Finding diameter-reducing shortcuts in trees

In the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices embedded in an unknown metric space. An oracle can report the cost of any edge in constant time, and we want to augment T with k shortcuts to minimize the resulting diameter. When k=1, O(nlog⁡n)-time algorithms e...

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Vydáno v:Journal of computer and system sciences Ročník 153; s. 103658
Hlavní autoři: Bilò, Davide, Gualà, Luciano, Leucci, Stefano, Pepè Sciarria, Luca
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.11.2025
Témata:
Approximation algorithms
Fast diameter computation
Time-efficient algorithms
Tree diameter augmentation
Approximation algorithms
Time-efficient algorithms
Tree diameter augmentation
Fast diameter computation
ISSN:0022-0000
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Shrnutí:In the k-Diameter-Optimally Augmenting Tree Problem we are given a tree T of n vertices embedded in an unknown metric space. An oracle can report the cost of any edge in constant time, and we want to augment T with k shortcuts to minimize the resulting diameter. When k=1, O(nlog⁡n)-time algorithms exist for paths and trees. We show that o(n2) queries cannot provide a better than 10/9-approximation for trees when k≥3. For any constant ε>0, we design a linear-time (1+ε)-approximation algorithm for paths when k=o(log⁡n), thus establishing a dichotomy between paths and trees for k≥3. Our algorithm employs an ad-hoc data structure, which we also use in a linear-time 4-approximation algorithm for trees, and to compute the diameter of (possibly non-metric) graphs with n+k−1 edges in time O(nklog⁡n).
ISSN:0022-0000
DOI:10.1016/j.jcss.2025.103658