycephalum ◽  
Minimum Spanning Tree ◽  
Nearest Neighbour ◽  
Minimal Spanning Tree ◽  
Proximity Graphs ◽  
Relative Neighborhood Graph ◽  
Plasmodium Of Physarum Polycephalum

Plasmodium of Physarum polycephalum spans sources of nutrients and constructs varieties of protoplasmic networks during its foraging behavior. When the plasmodium is placed on a substrate populated with sources of nutrients, it spans the sources with protoplasmic network. The plasmodium optimizes the network to deliver efficiently the nutrients to all parts of its body. How exactly does the protoplasmic network unfold during the plasmodium's foraging behavior? What types of proximity graphs are approximated by the network? Does the plasmodium construct a minimal spanning tree first and then add additional protoplasmic veins to increase reliability and through-capacity of the network? We analyze a possibility that the plasmodium constructs a series of proximity graphs: nearest-neighbour graph (NNG), minimum spanning tree (MST), relative neighborhood graph (RNG), Gabriel graph (GG) and Delaunay triangulation (DT). The graphs can be arranged in the inclusion hierarchy (Toussaint hierarchy): NNG ⊆ MST ⊆ RNG ⊆ GG ⊆ DT . We aim to verify if graphs, where nodes are sources of nutrients and edges are protoplasmic tubes, appear in the development of the plasmodium in the order NNG → MST → RNG → GG → DT , corresponding to inclusion of the proximity graphs.

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