Geometric Spanning Trees Minimizing the Wiener Index

A. Karim Abu-Affash, Paz Carmi, Ori Luwisch, Joseph S.B. Mitchell

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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.

Original languageEnglish
Title of host publicationAlgorithms and Data Structures - 18th International Symposium, WADS 2023, Proceedings
EditorsPat Morin, Subhash Suri
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages14
ISBN (Print)9783031389054
StatePublished - 1 Jan 2023
Event18th International Symposium on Algorithms and Data Structures, WADS 2023 - Montreal, Canada
Duration: 31 Jul 20232 Aug 2023

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14079 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference18th International Symposium on Algorithms and Data Structures, WADS 2023


  • Minimum routing cost spanning tree
  • Optimum communication spanning tree
  • Wiener Index

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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