TY - GEN

T1 - Geometric Spanning Trees Minimizing the Wiener Index

AU - Abu-Affash, A. Karim

AU - Carmi, Paz

AU - Luwisch, Ori

AU - Mitchell, Joseph S.B.

N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023.

PY - 2023/1/1

Y1 - 2023/1/1

N2 - The Wiener index of a network, introduced by the chemist Harry Wiener [30], is the sum of distances between all pairs of nodes in the network. This index, originally used in chemical graph representations of the non-hydrogen atoms of a molecule, is considered to be a fundamental and useful network descriptor. We study the problem of constructing geometric networks on point sets in Euclidean space that minimize the Wiener index: given a set P of n points in Rd, the goal is to construct a network, spanning P and satisfying certain constraints, that minimizes the Wiener index among the allowable class of spanning networks. In this work, we focus mainly on spanning networks that are trees and we focus on problems in the plane (d= 2 ). We show that any spanning tree that minimizes the Wiener index has non-crossing edges in the plane. Then, we use this fact to devise an O(n4) -time algorithm that constructs a spanning tree of minimum Wiener index for points in convex position. We also prove that the problem of computing a spanning tree on P whose Wiener index is at most W, while having total (Euclidean) weight at most B, is NP-hard. Computing a tree that minimizes the Wiener index has been studied in the area of communication networks, where it is known as the optimum communication spanning tree problem.

AB - The Wiener index of a network, introduced by the chemist Harry Wiener [30], is the sum of distances between all pairs of nodes in the network. This index, originally used in chemical graph representations of the non-hydrogen atoms of a molecule, is considered to be a fundamental and useful network descriptor. We study the problem of constructing geometric networks on point sets in Euclidean space that minimize the Wiener index: given a set P of n points in Rd, the goal is to construct a network, spanning P and satisfying certain constraints, that minimizes the Wiener index among the allowable class of spanning networks. In this work, we focus mainly on spanning networks that are trees and we focus on problems in the plane (d= 2 ). We show that any spanning tree that minimizes the Wiener index has non-crossing edges in the plane. Then, we use this fact to devise an O(n4) -time algorithm that constructs a spanning tree of minimum Wiener index for points in convex position. We also prove that the problem of computing a spanning tree on P whose Wiener index is at most W, while having total (Euclidean) weight at most B, is NP-hard. Computing a tree that minimizes the Wiener index has been studied in the area of communication networks, where it is known as the optimum communication spanning tree problem.

KW - Minimum routing cost spanning tree

KW - Optimum communication spanning tree

KW - Wiener Index

UR - http://www.scopus.com/inward/record.url?scp=85172721252&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/978-3-031-38906-1_1

DO - https://doi.org/10.1007/978-3-031-38906-1_1

M3 - Conference contribution

SN - 9783031389054

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 1

EP - 14

BT - Algorithms and Data Structures - 18th International Symposium, WADS 2023, Proceedings

A2 - Morin, Pat

A2 - Suri, Subhash

PB - Springer Science and Business Media Deutschland GmbH

T2 - 18th International Symposium on Algorithms and Data Structures, WADS 2023

Y2 - 31 July 2023 through 2 August 2023

ER -