scholarly journals Degree-bounded minimum spanning tree for unit disk graph

2012 ◽  
Vol 418 ◽  
pp. 92-105 ◽  
Author(s):  
Hongli Xu ◽  
Liusheng Huang ◽  
Wang Liu ◽  
Yindong Zhang ◽  
Yanjing Sun
Keyword(s):  
Unit Disk ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Unit Disk Graph

scholarly journals LOCAL CONSTRUCTION AND COLORING OF SPANNERS OF LOCATION AWARE UNIT DISK GRAPHS

2009 ◽  
Vol 01 (04) ◽  
pp. 555-588 ◽  
Author(s):  
ANDREAS WIESE ◽  
EVANGELOS KRANAKIS
Keyword(s):  
Unit Disk ◽  
Minimum Spanning Tree ◽  
Optimal Solution ◽  
Local Approximation ◽  
Vertex Coloring ◽  
Local Algorithm ◽  
Unit Disk Graph ◽  
Unit Disk Graphs ◽  
Vertex Coloring Problem ◽  
Local Construction

We look at the problem of coloring locally specially constructed spanners of unit disk graphs. First we present a local approximation algorithm for the vertex coloring problem in Unit Disk Graphs (UDGs) which uses at most four times as many colors as an optimal solution requires. Next we look at the colorability of spanners of UDGs. In particular we present a local algorithm for constructing a 4-colorable spanner of a unit disk graph. The output consists of the spanner and the 4-coloring. The computed spanner also has the properties that it is planar, the degree of a vertex in the spanner is at most 5 and the angles between two edges are at least π/3. By enlarging the locality distance (i.e. the size of the neighborhood which a vertex has to explore in order to compute its color) we can ensure the total weight of the spanner to be arbitrarily close to the weight of a minimum spanning tree. We prove that a local algorithm cannot compute a bipartite spanner of a unit disk graph and therefore our algorithm needs at most one color more than any local algorithm for the task requires. Moreover, we prove that there is no local algorithm for 3-coloring UDGs or spanners of UDGs, even if the 3-colorability of the graph (or the spanner respectively) is guaranteed in advance.

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.

Latency, Capacity, and Distributed Minimum Spanning Tree†

2020 ◽  
Author(s):  
John Augustine ◽  
Seth Gilbert ◽  
Fabian Kuhn ◽  
Peter Robinson ◽  
Suman Sourav
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree

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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